exp z =

Transcription

1

2 exp z = Jordan Weierstrass M / z, / z Cauchy-Riemann

3 3.4 Cauchy Green Cauchy () ( ) (residue theorem) e iax f(x) dx x α (log x) n (log x) n f(x)e iax dx π r(cos θ, sin θ)dθ x α f(x)dx f(x) dx g(x)(log x) n dx f(x)(log x) n dx ( ) ,4 ( )

4 Cauchy Goursat Morera Schwarz Poisson Schwarz (Schwarz reflection principle) Rouché Mittag-Leffler Runge Jordan Poisson Ascoli-Arzelra Laplace Dirichlet A 5 A A.., a a A A..3 a a A.2 arg z, log z A.2. arg z A.2.2 log z A.2.3 z α A A A.4 cot z A

5 A A.4.3 cot z = cos z/ sin z A.5 tan z B 63 B B C C. 8, C.. arg z, log z C.2 9, C C.3, C C C.4 2, C.4. (analytic continuation) C

6 . R, C R +, +, 3 = +.2 {a n } n N {a n } n N n N inf k n a k = inf{a k ; k n} {a k ; k } {a k ; k 2} {a k ; k 3} inf a k inf a k inf a k. k k 2 k 3 { } inf a k k n n N sup inf a k (, ] k n n N {a n } n N n N {a k ; k n} inf k n a k =. sup inf a k =. k n n N lim sup n {a n } {a n } n N a a n := sup inf a k [, ] k n n N lim a n = a n 5

7 (a) ( ε > ) ( n N) ( n N: n n ) a n < a + ε (b) ε > k N ( n N: n k) a n > a ε lim n a n = a ε > n a ε < a n < a+ε n a n < a+ε a ε < a n n a {a n }.3.3. n N, {a j } n j= C n, a, f(z) = a z n + a z n + + a n z + a n ( < ε < )( R R)( z C : z R) ( ε) a z n f(z) ( + ε) a z n. f(z) ( z R), lim f(z) =. z A := max{ a, a 2,..., a n } a z n + + a n z + a n a z n + + a n z + a n A ( z n + + z + ). z z j z n (j =,,..., n 2) a z n + + a n z + a n na z n ( z ). { } na R := max ε a, na ε a R a z n + + a n z + a n ε a R z n ( z ). a z n + + a n z + a n ε a R z n ε a z n ( z R). f(z) a z n + a z n + + a n z + a n a z n + ε a z n = ( + ε) a z n, f(z) a z n a z n + + a n z + a n a z n ε a z n = ( ε) a z n. 6

8 ( ) z f(z) z n = a + a z + a 2 z a n z n. m N f(z) ( z ) lim zm z z n ε >, R R s.t. lim z f(z) a z n =. ε f(z) a z n + ε ( z R). = a..3. n N, {a j } n j= C n+, a, f(z) = a z n + a z n + + a n z + a n ( M R)( M R)( R R)( z C : z R) M z n f(z) M z n..3. ε = /2 2 a z n f(z) 3 2 a z n ( z R). M := a /2, M := 3 a /2.4 exp z =.4. C exp z = z = 2nπi (n Z). i z = x + iy (x, y R) exp z = e x (cos y + i sin y) exp z = e x. exp z = e x = and cos y + i sin y = x = and n Z s.t. y = 2nπ n Z s.t. z = 2nπi. sin z = n Z s.t. z = nπ 7

9 .5 Jordan R 2 Jordan C ( ) C C R 2 \ C Ω i, Ω e Ω i Ω e C.6 k N {} c f ( ) k f(z) = (z c) k g(z), g(c) c g f(c) = f (c) = = f (k ) (c) =, f (k) (c) k N c f k f(z) = g(z) (z c) k, g(c) c g k k c f k Z c g l Z c f g k + l.7 Abraham de Moivre 667 (Virty-le-Fran ois, France) 754 (London) Brook Taylor 685 (Edmonton, Middlesex) 73 (Somerset Ho Johann Carl Friedrich Gauss 777 (Brunswick) 855 (Göttingen) Augustin Louis Cauchy 789 (Paris) 857 (Sceaux) Joseph Liouville 89 (Saint-Omer) 882 (Paris) Pierre Alphonse Laurent 83 (Paris) 854 (Paris) Karl Theodor Wilhelm Weierstrass 85 (Ostenfelde) 897 (Berlin) Georg Friedrich Bernhard Riemann 826(Breselenz) 866(Selasca) Felice Casorati 835(Pavia, Italy) 89(Casteggio, Italy) Charles Emile Picard 856 (Paris) 94 (Paris) [] ( ) E. T.,,, I, II, III, NF, (23). 8

10 [2],,, (995)..8 ().,, (98) 2., 3, (96) , II, (985) IX ( ) 4. L. V.,,, (982) A 5., 2, (965).9 (8 2 8 ) dx (Gelehrter Anzeiger) [ li(x) = log x ] (Ueberbein) a + b = a + bi 9

11 [ ] li(a + bi) x = a + bi φ(x)dx x ( α + βi ) x = a + bi φ(x) dx [] x a + bi [ ] : φ(x) dx φx = φ(x) x φx = φ(x) dx φ(x) dx x φ(x) = log x x dx x = +2πi 2πi log x φx x φ(x) = ex x e x dx x x : x + 4 x2 + 8 x x4 + etc. [ ] 88

12

13 Weierstrass M ( a n (z c) n dz = C n= n= C a n (z c) n dz ( ) Weierstrass M Weierstrass 9 Cauchy Weierstrass M-test ) 2.. ( ) {a n } n N a n a n k= n, m N, m > n m n a k a k = k= k= m k=n+ a k m k=n+ m a k = a k k= n a k. n n a k Cauchy a k Cauchy C n a k k= k= k= a n a n a n (absolutely convergent) 2

14 Proposition 2.. ( ) {a n } n N, {b n } n N (2.) a n b n (n N) b n a n n, m N, m n m m a k a k = m k= k= k=n+ a k = m k=n+ a k m k=n+ b k = m b k k= n b k. n b k Cauchy s n := k= Cauchy C a n (2.) b n k= n a k k= a n (majorant, majorant series, dominant series, dominating series) Ω ( ) n N f n : Ω C {f n } n N Ω z Ω {f n (z)} n N lim f n(z) n Ω Ω z f(z) := lim n f n(z) f : Ω z f(z) = lim n f n(z) C {f n } n N Ω f f lim f n n f n z Ω () f n (z) Ω Ω z S(z) := f n (z) = lim S : Ω z S(z) = n k= n f k (z) f n (z) C f n Ω (S ) ( ) S n := n k= f k {S n } n N S S f n 3

15 2.. Ω {f n } n N Ω () f : Ω C {f n } n N Ω f ({f n } is uniformly convergent to f on Ω) lim f(z) f n (z) = sup n z Ω (2) S : Ω C f n Ω S lim sup n n S(z) f k (z) = z Ω k= Ω f Ω f Example 2.. Ω := R, n N (x < /n) f n (x) := nx ( x /n) (x > /n), f(x) := (x < ) (x = ) (x > ) {f n } R f x R lim f n(x) = f(x) n {f n } R f sup f(x) f n (x) = x R lim sup f(x) f n (x) =. n x R f n f Proposition 2..2 () Ω C, Ω, {f n } n N Ω f : Ω C {f n } Ω f f Ω z, z Ω, N N f(z ) f(z ) f(z ) f N (z ) + f N (z ) f N (z ) + f N (z ) f(z ) sup z Ω f(z) f N (z) + f N (z ) f N (z ) + sup f(z) f N (z) z Ω = 2 sup f(z) f N (z) + f N (z ) f N (z ). z Ω 4

16 z Ω ε > {f n } f N N s.t. n N = sup f(z) f n (z) < ε z Ω 3. sup f(z) f N (z) < ε z Ω 3. f N z δ > s.t. z z < δ = f N (z ) f N (z ) < ε 3. z z < δ = f(z ) f(z ) < 2 ε 3 + ε 3 = ε. f z f n (z) = lim n ( ) lim f n(z), n ( ) f n (z) = f n (z) ( ) Example 2..2 ( ) (x < x > 2/n) f n (x) = n 2 x ( x /n) 2n n 2 x (/n x 2/n), f(x) = (x R) ( 2/n, n ) {f n } f 2 f(x) dx =, 2 f n (x) dx = 2 2 n n = 2 lim n f n (x) dx = = 2 f(x) dx. f n, f x R lim f n(x) = n Proposition 2..3 () Ω C C Ω C {f n } Ω f : Ω C {f n } Ω f lim n C f n (z) dz = C f(z) dz. 5

17 C z = φ(t) (t [α, β]) C f(z) dz C β β f n (z) dz = f (φ(t)) φ (t) dt f n (φ(t)) φ (t) dt α α β = (f (φ(t)) f n (φ(t))) φ (t) dt α β α (f (φ(t)) fn (φ(t))) φ (t) dt sup f(z) f n (z) z Ω = sup z Ω β α φ (t) dt f(z) f n (z) (C ). (Ω C {φ(t); t [α, β]} ) Proposition 2..4 (Weierstrass M (the Weierstrass M-test)) Ω, Ω {f n } n N M n n N, z Ω f n (z) M n f n (z) Ω (converges absolutely and uniformly on Ω) z Ω f n (z) (z ) ε >, N N s.t. n N, z Ω f k (z) k= n f k (z) ε. f n (z) z Ω ε >, N N s.t. f n (x) k= n f n (z) ε. k= z Ω f n (z) ( 2..) m > n m f k (z) k= n f k (z) = k= m k=n+ m f k (z) m n M k = M k M k M k k=n+ k= k= k= k= n M k. 6

18 M n m ( ) f k (z) k= n f k (z) M k k= sup f k (z) z Ω k= k= n M k. k= n f k (z) M k k= k= n M k. n f n (z) k= 2.. {a n } n c C a n (z c) n (z C) n= (power series) 2..2 ( ) 2.. {a n } n c C (2.2) a n (z c) n n= z = z ( c) < r < z c r (2.2) z D(c; r) (2.2) z D(c; z c ) ( : z z c > z c z ) n= a n (z c) n lim n a n(z c) n =. M R s.t. a n (z c) n M (n =,, 2,... ). < r < z c z D(c; r) a n (z c) n a n (z c) n z c n z c M M n= ( ) r n (n =,, 2,... ). z c ( ) r n Weierstrass z c M (2.2) D(c; r) 7

19 2..2 {a n } n c C a n (z c) n n= (i), (ii), (iii) (i) z c (ii) ρ (, ) s.t. z D(c; ρ) z {z C; z c > ρ} (iii) z C (i) (iii) (ii) C := {z C; }, D := {z C; }, ρ := sup{ z c ; z C} < ρ <, {z C; z c < ρ} C, {z C; z c > ρ} D C D = C, c C C (i) z C \ {c}. ρ = sup{ z c ; z C} z c >. (iii) z D. 2.. z C, z 2 D = z c z 2 c z 2 = z z C ρ z c. ρ <. z c < ρ z C s.t. z c < z c. 2.. z C. z c > ρ z C. z D. (i) ρ =, (iii) ρ = (i),(ii),(iii) {z C; z c < ρ} {z C; z c > ρ} 2 ρ [, ] (2.2) (radius of convergence) {z C; z c < ρ} (2.2) (circle of convergence) ( = ) ( = ) 2..3 f (z) = z + z z n +..., f (z) = z + z zn n +..., f 2 (z) = z 2 + z zn n z > z n, z n n, z n n 2 ( ) z < z n n 2 z n n zn = z n 2 ρ = {z C; z c < ρ} =, ρ = {z C; z c < ρ} = C, {z C; z c > ρ} = 8

20 z n 2.. () f (z), f (z), f 2 (z) z = z f (z) z n f 2 (z) z n /n 2 z n = n 2 n () n2 f (z) z = ( = ) z =, z z n t n := + z + + z n = n z k (n =,, 2,... ) k= t n = z n+ z + z n+ z /k n k= 2 z ( k ) = k + n + f (z) Abel 3 n k= z k k = n k= n = k= k (t k t k ) = t k ( k k + {t n } n t n n n t k k= ( k ) sup k + k k= n t k n k k= k= ) + t n n t n t k k=. t k k n = k= = n k= n t k k t k k + k= t k ( k k + ) + t n n. ( k ) 2 k + z = 2 z < n ( n t k k ) n k + [3] 75 n k= z k k 3 Abel summation, Abel s transformation, Abel s partial summation A n := n k= a k n k= a kb k = A n b n + n k= A k(b k b k+ ) 9

21 2.. (Abel (Abel s transformation)) {α n } n, {β n } n n (2.3) ( M R)( n N {}) α k M, (2.4) n= k= β n β n+ <, lim n β n = S = α n β n n= S A B, k=m+ α k β k A m+ B m+ n A m := sup α n m k, B m := β k β k+. k=m k=m {α n } {β n } {β n } (2.3) A m 2M < n n m n m α k = α k α k α k + α k M + M = 2M k=m k= k= k= k= A m 2M. n S n := α k β k, σ n := k= n α k S n Abel k= S n = α β + = σ β + n α k β k = σ β + k= n k= n (σ k σ k )β k k= n σ k β k σ k β k+ = k= n = σ k (β k β k+ ) + σ n β n. k= n k= n σ k β k σ k β k+ σ n β n = σ n β n A β n (n ). k= n n σ k β k β k+ A β k β k+ A B <. k= k= S = lim n S n S A B. 2

22 k = m + S m+,n := n k=m+ k=m+ α k β k, σ m+,n := α k β k A m+ B m+ n k=m+ k m + 2 α k = σ m+,k σ m+,k, α m+ = σ m+,m+. S m+,n = σ m+,m+ β m+ + = σ m+,m+ β m+ + = = n k=m+ n k=m+ σ m+,k β k n k=m+2 n k=m+2 n k=m+ α k (σ m+,k σ m+,k ) β k σ m+,k β k σ m+,k β k σ m+,k (β k β k ) + σ m+,n β n. n k=m+ σ m+,k β k σ m+,n β n A m+ β n, n k=m+ σ m+,k (β k β k ) lim n S m+,n n k=m+ k=m+ A m+ β k β k lim S m+,n A m+ B m+. n A m+ β k β k = A m+ B m+ 2.. (Abel ( 2..) ) a f(x)g(x) dx F (x) := a x a f(x)g(x)dx sup x [a, ) f(t) dt x a f(t) dt F (x) = f(x) F, M := a (x [a, )) sup F (x) < x [a, ) 2 g (x) dx

23 a g (x) dx <, lim g(x) = x F (a) = R a a a f(x)g(x) dx = R a R F (x)g(x) dx = [F (x)g(x)] R a F (x)g (x) dx = F (R)g(R) R a F (x)g (x) dx. F (R)g(R) M g(r) (R ), F (x)g (x) M g (x), f(x)g(x) dx f(x)g(x) dx M g (x) dx = a R a sup x [a, ) c [a, ) f(x)g(x) dx sup x f(t)dt c x [c, ) c a g (x) dx < x a c f(t) dt a g (x) dx. g (x) dx {β n } α n = ( ) n α n = e inθ (θ 2πZ) f(z) b z b f(z) f(b) Abel 2..2 (Abel (Abel s continuity theorem)) f(z) = a n z n z = R (R > ) K { } z/r Ω K := z C; z < R, z /R K f Ω K {R} f Ω K {R} n= lim f(x) = f(r). x [,R) x R f n (z) := n a k z k k= 22

24 k= sup f(z) f n (z) (n ) z Ω K {R} z = R lim f n(r) = f(r) ( n z ) n z Ω K α n := a n R n, β n := R n n α k = a k R k = f n (R) (n =,, 2,... ) k= β n β n+ = n= n= ( ) z n z R R = Abel n N A n+ = z R z R K. f(z) f n (z) A n+ B n+ A n+ B KA n+ (z Ω K ). sup m n+ m k=n+ f(z) f n (z) K α k = sup m n+ m k=n+ a k R k = sup f m (R) f n (R) m n+ sup f m (R) f n (R) (z Ω K, n N). m n+ { } sup z Ω K {R} f(z) f n (z) max K sup m n+ f m (R) f n (R), f(r) f n (R). n f n (R) f(r) {f n (R)} n Cauchy n {f n } n N Ω K {R} f α (, π/2) arg (z ) π < α z Stolz K := 2 sec α z z < K D z z = re iθ θ < α, cos θ > cos α, r = z < cos α z r ( + z ) = z z 2 = 2..5 r ( + z ) ( 2r cos θ + r 2 ) < 2 2 cos θ r < 2 = 2 sec α. 2 cos α cos α tan x = x x3 3 + x5 x2n + ( )n + ( x < ) 5 2n 23

25 x = Abel tan = π 4 π 4 = ( )n 2n +. log 2 = ( )n n 2..6 (Cauchy-Hadamard ) =, = ( ) 2..7 ρ = +. a n (z c) n ρ [, ] n= ( ) n lim sup an. n a n (z c) n n= lim n a n a n+ lim n a n a n+ = ( ) 24

26 2..8 (2 ) {a n } n c C f(z) := a n (z c) n r (, ] n= g(z) := na n (z c) n = (n + )a n+ (z c) n n= r f D(c; r) f (z) = g(z) (z D(c; r)) f D(c; r) a n = f (n) (c) n! (n =,,... ) B 2n e z z z e z + z 2 = + ( ) n B 2n (2n)! z2n B 2 = 6, B 4 = 3, B 6 = 42, B 8 = 3, B = 5 66, B 2 = , B 4 = 6 7, B 6 = 367 5, B 8 = , B 2 = , ( B 2k+ = (k =, 2, 3,... ) B B = 2 ( Bernoulli) B = 2 () ) (2.5) z cot z = n B 2n z 2n. (2n)!

27 tan z = 2 2n (2 2n )B 2n z 2n. (2n)! Mathematica BernoulliB[n], Maple bernoulli(n) Bernoulli BASIC (exp, cos, sin, ( + x) α, log( + x) ) Taylor tan Taylor tan Taylor 2 ( Bernoulli ) Taylor,2 ( f(x) = tan x f (n) () OK BASIC ) cos z, sin z C cos z ( z < π/2) z < π 2 F (z) := tan z = sin z cos z F (z) = c n z n ( z < π 2 ) n= Taylor (n c n ) ( ) F (z) = g(z) f(z) f(z) = a n z n, g(z) = n= (2.6) c = b a, c n = b n z n, F (z) = n= c n z n n= ( f() ) b n n a k c n k k= a (n N) g(z) = f(z)f (z) (A + A + A 2 + ) (B + B + B 2 + ) = A B + (A B + A B ) + (A B 2 + A B + A 2 B ) + ( ) ( ) ( n ) A n B n = A k B n k n= n= n= k= 26

28 ( ) ( ) ( n ) f(z)f (z) = a n z n c n z n = a k c n k z n. n= n= n= k= g(z) = b n z n n= b = a c, b = a c + a c, b 2 = a c 2 + a c + a 2 c,.. n b n = a c n + a k c n k (n N) k= c = b a, c = b a c a, c 2 = b 2 a c a 2 c a,, c n = b n n a k c n k k= a (n N). (2.6) cos, sin Taylor cos z = n= ( ) n (2n)! z2n, sin z = n= ( ) n (2n + )! z2n+ (z C) tan z Taylor BASIC OPTION ARITHMETIC RATIONAL cos, sin Taylor ^ FACT() tan z = z + 3 z z z7 + tan z 27

29 cotangent.bas OPTION ARITHMETIC RATIONAL DECLARE EXTERNAL SUB INverse DECLARE EXTERNAL SUB conv LET maxn=2 OPTION BASE DIM c( TO maxn),s( TO maxn),is( TO maxn),cotangent( TO maxn) FOR n= TO maxn LET c(n)= LET s(n)= NEXT n FOR j= TO maxn/2 LET c(2*j)=(-)^j/fact(2*j) LET s(2*j)=(-)^j/fact(2*j+) NEXT j PRINT "z/sin(z)" CALL INverse(s,IS,maxn) CALL PRINTc(IS,maxn) PRINT "z cos(z)/sin(z)" CALL CONv(IS,c,cotangent,maxn) CALL printc(cotangent,maxn) END REM EXTERNAL SUB printc(a(),maxn) OPTION ARITHMETIC RATIONAL FOR n= TO maxn PRINT a(n) NEXT n END SUB REM EXTERNAL SUB INverse(a(),b(),maxn) OPTION ARITHMETIC RATIONAL LET b()=/a() FOR TO maxn LET s= FOR k= TO n LET s=s+a(k)*b(n-k) NEXT k LET b(n)=-s/a() NEXT n END sub REM EXTERNAL SUB conv(a(),b(),c(),maxn) OPTION ARITHMETIC RATIONAL FOR n= TO maxn LET s= FOR j= TO n LET s=s+a(j)*b(n-j) NEXT j LET c(n)=s next n END sub 28

30 OPTION ARITHMETIC RATIONAL LET maxn=4 DIM a( TO maxn+),b( TO maxn) REM sin(z) FOR n= TO maxn+ LET a(n)= NEXT n FOR j= TO maxn/2 LET a(2*j+)=(-)^j/fact(2*j+) NEXT j REM f(z)=sin(z)/z FOR j= TO maxn+ LET a(j-)=a(j) NEXT j REM /f(z) LET b()=/a() FOR TO maxn LET s= FOR k= TO n LET s=s+a(k)*b(n-k) NEXT k LET b(n)=-s/a() NEXT n FOR n= TO maxn PRINT b(n) NEXT n END 29

31 / z, / z Cauchy-Riemann C Ω f : Ω C (3.) (3.2) Ω := {(x, y) R 2 ; x + yi Ω}, u(x, y) := Re f(x + iy), v(x, y) := Im f(x + iy) ((x, y) Ω) f Ω u x = v y, u y = v x ((x, y) Ω). Cauchy-Riemann (the Cauchy-Riemann equations), Cauchy-Riemann (the Cauchy-Riemann differential equations), Cauchy-Riemann (the Cauchy- Riemann relations) 3.2. (, ) ( ) f : Ω C f(z) R (z Ω) (3.), (3.2) Ω, u, v v(x, y) = ((x, y) Ω). v x (x, y) = v y (x, y) = ((x, y) Ω). Cauchy-Riemann u x (x, y) = u y (x, y) = ((x, y) Ω). C R s.t. u = C on Ω. f(x + iy) = u(x, y) + iv(x, y) = C + i = C. 3

32 3.2.2 ( ) f f f ( ) C R s.t. f(z) = C (z Ω) C = f(z) = (z Ω) f C > (3.), (3.2) Ω, u, v Cauchy-Riemann = x C2 = x = y C2 = y ( u 2 + v 2) = 2uu x + 2vv x = 2(uu x vu y ), ( u 2 + v 2) = 2uu y + 2vv y = 2(uu y + vu x ) ( u v v u ) ( ) ux = u y ( ). u u ( v) v = u 2 + v 2 = C 2 > u x = u y =. Cauchy-Riemann v x = v y =. u v f(x + iy) = u(x, y) + iv(x, y) Cauchy 3.4. Green Cauchy

33 4 (24 22 ) ( ) : D(c; R), D(c; R), z a = r z = a + re iθ (θ [, 2π]) ( Cauchy ) Ω C f : Ω C c C, R >, Ω D(c; R) z D(c; R) (4.) f(z) = f(ζ) 2πi ζ c =R ζ z dζ. ζ z = ε D(c; R) ε > ζ z = ε, ζ c = R ζ f(ζ) Cauchy ( ζ z ) f(ζ) (4.2) 2πi ζ c =R ζ z dζ = f(ζ) 2πi ζ z =ε ζ z dζ. ζ = z + εe iθ (θ [, 2π]) f(ζ) 2πi ζ z =ε ζ z dζ = 2π f(z + εe iθ )dθ 2π ε f(z) 2π f(z + εe iθ )dθ f(z) 2π = 2π [ ] f(z + εe iθ ) f(z) 2π 2π f(z + εe iθ ) f(z) dθ 2π max θ [,2π] f(z + εe iθ ) f(z) dθ (ε ). f(ζ) dζ = f(z). 2πi ζ c =R ζ z 32

34 4..2 (, =, 4.3, 4.4 ) c C, R >, f : D(c; R) C z D(c; R) (4.3) 2πi ζ c =R f(ζ) ζ z dζ = n= a n (z c) n, a n := 2πi ζ c =R f(ζ) dζ (n ). (ζ c) n+ f ( 4.. ) (4.4) f(z) = a n (z c) n n= (z D(c; R)). (4.5) f (n) (z) = n! 2πi ζ c =R f(ζ) dζ (n =,,... ). (ζ z) n+ ((4.) ) Weierstrass M < r < R z c r z ζ c = R ζ z c z c ζ c = ζ c r R. r/r ζ z = ζ c (z c) = ζ c z c ζ c = ζ c n= ( ) z c n = ζ c n= (z c) n (ζ c) n+. (4. ) f(ζ) ζ z = n= f(ζ)(z c) n (ζ c) n+. ζ c = R ζ M := n= M R ( r R f(ζ)(z c) n (ζ c) n+ M ( r n R R) max ζ c =R f(ζ) ) n Weierstrass M test (4. ) ζ c = R ζ c =R f(ζ) ζ z dζ = n= ζ c =R f(ζ)(z c) n dζ. (ζ c) n+ 2πi (4.3) 4.. (4.4) ζ c = R ζ f(ζ) 33

35 (4.5) ζ c = R ζ z = R 4..3 Taylor f(z) = n= f (n) (c) (z c) n n! 4..5 (Cauchy ( )) Ω C f : Ω C D D = D D Ω D (4.6) f(z) = 2πi D f(ζ) ζ z dζ (z D). (4.7) f (n) (z) = n! f(ζ) dζ 2πi D (ζ z) n+ (z D). z D D(z ; R) D R > z D(z ; R) f(ζ) f(ζ) dζ = dζ 2πi D (ζ z) n+ 2πi ζ z =R (ζ z) n+ (n =,, 2, ) 4..2 f(ζ) 2πi (ζ z) n+ dζ = f (n) (z). n! D ( (the mean-value property)) Ω C f : Ω C c Ω D(c; r) Ω r > f(c) = 2π ( f c + re iθ) dθ. 2π ( z c = r f ) Cauchy z = c f(c) = f(ζ) 2πi ζ c dζ. ζ c =r ζ = c + re iθ (θ [, 2π]) f(ζ) 2πi ζ c dζ = 2π f(c + re iθ ) 2πi re iθ ire iθ dθ = 2π ζ c =r 2π f ( c + re iθ) dθ. 34

36 4.2. ( ) u = u(x) = u(x,..., x n ) (n = ) u u = (u ) u u (x) = u. u(a + r) + u(a r) a u(a) a ± r 2 f u, v (f(x + iy) = u(x, y) + iv(x, y)) u = v = ( ) Proposition 4.2. (() (the maximum principle, maximum-modulus theorem)) Ω C f : Ω C z Ω, z Ω f(z) f(z ) ( f(z ) f ) C C s.t. z Ω f(z) = C. ( ) (( ) ) c C, R >, f : D(c; R) C D(c; R) f z f(z) D(c; R) 4.2. ( ) ( ) 2 [] max f(ζ) max ζ D(c;R) ) ζ D(c;R) [2] max ζ D(c;R) f(ζ) z D(c; R) f(z) = z D(c; R) f(z) < f(ζ) ( max ζ D(c;R) max f(ζ) () ζ D(c;R) f(ζ) 2 [] [2] [] ( z D(c; R) f(z ) = z D(c; R) f(z) = max f(ζ). ζ D(c;R) max f(ζ) f(z ) = ζ D(c;R) [] ) [2] max ζ D(c;R) max f(ζ) < ζ D(c;R) max ζ D(c;R) max f(ζ) ζ D(c;R) f(ζ) f(ζ) f [] 2 35

37 M := f(z ) Ω ε > s.t. D(z ; ε) Ω. ρ := ε/2 D(z ; ρ) Ω. < r ρ r f(z ) = 2π ( f z + re iθ) dθ. 2π M = f(z ) 2π ( f z + re iθ) 2π dθ M dθ = M. 2π 2π 3 2π 2π ( f z + re iθ) 2π dθ = M dθ. 2π f(z + re iθ ) M θ f ( z + re iθ) r ( f z + re iθ) = M (θ [, 2π]) i.e. f(z) = M ( z z = r). f(z) = M ( z z ρ) (p. 3, ( p. 56, 3., ) f D(z ; ρ) : C C s.t. f = C on D(z ; ρ). (identity theorem, p. 4, 2.38) Ω f = C. ( ) z Lemma 4.2. ( ) n N, {a j } n j= Cn+, a, f(z) = a z n +a z n + + a n z + a n ( < ε < )( R R)( z C : z R) ( ε) a z n f(z) ( + ε) a z n. f(z) ( z R), n lim f(z) =. z z f(z) a z n = + a a z + a 2 a z a n a z n. m N f(z) ( z ) lim =. zm z a zn lim z f(z) a z n =. 3 f g on [a, b] b f(x) dx b a a ( ) b f(x) dx = a a g(x) dx. f g x [a, b] f(x ) < g(x ) b f(x) dx < b a a b g(x) dx f g. 36 g(x) dx. f g

38 ε >, R R, ( z C: z R) f(z) a z n f(z) < ε, i.e. ε a z n + ε ( (fundamental theorem of algebra) ) ( ) P (z) n P (z) z C P (z) f(z) := f P (z) Lemma 4.2.(p. 36) R R M >, R R, ( z C : z R ) P (z) M z n. (4. ) ( z C : z R) f(z) M z n MR n. f D(; R) z = R (4. ) MR n f(z) (z C). R f(z) =. MRn f(z) ( ) p. 8 ( ) (R M R R M > ) Liouville Definition 4.2. () C (entire function) Example 4.2. ( ) exp z, cos z, sin z tan z, log z, + z (Liouville (, Liouville s theorem)) f : C C M R s.t. f(z) M (z C) f Taylor f(z) = a n z n z C n N n= a n = R > a n = f (n) () = f(ζ) dζ = n! 2πi ζn+ 2πi ζ =R Cauchy a n 2π 2π 2π f(re iθ ) (Re iθ ) n+ ireiθ dθ = 2π f(re iθ ) 2π R n e inθ dθ. f(re iθ ) R n dθ 2π M 2π R n dθ = M R n 37

39 4 R a n = (n N). f(z) = a ( z C). Example ( ( )) P (z) n P (z) z C P (z) f(z) := P (z) Lemma 4.2.(p. 36) M >, R R, ( z C : z R) P (z) M z n f(z) M z n MR n ( z R). D(; R) = {z C; z R} f M R, z D(; R) : f(z) M. { } f(z) max MR n, M (z C). Liouville f P n 5 Example f(z) := z, g(z) := (z )(z 2) C \ {}, C \ {, 2} f g z < h := f + g z < h z < 2 g(z) g(z) = z + z 2 Taylor Ω C f : Ω C c Ω R := {R > ; f D(c; R) } f c Taylor ρ sup R 4 : ) n ( R) ratio test Cauchy-Hadamard n ( z R Cauchy 5 a z n + a z n + + a n z + a n (a, n ) a z n + a z n a n ( )a n = a n 2 = = a = a = a 6 n= a n (z c) n R = lim n ( a n n R (ratio test), R = lim sup an ) a n+ n R (Cauchy-Hadamard) {a n } (a n = a n ) a n (z c) n = (a(z c)) n z c < / a z c > / a R = / a. n= n= 38

40 4.2.3 ( ) R f D(c; R) D(c; R) Ω (f D(c; R) ) D(c; R) Ω f : D(c; R) C s.t. f = f in D(c; R) Ω (R D(c; R) Ω ) (2 ) f : D(c; R) C s.t. f = f in D(c; R) Ω sup R c f ( f f ) ρ R ρ sup R. R R, < ε < R f c Taylor D(c; R ε) 7 ( 4..2) R ε ρ. ε R ρ. sup R ρ. ρ = sup R. Example ( Taylor ) f(z) = Q(z) (P (z), Q(z) C[z], P (z) P (z) Q(z) ) P (z) α, α 2,..., α n Ω := C \ {α j ; j n} z f(z) c Ω f c Taylor min α j c j n ( p.8) 2.29 ( pp.33 34) (Bernoulli ) f(z) g(z) := + z 2! + z2 3! + = z n (n + )! z C g : C C g(z) = ez z n= (z C \ {}) g(z) = (z e z = ) n Z \ {} s.t. z = 2nπi. g(z) f(z) : f(z) := g(z). Ω := C \ {2nπi; n Z \ {}} (Ω ) D(; 2π) Taylor B n Bernoulli f(z) = z 2 + ( ) n B 2n (2n)! z2n ( pp.33 34) B 2 = 6, B 4 = 3, B 6 = 42,. 7 ε Taylor D(c; R) 4..2 R ε 39

41 2π 8 f z < 2π f R > 2π R z < R sup R = 2π () f Taylor Mathematica Series[z/(Exp[z]-),{z,,}] Maple taylor(z/(exp(z)-),z=,) f(z) = z 2 + z2 2 z z6 324 z z exp z = n Z s.t. z = 2nπi (p.7 ) sin z = ( p.8) f(x) := + x 2 R ( x R f x : r >, {a n } n s.t. f(x) = a n (x x n ) (x r < x < x + r)) x = Taylor n= f(x) = x 2 + x 4 x 6 + = ( ) n x 2n < x < ( ) f(z) = z 2 + = (z + i)(z i) z < z = ±i z < R ( R > ) R sup R = n= () ( ) (Taylor ) ( 4.) ( ) Laurent ( 4.3.4) b n (z c) n ( ) 8 ratio test Hadamard 4

42 Example 4.3. ( Laurent ) f(z) := C \ {3} z 3 c = D(; 2) (Taylor ) f(z) = z 3 = (z ) 2 = ( 2 z ) 2 = ( ) z n (z ) n = n+ (z D(; 2)). n= f D(; 2) D := {z C; z > 2} n= f(z) = z 3 = (z ) 2 = ( (z ) 2 ) z = ( ) 2 n 2 n = z z (z ) n+ n= n= 2 n = (z ) n ( 2 z < z > 2). ( z 3 ) 4.2 Prop ( (Lemma 4.3.) ( )) Lemma 4.3. ( ( )) a n (z c) n 3 n= (i) z C < ρ < z c ρ (ii) < ρ < s.t. z c < ρ z c > ρ < ρ < ρ z c ρ (iii) z C \ {c} 4

43 Proposition 4.3. () b n 3 (z c) n (i) z C \ {c} < R < z c R (ii) < R < s.t. z c > R z c < R R < R < z c R (iii) z C \ {c} z c z c = ζ b n (z c) n = b nζ n. b n ζ n Lemma 4.3. (i) b n ζ n b n ζ C z C \ {c} (z c) n < R < ρ = /R b n ζ n ζ ρ b n (z c) n z c R (ii) < ρ < s.t. b n ζ n ζ < ρ ζ > ρ b n R := /ρ (z c) n z c > R z c < R R < R < R /R < /R = ρ b n ζ n ζ /R b n (z c) n z c R (iii) b n ζ n b n ζ z c (z c) n (i) R =, (iii) R = ( ) Prop R s.t. b n (z c) n z c > R z c < R R < R < b n (z c) n z c R a n (z c) n a n + (z c) n n= 42

44 Theorem 4.3. (Laurent ) a n (z c) n a n + (z c) n ρ n= ρ s.t. ρ < z c < ρ (ρ = ρ ) z c < ρ z c > ρ ρ < R < R < ρ R, R R z c R 4.3. ( ) 4.3. c f (isolated singularity) R > s.t. f < z c < R ( {z; < z c < R} ) f c c c 3 (i) f c (ii) f c c (iii) f c c ( f z c < R ) (iii) c f ( f D(c; ε) ) Ω f Ω f f (f < z < ) (a) f(z) := sin z z (b) f(z) := z(z 2 ) ( R < z < R ) ( < z < ) (c) f(z) := exp z ( R < z < R ) Example ( ) f(z) = { } sin C\ nπ ; n Z z z = (n Z) c = R > f < z c < R nπ c = f ( ) ( p.82) f(z) = Q(z) (P (z), Q(z) C[z]) P (z) P (z) n P (z) n n P (z) f f 43

45 c f c (c ) ( ) c f [4] c ( < z c < R) c (annulus) A(c; R, R 2 ) := {z C; R < z c < R 2 } R = c ( Laurent ) c C R, R 2 R < R 2 (R R 2 R ) f R < z c < R 2 {a n } n Z s.t. (4.8) f(z) = a n (z c) n a n + (z c) n = n= n= a n (z c) n (R < z c < R 2 ). (4.8) R < r < r 2 < R 2 r, r 2 r z c r 2 ( ) () (4.8) R < r < R 2 r (4.9) a n = f(z) dz 2πi z c =r (z c) n+ (n Z). () m Z (4.8) (z c) m+ f(z) (z c) m+ = n= a n (z c) n m R < r < R 2 r z c = r f(z) dz = 2πi z c =r (z c) m+ 2πi z c =r = = n= n= 2πi a n n= z c =r a n (z c) n m dz 2πi a n 2πiδ nm = a m. (z c) n m dz (4.9) ( ) R < z c < R 2 z R < r < z c < r 2 < R 2 r, r 2 C : ζ c = r, C 2 : ζ c = r 2, C := C 2 C C D = {ζ; r < ζ c < r 2 } D D = {ζ C; r ζ c r 2 } R < ζ c < R 2 f Cauchy f(z) = 2πi = 2πi C f(ζ) ζ z dζ = f(ζ) 2πi C 2 f(ζ) ζ z dζ 2πi ζ c =r 2 ζ z dζ 2πi ζ c =r C f(ζ) ζ z dζ. f(ζ) ζ z dζ 44

46 : S(ζ) := f(ζ) ζ z = f(ζ) ζ c z c ζ c S N (ζ) := f(ζ) ζ c ζ c = r 2 ζ S(ζ) S N (ζ) max f(ζ) ζ c =r 2 ζ c N n= = f(ζ) ζ c n= ( ) z c n = f(ζ) ζ c ζ c S(ζ) S N (ζ) = f(ζ) ζ c z c N ζ c z c = ζ c ( ) z c N ζ c z c ζ c max f(ζ) ζ c =r 2 r 2 {ζ C; ζ c = r 2 } f(ζ) ζ z = f(ζ) (ζ c) (z c) = f(ζ) z c = f(ζ) n= (ζ c) n (z c) n+ = f(ζ) ζ c z c (ζ c) n (z c) n ( ) z c n. ζ c ( ) z c N ζ c z c ζ c ( z c r 2 = f(ζ) z c ) N z c r 2 n= ( z c = const. ( ) ζ c n z c {ζ C; ζ c = r } ( ζ c z c r z c < ) f(z) = = a n := 2πi (z c) n 2πi a n (z c) n + n= n= ζ c =r 2 ζ c =R 2 f(ζ) (ζ c) n+ dζ + a n (z c) n. (z c) n f(ζ) (ζ c) n+ dζ (n =,, ), a n := 2πi ζ c =r R < r < R 2 r a n = f(ζ) dζ (n Z). 2πi (ζ c) n+ ζ c =r 45 r 2 ) N f(ζ)(ζ c) n dζ 2πi ζ c =R f(ζ) dζ (n =, 2, ). (ζ c) n+

47 r ζ c r 2 Lemma 4.3., Prop ( self-contained ) {a n } (4.9) R < ρ < ρ 2 < R 2 ρ, ρ 2 f(z) = a n (z c) n a n + (z c) n (ρ z c ρ 2 ) n= R < r < ρ < ρ 2 < r 2 < R 2 r, r 2 C : ζ c = r, C 2 : ζ c = r 2, C := C 2 C, D := A(c; r, r 2 ), Ω := A(c; R, R 2 ), M := max f(ζ), M 2 := max f(ζ) ζ c =r ζ c =r 2 D = C ( ), D Ω Cauchy (4.) f(z) = f(ζ) 2πi ζ z dζ = f(ζ) 2πi ζ z dζ f(ζ) 2πi ζ z dζ (z D). C C 2 z A(c; ρ, ρ 2 ) (4.) ζ C2 ζ c = r 2, z c ρ 2 z c ζ c ρ 2 <. r 2 C ζ z = (ζ c) (z c) = ζ c z c ζ c = ζ c n= ( ) z c n ζ c (4.) f(ζ) ζ z = n= f(ζ)(z c) n (ζ c) n+ (ζ C 2 ). f(ζ)(z c) n (ζ c) n+ M 2 ρ n 2 r2 n+ = M 2 r 2 ( ρ2 r 2 ) n, n= M 2 r 2 ( ρ2 r 2 ) n = M 2 r 2 ( ρ 2 /r 2 ) < (4.) ζ C 2 2πi C 2 f(ζ) ζ z dζ = 2πi C 2 f(ζ)(z c) n (ζ c) n+ dζ = (z c) n 2πi n= n= C 2 f(ζ) dζ. (ζ c) n+ (4.) 2 ζ C ζ c = r, z c ρ ζ c z c r <. ρ ζ z = (ζ c) (z c) = z c ζ c z c = z c n= ( ) ζ c n = z c (ζ c) n (z c) n 46

48 (4.2) f(ζ) ζ z = f(ζ)(ζ c) n (z c) n (ζ C ). f(ζ)(ζ c) n (z c) n M r n r ρ n = M r ( r ρ ) n, M r ( r ρ ) n = M r ( r /ρ ) < (4.2) ζ C 2πi C f(z) = n= a n = 2πi f(ζ) ζ z dζ = 2πi C a n (z c) n + f(ζ)(ζ c) n (z c) n dζ = a n (z c) n (z A(c; r, r 2 )), a n = 2πi C 2 C f(ζ)(ζ c) n dζ (n =, 2,... ). n= (z c) n f(ζ)(ζ c) n dζ. 2πi C f(ζ) dζ (n =,,... ), (ζ c) n+ 2 Weierstrass M-test a n (z c) n = f(ζ)(z c) 2πi C2 n dζ (ζ c) n+ ( ) n ( ) n 2π 2πr 2 M2 ρ2 ρ2 = M 2 (n =,, ), r 2 r 2 r 2 a n (z c) n = f(ζ)(ζ c) 2πi C n (z c) n dζ ( ) n ( ) n 2π 2πr M r r = M (n =, 2, ), r ρ ρ ( ) n ρ2 M 2 ( ) n r M r M 2 = <, M = r 2 ρ 2 /r 2 ρ ρ ( r /ρ ) <. C, C 2 ζ c = r (R < r < R 2 ) a n = f(ζ) dζ (n Z). 2πi (ζ c) n+ ζ c =r 4.3. ( n= ) S = a n n= k=n a k n k= n S = lim S = a n + a n = lim n= m k= m a k + lim n k= n a k, S = lim n m k=m k= n a k 47

49 ( a lim R R 2 f(x) dx + R2 a f(x) dx = lim R 2 R2 a f(x) dx = lim f(x) dx + lim R R f(x) dx ) n m R a R ε >, N N, M N, ( n N : n N), ( m N : m M) R f(x)dx R f(x) dx, f(x) dx = f(x) dx = ε-n S m k= n a k < ε, ε >, N N, ( n N : n N), ( m N : m N) S m k= n a k < ε, () (4.8) f R < z c < R 2 (Laurent expansion, Laurent series) R = c ( c ) Example () Laurent Taylor (c Taylor Laurent ) Taylor f(z) = a n (z c) n, n= a n = f (n) (c) n! (n =,, 2,... ) a n = f(ζ) 2πi n+ dζ (ζ c) ζ c =r a n () f(z) = n Z a n (z c) n (R < z c < R 2 ) {a n } Laurent 3 f(z) = Laurent (z ) 2 {a n } f(z) = a 2 := 3, a n := (n Z \ { 2}) n= a n (z ) n ( < z < ) 48

50 Taylor sin z z 2 = exp z = n= n= exp z = n! sin z = n= ( ) n = + z n= n! zn (z C) n! z n ( ) n (2n + )! z2n+ (z C) ( < z < ). ( ) n (2n + )! z2n = z + ( ) n (2n + )! z2n ( < z < ) : Laurent 4.3. f (z a) n Laurent f Laurent n = z a r = r n ( r < ) Laurent Laurent z a = a( z/a) = a ( z n = a) z n a n+ n= n= n= ( z/a < z < a ). a < z a < Laurent z a = z( a/z) = ( a ) n = z z n= n= a n z n+ = a n z n ( a/z < a < z < ). c C Laurent z a = z c (a c) = a c (z c)/(a c) = a c = n= n= ( ) z c n a c (z c) n (a c) n+ ( (z c)/(a c) < z c < a c ). a c < z c < Laurent z a = z c (a c) = z c (a c)/(z c) = z c = n= ( ) a c n = z c (a c) n (z c) n ( (a c)/(z c) < a c < z c < ). n= (a c) n (z c) n+ 49

51 4.3.2 f c ( R < z c < R 2 ) Laurent (z c) a n (z c) n f (, principal part) ( p.84) cos z () z 2 sin z (z = ) () z 2 (z 2 (z = ) ) 3 Definition 4.3. ( ( ) ) Ω C f : Ω C, c C f ( ) R R s.t. f < z c < R R =, R 2 = R {a n } n Z s.t. f(z) = a n (z c) n n= ( < z c < R). (a) c f (removable singularity) (f c ) n N a n = 9 f c (b) c f (pole) (f c ) k N a k [ n > k a n = ] f c k c c k (c) c f ( ) (isolated essential singularity) k N n > k a n f c 4.3. X ( ) X (a) c f {n N; a n } = ( ) 5

52 (b) c f {n N; a n } < ( ) (c) c f {n N; a n } = () Example f(z) = sin z (z C \ {}) f z Laurent ( ) n sin z (2n + )! z2n+ n= ( ) n = = z z (2n + )! z2n = z2 3! + z4 5! z6 + ( < z < ) 7! Example f(z) = n= 2 (z C \ {3}) 3 4 (z 3) 4 f(z) = 2 (z 3) 4 3 Laurent (a 4 = 2, a n = (n Z \ { 4}) f(z) = n Z a n (z 3) n ) 2 Laurent (z 3) 4 Example f(z) = exp z Laurent f(z) = (z C \ {}) f n= n! ( ) n = + z n! z n ( < z < ) f Laurent = n! z n Proposition ( ) c f (), (2) () lim f(z) ( ) z c z c (2) R (, ], f : D(c; R) C s.t. f(z) = f(z) ( < z c < R). f c c f R > s.t. f < z c < R {a n } s.t. f(z) = a n (z c) n a n + (z c) n ( < z c < R). n= 5

53 c f f(z) = n N a n =. a n (z c) n n= z = c ( a ) f(z) := ( < z c < R). a n (z c) n ( z c < R) n= f : D(c; R) C ( ) z = c lim z c z c f(z) = lim z c z c f(z) = f(c) = a c f f D(c; R) f f f(z) ( < z c < R) f(z) := f(z) (z = c) lim z c z c c f k {a n } n k, R > s.t. k a j f(z) = (z c) j + a n (z c) n ( < z c < R) j= n= Laurent a k+n = n! lim z c z c ( ) d n [ (z c) f(z)] k dz (n =,, 2,... ) Proposition ( ) c f lim f(z) =. z c z c c f R >, {a n } s.t. f(z) = a n (z c) n a n + (z c) n ( < z c < R). n= k a k ( n N: n > k) a n = f(z) = a n (z c) n + n= k a n (z c) n ( < z c < R). 52

54 ζ = z c lim z c z c a n (z c) n = a. n= z c, z c ζ k a n k (z c) n = a n ζ n. Lemma 4.2. lim ζ k a n ζ n =. lim f(z) = a + =. z c z c Lemma (Riemann) f < z c < R c f lim f(z) c f z c z c f < z c < R {a n } s.t. f(z) = a n (z c) n a n + (z c) n ( < z c < R). n= a n (n Z) < r < R r a n = 2πi ζ c =r f(ζ) 2π dζ = (ζ c) n+ 2πi f(c + re iθ ) r n+ e i(n+)θ ireiθ dθ = 2π 2πr n f(c + re iθ )e inθ dθ. f M R s.t. f(z) M ( < z c < R). a n 2π 2πr n f(c + re iθ )e inθ 2π dθ 2πr n Mdθ = 2πM 2πr n = M r n. a n M = Mrn (n r n N). r a n = (n N). f c c f ( ) f A(c;, R) { (z c) g(z) := 2 f(z) ( < z c < R) (z = c) Riemann Liouville Cauchy 53

55 g < z c < R g g(z) g(c) (z c) 2 f(z) (c) = lim = lim = lim(z c)f(z) = z c z c z c z c z c ( f ) g c z c < R : {a n } n s.t. g(z) = a n (z c) n n= ( z c < R). g(c) =, g (c) = a = a =. g(z) = a n (z c) n = (z c) 2 a n (z c) n 2 = (z c) 2 n=2 n=2 n= a n+2 (z c) n ( z c < R). f(z) = a n+2 (z c) n n= c f ( < z c < R). Theorem (Casorati-Weierstrass) c f β C, {z n } n N s.t. z n c ( n N), lim z n = c, lim f(z n) = β. n n ( β = ) f < z c < R β C ε >, r (, R), z A(c;, r) s.t. f(z) β < ε. n =, 2, ε = r = n {z n} n N s.t. n N z n B(c; /n), f(z n ) β < n. lim z n = c, n lim f(z n) = β. n 2 ε > r > z A(c;, r) f(z) β ε. g(z) := f(z) β (z A(c;, r)) ( ) g A(c;, r) c g g(z) ε (z A(c;, r)) ε-δ 2 (proof by contradiction, reductio ad absurdum) 54

56 Riemann c g B(c; r) g(z) (z A(c;, r)) f(z) = β + g(z) = βg(z) + g(z) c f (c g c f, c g k c f k ) c f Corollary 4.3. ( lim ) c f (), (2), (3) () c f lim f(z) z c z c (2) c f lim f(z) = z c z c (3) c f lim f(z) lim f(z) = z c z c z c z c Example f(z) := exp ( z ) 2 f z f(z) lim f(x) =, x R x lim f(iy) = y R y Casorati-Weierstrass Theorem (Picard ) c f e C, ( U: c ) v C \ {e}, z U s.t. f(z) = v. c ( Ahlfors [5] ) Picard [6] 2 ( p.85) f(z) = exp z z n () f(z) (2) f(z n ) (3) α f(z n ) α. (), (2), (3) () a C \ {}, ε >, z A(;, ε) s.t. exp z = a. (2) {z n } n N s.t. lim n z n = lim n exp z n =. 55

57 (3) {z n } n N s.t. lim n z n = lim n exp z n =. ( : a = re iθ (r >, θ [, 2π)) exp z = a = reiθ n Z s.t. z = log r + i(θ + 2nπ) n Z s.t. z = log r + i(θ + 2nπ) n (2), (3) w n = /z n w n ) () lim f(z) z c z c ( ) f(z) = (z c) n g(z) ( < z c < r), n Z, g z c < r, g(c) n c 3 n < c n ( p.85) f a a f (f(z) := sin(/z) z = ) (multifunction, multi-valued function) (transcendental branch point) 3 f(z) = Log ( z) z =. ( ) () f(z) = cos z z 2 (2) f(z) = sin(z3 ) z( cos z) (3) r > A(c; ; r) f c f def. R > s.t. f A(c;, R) := {z C; < z c < R} f A(c; ; R) 3 n > lim z c 4 lim z c z c z c f(z) =, n = lim f(z) = g(c) z c z c f(z) = 56

58 c f = R >, {a n } s.t. f(z) = n= a n (z c) n (z A(c;, R)) ( a n a n = f (n) (c) = f(z) 2πi z c =r (z c) n+ dz ( < r < R), < r < r 2 < R A(c; r, r 2 ) = {z C; r z c r 2 } ) c f def. f c = c f lim f(z) z c z c c f = f(z) f(z) (z A(c;, r)) := f(z) (z = c) D(c; r) lim z c a c c f k def. f c k a n (z c) n, a k c f def. k N s.t. c f k c f lim f(z) = z c z c c f def. f c ( ) ( {n N; a n } = ) : f(z) 5 ( ) Proposition (k ) f(z) C[z], c C, k N 3 (i) c f(z) k ( f(z) (z c) k k = ) (ii) g(z) C[z] s.t. f(z) = (z c) k g(z) g(c). (iii) f(c) = f (c) = = f (k ) (c) = f (k) (c). f c C f(z) 5 f(z) = a n (z α j ) f(z) α j (j =, 2,..., n) j= f(z) ( f(z) = ) (root) α f(z) = f(α) = α f(z) α f(z) = 57

59 Proposition (k ) c C, f c U k N 2 (i) U g f(z) = (z c) k g(z) (z U) g(c). (ii) f(c) = f (c) = = f (k ) (c) = f (k) (c). k (i) (ii) (i) = (ii) f(z) = (z c) k g(z), g(c) h(z) := (z c) k f(z) = g(z)h(z). m k Leibniz m ( ) m f (m) (z) = h (r) (z)g (m r) (z). r r= r k h (r) (c) = m k h (r) (c) = ( r m) h (k) (z) k! f (m) (c) = f (k) (c) = m ( m r ) k!g(c) = k!g(c). r= ( k k ) g (m r) (c) = ( m k ), (ii) = (i) f c c Taylor R >, {a n } s.t. f(z) = a n (z c) n ( z c < R). n= a n = f (n) (c) a = a = = a k =, a k. n! f(z) = a n (z c) n = (z c) k a n (z c) n k = (z c) k a n+k (z c) n. n=k g(z) := n=k z c < R ( Prop. ) f(c) = f a n+k (z c) n n= f(z) = (z c) k g(z), g(c) = a k. n= f(c) = f (c) = = f (k ) (c) = f (k) (c) k N n N {} f (n) (c) = c f(x) = n= f (n) (c) (z c) n = n! f 58 n= n! (z c)n =

60 Definition (, ) c C, f c () c f (zero) f(c) = (2) c f f ( ) f(c) = f (c) = = f (k ) (c) = f (k) (c) k N f c (oredr) Example f(z) ( ) Example (a) f(z) = sin z kπ (k Z) f(kπ) = sin kπ =, f (kπ) = cos kπ = ( ) k kπ f Taylor sin z = ( ) k sin(z kπ) = ( ) k n= sin z = (z kπ)g(z), g(z) := ( ) k g() = ( ) k (b) f(z) = cos z 2kπ (k Z) 2 ( ) n (z kπ)2n+ (2n + )! n= f(2kπ) = cos 2kπ = =, f (z) = sin z, f (2kπ) = sin 2kπ =, ( ) n (z kπ)2n (2n + )! f (z) = cos z, f (2kπ) = cos 2kπ = 2kπ (k Z) 2 Taylor cos z = cos(z 2kπ) = n= ( ) n (z 2kπ)2n (2n)! cos z = (z kπ) 2 g(z), g(z) := ( ) n (z 2kπ)2(n ) (2n)! g(2kπ) = 2 2: Laurent Laurent 59

61 Proposition (ii) = (i) Laurent Taylor ( ) Laurent Proposition () c f, k N (i), (ii) (i) c f k (ii) c U g f(z) = g(z) (z c) k (z U \ {c}), g(c). c f c Laurent R (, ], {a n } n Z s.t. f(z) = a n (z c) n ( < z c < R). n= (i) = (ii) c f k f(z) = n= k a n (z c) n a k ( n N : n > k) a n =. = a k (z c) k + + a z c + a + a (z c) + a 2 (z c) 2 + ( < z c < R) (z c) k f(z) = a k + a k+ (z c) + = a n k (z c) n n= ( < z c < R). ( < z c < R ) R g(z) := g D(c; R) f(z) = a n k (z c) n ( z c < R) n= (z c) k f(z) = g(z) ( < z c < R). g(z) (z c) k ( < z c < R) g(c) = n= k a n k (c c) n = a k. (ii)= (i) (ii) R >, g : D(c; R) C s.t. g f(z) = g(z) (z c) k ( < z c < R), g(c). 6

62 g D(c; R) Taylor {a n } n s.t. g(z) = a n (z c) n = a + a (z c) + a 2 (z c) 2 + n= ( z c < R). < z c < R z f(z) = g(z) (z c) k = a (z c) k + + a k z c + a k + a k+ (z c) + a k+2 (z c) 2 + k = a n+k (z c) n a k n + (z c) n. n= a = g(c) c f k ( ) Proposition c f k f(z) = (z c) k g(z) ( z c < R), g(c) g : D(c; R) C Proposition c f k f(z) = g(z) (z c) k ( < z c < R), g(c) g : D(c; R) C Example 4.3. f(z) = z 2 f 2 ( f(z) = z 2 f ) f(z) = sin z f z ( ) n sin z = (2n + )! z2n+ = z z3 3! + z5 5! z7 + (z C) 7! n= sin z z = n= ( ) n (2n + )! z2n = z2 3! + z4 5! z6 + ( < z < ) 7! f exp z = n! z n = + z + 2!z 2 + 3!z 3 + n= exp z n! z n = z + 2 z 2 + 3! z 3 + exp z 6

63 Corollary P Q c c P k Q(c) c f := Q P k c P k c R P (z) = (z c) k R(z), R(c). g(z) := Q(z) Q(c) g c g(c) =, f(z) = g(z) R(z) R(c) (z c) k (c ) Proposition c f k k N, c C, f c c f k c f 4.3. f(z) := sinh z sin z ( ) Q(z) := sinh z, P (z) := sin z C (sinh z = (exp z exp( z))/2, sin z = (exp(iz) exp( iz))/(2i) Taylor ) c C P (c) = sin c = n Z s.t. c = nπ. P (c) = cos nπ = ( ) n c = nπ P (i) n Q(nπ) = sinh nπ (sinh nπ > ) Cor. nπ f = Q/P (ii) P P s.t. P (C OK) P (z) = zp (z), P (z) ( < z < ). Q Q s.t. Q (C OK) Q(z) = zq (z). ( < z < ) f(z) = Q(z) P (z) = zq (z) zp (z) = Q (z) P (z). z < f f (residue theorem) ( ) c f f c Laurent n= a n (z c) n a f c (residue) Res(f; c) Res f(z) dz : Res(f; c) = Res z=c f(z) dz := a. z=c Laurent ( ) Laurent 62

64 Example 4.3. f(z) = (z C\{}) f Laurent z (a =, a n = (n ) f(z) = a n z n ( < z < )) Res(f; ) =. n= Example f(z) := exp z (z C \ {}) f(z) = n= n! ( ) n = + z z + 2! z 2 + 3! z 3 + Res(f; ) =. Res (zf(z); ) = 2, Res(z2 f(z); ) = 3! = 6. f < z c < R 4.6 ( Laurent ) a n = f(ζ) dζ (n Z, < r < R) 2πi ζ c =r (ζ c) n+ ( ) Res(f; c) = a = 2πi ζ c =r f(ζ)dζ ( < r < R). a (f(ζ)/(ζ c) n+ ) f { (z c) n 2πi (n = ) dz = (n ) z c =R f ( f ) Res(f; c) = a (Res(f; c) f ) () Res(f(z) + cos z; c) (2) Res(3f(z); c) ( (the residue theorem)) C C D C Ω D Ω {c j } N j= D f : Ω\{c,..., c N } C N f(z) dz = 2πi Res(f; c j ). C j= C (C D ) () j c j r j C j C := C C C 2 C N, D := D \ N D(c j ; r j ) j= 63

65 D = C ( ), D Ω \ {c,, c n } 3.2 (Cauchy ) N f(z) dz = f(z) dz. C C j (f C j ) ( ) f(z) dz = f(z) dz = 2πi Res(f; c j ). C j z c j =r j j= C N f(z) dz = 2πi Res(f; c j ). j= () (2) (3) (4) (5) C D D (6) Example z = C C C D = {z C; z < } D = {z C; z } Ω := C C f(z) := z Ω \ {} N =, c = dz = 2πi Res(f; ) = 2πi = 2πi. z z = Mathematica, Maple Maple series(/(z*sin(z)),z=,) Laurent z z z z6 + Mathematica Series[/(z Sin[z]),{z=,}] Res(f; c) = a f(z) = a n (z c) n f n= a n = f(z) dz 2πi (z c) n+ z c =r Res(f; c) 64 z c =r f(z) dz

66 a n = f (n) (c) n! Cf. Taylor ( ) f(z) = a n (z c) n = a + a (z c) + a 2 (z c) 2 + a 3 (z c) 3 + n= ( z c < R). z = c f(c) = a. z = c f (c) = a. f (z) = a + 2a 2 (z c) + 3a 3 (z c) 2 +. f (z) = 2a a 3 (z c) + 4 3a 4 (z c) 2 +. z = c f (c) = 2a 2. a 2 = f (c) 2. f (z) = 3 2a a 4 (z c) a 5 (z c) 2 +. z = c f (c) = 3!a 3. a 3 = f (c). 3! f (n) (z) = n(n ) 2 a n + (n + )n(n ) 2a n+ (z c) + = k(k ) (k n + )a k (z c) k n k! = (k n)! a k(z c) k n = k= m= (n + m)! a n+m (z c) m m! f (n) (c) = n!a n. a n = f (n) (c). n! c f k f(z) = a k (z c) k + a (k ) (z c) k + + a z c + a n (z c) n ( < z c < R) n= (z c) k f(z) = a k + a (k ) (z c) + a (k 2) (z c) 2 + = k=n a n k (z c) n b n := a n k, g(z) := b n = g(n) (c) n! = lim z c z c n= ( ) d n [ (z c) f(z)] k. n! dz 65 ( < z c < R). b n (z c) n n=

67 (4.3) a n = b n+k = lim z c z c ( ) d n+k [ (z c) f(z)] k. (n + k)! dz a = lim z c z c ( ) d k [ (z c) f(z)] k. (k )! dz c c f R, {a n } s.t. f(z) = n= a n (z c) n + a z c ( < z c < R). (z c) (z c)f(z) = a + a (z c) + a (z c) 2 + ( < z c < R) Res(f; c) = a z = c z c Proposition c f (4.4) Res(f; c) = lim(z c)f(z). z c 2 c f (4.4) ( ) f f = Q/P (P, Q c ) c P (Q ) c f c f f Example f(z) = z 2 + Res(f; i) f(z) = (z + i)(z i) i f Res(f; i) = lim(z i)f(z) = lim z i z i z + i = z + i = z=i 2i = i 2. Proposition f(z) = Q(z), P (z) Q(z) c c P (z) P (z) (P (c) = P (c) ), Q(c) c f (4.5) Res(f; c) = Q(c) P (c). 66

68 c P g s.t. g c P (z) = (z c)g(z), g(c). f(z) = Q(z) P (z) = Q(z) (z c)g(z) = h(z) z c, Q(z) h(z) := g(z). h c h(c) = Q(c) g(c) (4.6) Res(f; c) = lim z c z c c f Prop (z c)q(z) (z c)f(z) = lim = lim z c P (z) z c z c Q(z) P (z) P (c) z c = Q(c) P (c). P (c) = f = Q/P, P Q c c P c f Res(f; c) = Q(c) P. c P Prop (x) Example f(z) = z 4 Res(f; i) ( ) f(z) = i (z )(z + )(z + i)(z i) Res(f; i) = (z 4 ) = z=i 4i 3 = i 4i 4 = i n N, f(z) = z n. z = ωk (ω := exp 2πi, k =,,, n ) n z n ( ( ω k) n = ) Res(f; ω k ) = (z n ) = z z=ω nz n = ωk k n. z=ω k c k c f k f(z) = a k (z c) k + + a z c + a + a (z c) + ( < z c < R). (z c) k f(z) = a k + a (k ) (z c) + + a (z c) k + a (z c) k + a (z c) k+ + a k ( ) d k [ (z c) f(z)] k = (k )!a + k! dz! a (k + )! (z c) + a (z c) ! 4.3. c f k Res(f; c) = lim z c z c ( ) d k [ (z c) f(z)] k. (k )! dz ( k ) 67

69 Example f(z) = 3 f 2 z (z 3) 2 (z + ) ( ) d 2 [ Res(f; 3) = lim (z 3) 2 f(z) ] ( ) z (z + ) z = lim = lim z 3 (2 )! dz z 3 z + z 3 (z + ) 2 = lim z 3 (z + ) 2 = (3 + ) 2 = f(z) =. z = 2 Laurent z sin z f(z) = z Res(f; ) =. f f < z < ρ f( z) = f(z) ( < z < ρ) Res(f; ) = c f Res(f; c) = a φ c Res(φ(z)f(z); c) 4.3. () f(z) = π cot πz ( ) (Prop ) P (z) := sin πz, Q(z) := π cos πz f(z) = Q(z) P (z). P (z) = n Z s.t. z = n. P (z) = Q(z) Q(n) = P (n) = π cos nπ = π( ) n. n f ( 2: p.88 ) 4.3. ( pp.88 89) f(z) = Res(f; n) = Q(n) P (n) =. 8z 2 2z 68

70 Mathematica, Maple Mathematica Residue[,{, }] Series[,{,, }] Infinity Apart[ ] Residue[z/((z-2)(z-)^3),{z,}] Series[z/((z-2)(z-)^3),{z,,}] Maple residue(z/((z-2)*(z-)^3)),z= residue(z/((z-2)*(z-)^3)),z=, c f or f c (c f ) = Res(f; c) = c f k = Res(f; c) = ( (k )! lim d ) k [ z c dz (z c) k f(z) ]. z c f = Q P, P (c) =, P (c) Res(f; c) = Q(c) P (c) c f = 4.4 2t sin x x dx = π, e x2 /4t e ixy dx = e ty2 ( ) 9C Cauchy (Cauchy ) ( ) 2 [7], [8], [9], Bak and Newman [] ( Ahlfors [5] ) [8] 69

71 4.4., Cauchy I C ( ) f(x) dx B lim f(x) dx = I, A,B + A ( ε > ) ( R R) ( A R : A R) ( B R : B R) B I f(x) dx < ε A ( ) A lim f(x) dx = I A A ( ) ( ) ( ) Cauchy p.v. f(x) dx v.p. f(x) dx Example 4.4. f(x) := p.v. dx + x 2 = π. x ( x ) ( x < ) f(x) dx = f(x) dx ( ) Cauchy f c c (a, b) b a f(x) dx c f ( ) Cauchy 7

72 f [a, b] C c (a, b) ( c ε lim ε + a f(x) b x c dx + c+ε ) f(x) x c dx b p.v. c a Cauchy f(x) b dx p.v. x c a f(x) x c dx f(x) dx Proposition 4.4. f(x) = Q(x), P (z), Q(z) C[z], deg P (z) deg Q(z)+2, x R P (x) P (x) f(x) dx = 2πi Res(f; c). ( Im c> Im c> f c ) 4.4. ( ) C Ĉ := C { } Ĉ n := deg P (z), m := deg Q(z) P (z) = a z n + a z n + + a n (a ), Q(z) = b z m + b z m + + b m (b ) R [, ) ( z C : z R ) P (z) a 2 z n, Q(z) 3 b 2 z m. P (z) z < R n m + 2 ( ) f(z) = Q(z) P (z) 3 2 b z m 3 b = 2 a z n a z n m M z 2 ( z R ), M := 3 b a. P (x) (x R) f R R f(x) dx = lim f(x) dx R R Γ R : z = x (x [ R, R]), C R : z = Re iθ (θ [, π]), γ R := Γ R + C R f(x) dx 7

73 R R f(x) dx = f(z) dz. Γ R ( ) R > R π f(z) dz = f(re iθ ) ire iθ dθ max C R θ [,π] f ( lim f(z) dz =. R C R Re iθ) M Mπ R π πr = R2 R P (z) z < R f(z) f ( c j γ R ) γ R f(z) dz = 2πi Res(f; c) = 2πi Res(f; c). γ R c Γ R Im c> 2πi Im c> R Res(f; c) f(x) dx = R f(z) dz γ R f(z) dz = Γ R f(z) dz C R (R ). f(x) dx = 2πi Im c> Res(f; c). f Prop πi Im c< Res(f; c) ( p.76) Example I := x 2 dx. Prop f(z) := + z 2 + z2 + c = ±i Im c > i I = 2πi Res(f; c) = 2πi Res(f; i). i f Im c> Res(f; i) = lim z i z i (z i)f(z) = lim z i z i z + i = 2i. I = 2πi 2i = π Mathematica Integrate[/(x^2+),{x,-Infinity,Infinity}] Maple int(/(x^2+),x=-infinity..infinity) 72

74 4.4. f Example (Ahlfors [5] p.73) I := x 2 f(x) dx = 2 I = 2 f(x) dx (, ) x 2 x 4 + 5x 2 dx. + 6 x 2 x 4 + 5x dx. f(x) := x 4 + 5x 2 + 6, P (z) := z4 + 5z 2 + 6, Q(z) := z 2 f(x) = Q(x), deg P = deg Q + 2, P (x) P (z) = (z 2 + 2)(z 2 + 3) = (z + 2i)(z 2i)(z + 3i)(z 3i), x R P (x) I = 2 2πi ( ( Res (f; c) = πi Res f; ) ( 2i + Res f; )) 3i. Im c> ( Res f; ) 2i = Q ( 2i ) P ( 2i ) = z 2 4z 3 + z = z= 2i 4z 3 + z = z= 3i ( Res f; ) 3i = Q ( 3i ) P ( 3i ) = z 2 z 2i 2i 4z 2 + = z= 2i 8 + = 2, z 3i 3i 4z 2 + = z= 3i 2 + = 2. 2i 3i 3 2 I = πi = π. 2 2 ( Mathematica Maple ) 4.4. ( pp.89 9 ) I := dx + x 4. f(x) := + x 4, P (z) := z4 +, Q(z) := f = Q, deg P deg Q + 2, x R P (x). P P (z) = z = c c = exp ( πi 4 + k 2πi ) ± ± i 4 (k =,, 2, 3) c = ( ). 2 c = + i, + i, i, i ( P (z) = z 4 + 2z 2 + 2z 2 = (z 2 + ) 2 ( 2z ) 2 = (z 2 + 2z + )(z 2 2z + ) c = 2 ± ± 2 4, 2 2 = ± i, ± i i. P (z) c c 4 = 2 Res(f; c) = Q(c) P (c) = 4z 3 I = 2πi = 2πi Im c> = z=c ) + i 2 z 4z 4 z=c = c 4 ( ) = c 4 ( ( Res(f; c) = 2πi Res f; + i ) ( + Res f; + i )) 2 2 ( ) ( + i + + i ) = π

75 4.4.3 Mathematica Integrate[/(x^4+), {x,-infinity,infinity}] Maple int(/(x^4+),x=-infinity..infinity) 4.4. n N I = dx + x 2n = π n sin π. 2n Maple assume(n::integer, n>) int(/(x^(2n)+),x=-infinity..infinity) Mathematica n N I := dx (2n )!! ( + x 2 = π. ) n+ (2n)!! ( ) Mathematica Integrate[/(+x^2)^(n+), {x,-infinity,infinity}, Assumptions-> n>] Maple assume(n::integer, ( n>) ) int ( + x 2, x = infinity..infinity ) n+ ( πγ n + ) 2 I = Γ (n + ) Γ (x) = xγ (x ) ( Γ n + ) ( = n ) ( n 3 ) Γ (/2) = (2n )!! = π, 2 n I = Γ (n + ) = n! = (2n)!! 2 n π (2n )!! π/2 n (2n)!!/2 n = ( n 2 π(2n )!!. (2n)!! ) ( n 3 ) 2 π 2 () a I = (2) a I = dx x 2 + a 2. ( : π a ) dx x 4 + a 4 dx. ( : π a 3 2 ) 74

76 (3) a I = (4) a I = (5) a I = (6) a I = (7) a, n I = (8) a, n I = dx (x 2 + a 2 ) 2. ( : π 2a 3 ) dx x 6 + a 6 dx. ( : 2π 3a 5 ) x 4 (x 2 + a 2 ) 4 dx. ( : π 6a 3 ) x 2 (x 4 + a 4 ) 5 2π dx. ( : 3 64a 9 ) dx (x 2 + a 2 ) n. ( : π(2n 3)!! a 2n (2n 2)!! ) dx x 2n + a 2n. ( : π na 2n sin π ) 2n Maple assume(n::integer, n>) assume(a>) f:= n-> int(/(x^2+a^2)^n,x=-infinity..infinity) f(n) (a 2 ) n a πγ (n /2) Γ (n) seq(f(j),j=..) = πγ (n /2) a 2n (n )! = π a, π 2a 3, 3π 8a 5, 5π 6a 7, 35π 28a 9, dx x 2n + a 2n = π π(2n 3)!!/2 n a 2n (2n 2)!!/2 n = 63π 256a, 23π 24a 3, π na 2n sin π 2n. 429π 248a 5. π(2n 3)!! a 2n (2n 2)!!. () x 2 (Ahlfors p.73) ( ) () π( 3 2) 2 x 4 + 5x 2 dx (2) + 6 (2) 5π 2 (3) π 6 a 3 x 2 x + 2 x 2 x 4 + x 2 dx (3) + 9 (x 2 + a 2 ) 3 dx, a R Cauchy P (z), Q(z) C[z], deg P (z) deg Q(z) +, x R P (x) c R Q(x) p.v. dx = πiq(c) (x c)p (x) P (c) + 2πi ( ) Q(z) Res (z c)p (z) ; ζ. Im ζ> 75

77 p.72 f(x) dx e iax Fourier Fourier (F) f(y) := 2π f(x)e iax dx f(x)e ixy dx (y R) (F ) g(x) := 2π g(y)e ixy dy (x R) Fourier (F) y < a = y y f(x) x R f(x) R f(x) cos(ax) dx = Re f(x)e iax dx, f(x) sin(ax) dx = Im f(x)e iax dx Proposition f(x) = Q(x), P (z), Q(z) C[z], deg P (z) deg Q(z)+, x R P (x), P (x) a > (4.7) f(x)e iax dx = 2πi Res(f(z)e iaz ; c). Im c> (4.7) f(z)e iaz ( f ) c Im c> ( ) M R, R [, ) s.t. f(z) P (z) z < R f(z) M z ( z R ). ( ) A, B [R, ) C : z = x (x [ A, B]), C 2 : z = B + iy (y [, A + B]), C 3 : z = x + i(a + B) (x [ A, B]), C 4 : z = A + iy (y [, A + B]), C A,B := C + C 2 + C 3 + C 4 76

78 P (z) z < R f(z)e iaz f(z)e iaz Im c > c C A,B f(z)e iaz dz = 2πi Res(f(z)e iaz ; c) = 2πi Res(f(z)e iaz ; c). C A,B C B A c C A,B C f(z)e iaz dz = B A f(x)e iax dx. f(z)e iaz dz = f(z)e iaz dz = C f(z)e iaz dz C A,B = 2πi Im c> Res(f(z)e iaz ; c) 4 j=2 Im c> 4 j=2 C j f(z)e iaz dz. C j f(z)e iaz dz C 2 z C 2 z = z(y) = B + iy (y [, A + B]) dz = i dy, z = B 2 + y 2 B R, f(z) M z M B, Re (iaz) = a Re (iz) = a Im z = ay, e iaz = e Re(iaz) = e ay f(z)e iaz dz f(z) e iaz M dz C 2 C 2 B A+B e ay dy M B e ay dy = M ab. C 3 C 3 z = z(x) = x + (A + B)i (x [ A, B]) dz = dx, z = (A + B) 2 + x 2 A + B R, f(z) M z M A + B, Re (iaz) = a Re (iz) = a Im z = a(a + B), f(z) dz f(z) e iaz dz C 3 C 3 e iaz = e Re(iaz) = e a(a+b) M B A + B e a(a+b) dx = Me a(a+b). A C 4 C 4 z = z(y) = A + iy (y [, A + B]) dz = i dy, z = A 2 + y 2 A R, f(z) M z M A, Re (iaz) = a Re (iz) = a Im z = ay, e iaz = e Re(eiaz ) = e ay 77

79 f(z)e iaz dz f(z) e iaz M dz C 4 C 4 A A+B e ay dy M A e ay dy = M aa. A, B M ab, e a(a+b), M, ε > aa R R s.t. A R, B R = f(z)e iaz dz C j < ε (j = 2, 3, 4) 3 f(x)e iax dx (4.7) 2 ( deg P deg Q + 2 ()) f(x)e iax dx R (4.8) lim f(x)e iax dx = 2πi Res(f(z)e iaz ; c) R R Im c> P (x), Q(x) R[x] deg P (x) = n, m := deg Q(x) P (x) = a x n + + a n (a ), Q(x) = b x m + + a m (b ) n m + 2 ( P (x)e iax = P (x) Prop.4.4. ) n = m + h(x), M R s.t. x f(x) = b a x ( + h(x)), h(x) M x. b a x eiax dx b a x h(x)eiax dx f(x)e iax dx = R R f(x)e iax dx. f(x)e iax dx 78

80 (4.8) M R, R [, ) s.t. f(z) P (z) z < R f(z) M z ( z R ). R [R, ) Γ R z = x (x [ R, R]), C R z = Re iθ (θ [, π]) γ R := Γ R +C R P (z) z < R ( γ R ) f(z)e iaz f(z)e iaz γ R f(z)e iaz dz = 2πi Res(f(z)e iaz ; c) = 2πi Res(f(z)e iaz ; c). γ R c γ R Im c> C R z = Re iθ (θ [, π]) dz = ire iθ dθ π f(z)e iaz dz = f(re iθ )e iareiθ ire iθ dθ, γ R iare iθ = iar(cos θ + i sin θ) = ar sin θ + iar cos θ, [ Re iare iθ] = ar sin θ, e iareiθ = e Re[iaRe iθ ] = e ar sin θ f(z)e iaz π dz f(re iθ ) e iareiθ ire iθ dθ M π γ R R R e ar sin θ dθ = M π (4.9) lim R e ar sin θ dθ. π e ar sin θ dθ = R f(x)e iax dx = lim f(x)e iax dx = f(z)e iaz dz R R Γ R = f(z)e iaz dz f(z)e iaz dz γ R C R = 2πi Res(f(z)e iaz ; c) f(z)e iaz dz Im c> C R 2πi Res(f(z)e iaz ; c) (R ) Im c> (4.9) 3 79

81 (4.9) Lebesgue π e ar sin θ, dθ = π <, lim R e ar sin θ = < δ < π/2 π ( π/2 δ e ar sin θ dθ = 2 e ar sin θ dθ = 2 e ar sin θ dθ + ( ) π/2 2 δ + e ar sin δ δ ar sin δ 2δ + πe { (θ (, π)) (θ =, π) π/2 δ e ar sin θ dθ = (a.e.) δ R 2 Jordan 6 (4.2) sin θ 2θ π (4.2) < π/2 e R sin θ dθ π/2 e 2Rθ/π dθ = ( θ π 2 ) [ π 2R e 2Rθ/π] π/2 = π 2R ( e R ) < π 2R a < () Corollary 4.4. f(x) = Q(x), P (z), Q(z) C[z], deg P (z) deg Q(z) +, x R P (x), P (x) a < (4.22) f(x)e iax dx = 2πi Im c< Res(f(z)e iaz ; c) f(x) = Q(x) deg P (x) deg Q(x) + 2 a P (x) ( a = OK 7 ) lim f(z)e iaz dz = R C R π lim R R (e ar sin θ π e ar sin θ dθ = e ar sin θ dθ π ) 6 y = sin θ y = 2θ π Jordan [] (p.7) 7 ) 8

82 Example ( a ) a e iax ( ) e iaz (4.23) dx = 2πi Res + x2 + z 2 ; i = 2πi lim(z i) eiaz eia i = 2πi + z2 i + i = πe a. z i z i (4.24) cos ax + x 2 dx = πe a, sin ax dx =. + x2 a < e iax = e i( a)x e iax + x 2 dx = e i( a)x + x 2 dx = e i( a)x + x 2 dx = πe ( a) = πe a = πe a. a = 2 e 2ix = e 2ix e 2ix + x 2 dx = a e 2ix + x 2 dx = e 2ix + x 2 dx = πe 2 = πe 2. (4.25) e iax + x 2 dx = πe a (a R) ( a a =, ) (4.25) (4.23) (4.24) a < cos ax + x 2 dx = cos( a x) + x 2 dx = πe a, sin ax + x 2 dx = sin( α x) + x 2 dx = (4.25) assume(a>) int(cos(i*a*x)/(+x^2),x=-infinity..infinity) α > I = () x sin αx x 4 + dx = π 2 2 e α/ sin α. 2 Example a >, deg P (z) deg Q(z) +, P (z) 2 p.v. Q(x) P (x) eiax dx = 2πi Im c> Res ( ) Q(z) P (z) eiaz ; c 8 + πi Im c= ( ) Q(z) Res P (z) eiaz ; c

83 e ix e ix ( ) e iz p.v. dx = lim dx = 2πi x δ x 2 Res z ; x >δ sin x x ( ) dx = ( 2 Im e ix ) p.v. x dx = π 2. I = sin x x dx = π 2. = πi lim z eiz z z z = πi. ( ) f(z) := eiz z, Γ ε,r : z = x C R : z = Re iθ Γ R, ε : z = x C ε : z = εe iθ (x [ε, R]), (θ [, π]), (x [ R, ε]), (θ [, π]), γ ε,r := Γ ε,r + C R + Γ R,ε + ( C ε ) f f γ ε,r ( ) Cauchy ( ) = f(z) dz = γ ε,r f(z) dz + Γ ε,r f(z) dz + Γ R, ε f(z) dz C R f(z) dz. C ε ( ) lim f(z) dz =. R C R C R f(z) dz = (4.2) π f(z) dz C R Γ R, ε f(z) dz = π e ireiθ Re iθ e ir(cos θ+i sin θ) dθ = π ε R ( ) f(z) dz + Γ ε,r f(z) dz = Γ R, ε π ireiθ dθ = i e ir(cos θ+i sin θ) dθ e R sin θ dθ < 2 π 2R = π R. e ix ε x dx = e it R R t ( )dt = ε R ε 82 e ix R x dx ε e it t e it t dt R sin x dt = 2i ε x dx.

84 ( ) lim f(z) dz = iπ ε C ε (Cauchy 2πi C ε ) 2 ( ) Lebesgue z = εe iθ (θ [, π]) π e i(εeiθ ) π π f(z) dz = C ε εe iθ iεe iθ dθ = i e i(εeiθ) dθ i dθ = iπ (ε ). e i(εeiθ ) = e ε sin θ, e i(εeiθ) e = (ε +) ( 2) Lebesgue () ( ) ( ) f(z) = eiz z = z + (iz) n g n! C ε z dz = = z + π f(z) dz iπ = C ε i n n! zn = z + g(z), g(z) := π εe iθ iεeiθ dθ = i dθ = iπ C ε ( f(z) ) z dz = g(z) dz. C ε n= i n+ (n + )! zn g z M := max g(z) ε < z f(z) dz iπ g(z) dz M dz = Mπε. C ε C ε C ε lim f(z) dz = iπ. ε C ε ( ), ( ), ( ), ( ) R sin x 2i ε x dx = f(z)dz + f(z)dz πi + C ε Γ R (ε, R ). sin x x dx = π 2. f c C ε : z = c + εe iθ (θ [, π]) lim ε + C ε f(z) dz = πif(c). z c 83

85 () a, α I = (2) a, α I = (3) a, α I = cos αx x 2 + a 2 dx. e aα ( : a ) cos αx π( + aα)e aα (x 2 + a 2 dx. ( : ) 2 2a 3 ) cos αx x 4 + a 4 dx. πe aα/ 2 ( : 2a 3 ( cos aα 2 + sin aα 2 )) (4) a, α I = x sin αx x 4 + a 4 dx. πe aα/ 2 ( : a 2 sin ab 2 ) x sin x x 2 dx (a R) + a2 πe a ( : ) π r(cos θ, sin θ)dθ r(x, Y ) X, Y r(cos θ, sin θ) dθ [, 2π] ( ) Proposition () r(x, Y ) X, Y 2π r(cos θ, sin θ) dθ = 2πi Res(f; c). c < f(z) := ( z 2 ) iz r +, z2 2z 2iz f(z) z = z = e iθ (θ [, 2π]) cos θ = (z + /z)/2, sin θ = (z /z)/(2i), dz = ie iθ dθ dθ = dz iz I := 2π r(cos θ, sin θ)dθ 84

86 z = e iθ (θ [, 2π]) dz = ie iθ dθ, cos θ = eiθ + e iθ 2 sin θ = eiθ e iθ 2i dθ = dz iz, = 2 = 2i ( z + z ) = z2 +, 2z ( z ) = z2 z 2iz I = z = ( z 2 ) + r, z2 dz 2z 2iz iz = f(z) dz = 2πi Res(f; c). z = c < () cos θ = eiθ + e iθ 2 = z + z, sin θ = eiθ e iθ 2 2i = z z 2i cos θ = eiθ + e iθ = z + z 2 2, sin θ = eiθ e iθ = z z 2i 2i ( ) f(z) := ( z + z iz r 2, z z ) 2i f () < r < R I := 2π ) ( ) z = e iθ (θ [, 2π]) I = z = R 2 + r 2 2Rr z2 + 2z = i z = I = i 2πi c < = 2π lim z r/r dθ R 2 + r 2 2Rr cos θ dz Rrz 2 (R 2 + r 2 )z + Rr = i ( ) Res (Rz r)(rz R) ; c z r/r (Rz r)(rz R) = 2π R ( dz iz = dz i z = (R 2 + r 2 )z Rr(z 2 + ) z = dz (Rz r)(rz R). ( = 2π Res r r/r R = Example a > 2π dθ 2a cos θ + a 2 = 2π a (Rz r)(rz R) ; r R 2π R 2 r 2. )

87 Example a > π Example e < 2π dθ a + cos θ = π a 2. dθ ( + e cos θ) 2 = 2π ( e 2 ) 3/2. (Mathematica Integrate[/(+e Cos[x])^2,{x,,2Pi}, Assumptions->e> && e<] Kepler ) [2] pp.8 8 () n I = 2π 2π cos 2n θ dθ. (2) n I = 2π sin 2n θ dθ. dθ dθ (3) a, b I = a 2 cos 2 θ + b 2 sin 2. (4) () I = θ 5 4 cos θ. (5) 2π dθ ( ) I = R 2 + r 2 ( < r < R). (6) () I = 2Rr cos θ 2π ( ) R + re iθ 2π dθ Re 2π R re iθ dθ ( < r < R). (7) I = (R 2 + r 2 2Rr cos θ) 2 ( < r < R). (8) 2π dθ ( ) I = (a > b > ). (9) m, r, R (r < R) I + ij a + b cos θ I, J I = 2π 2π cos mθ 2π R 2 + r 2 2Rr cos θ dθ, J = sin mθ R 2 + r 2 2Rr cos θ dθ. () a, n I + ij I, J I = 2π e a cos θ cos (a sin θ nθ) dθ, J = 2π e a cos θ sin (a sin θ nθ) dθ. () 2π(2n)! 4 n (n!) 2 (2) 2π(2n)! 4 n (n!) 2 (3) 2π (4) 2π 2π (5) ab 3 R 2 r 2 (6) (7) 2π(R2 + r 2 ) (R 2 r 2 ) 3 2π (8) (9) I = 2π ( r ) m, 2πa n J = () I = a 2 b 2 R 2 r 2, J = R n! a 2 > b 2 + c 2 a, b, c dx a + b cos x + c sin x ( : 2π a 2 b 2 c 2 ) (Ahlfors [5], p.73) π/2 dx π a + sin 2, a > ( : sign a x 2 a 2 + a ) 86

88 4.4.5 x α x α f(x)dx f x α ( < α < ) [, ) I = x α f(x) dx z α z α α x α x < z α z α = exp (α log z) a (log z ) ( ) z = re iθ (r >, θ [, 2π)) log z = log r + i(θ + 2nπ) (n Z) ( log r log ) z α = exp (α (log r + i(θ + 2nπ))) = r α e iαθ e i2αnπ. α Z n Z nα Z e i2αnπ = n α Z ( ) z α = r α = z α. ( z α α ) a x = exp(log x), x α = exp(α log x) Ω := C \ [, ) log z (, 2π) z z = re iθ (r >, θ (, 2π)) log z = log r + iθ. z α := exp (α log z) = exp(α(log r + iθ)) = r α e iαθ. ( ) z α ( α (, )) z = re iθ (r >, θ (, 2π)) z α = r α e iαθ, z α = z α. z (, ) ( ) (z α ) = (z α ) e 2παi ( ) 87

89 4.4. ( ) f(z) = Q(z), P (z), Q(z) C[z], deg P (z) P (z) deg Q(z) + 2, x (, ) P (x), f < α < x α 2πi f(x) dx = e 2παi Res(z α f(z); c). c < ε < R, < δ < π ε, R, δ ( ε, δ, R ) C := C + C 2 + C 3 + C 4, C z = te iδ (t [ε, R]), C 2 z = Re iθ (θ [δ, 2π δ]), C 3 z = te i(2π δ) (t [ε, R]), C 4 z = εe iθ (θ [δ, 2π δ]), ε, δ, R ( ) z α f(z) dz + z α f(z) dz + z α f(z) dz + z α f(z) dz = 2πi Res(z α f(z); c). C C 2 C 3 C 4 c C C z α f(z) dz = R ε (te iδ) α R f(te iδ ) dt = e iαδ t α f(te iδ ) dt. δ t [ε, R] t α f(te iδ ) t α f(t), e iαδ R z α f(z) dz t α f(t) dt. C ε C 2 z α f(z) dz = C 2 δ 2π δ (Re iθ ) α f(re iθ ) ire iθ dθ 2π ε (Re iθ ) α f(re iθ ) ire iθ dθ (δ ). θ = (Re iθ ) α = R α, θ = 2π (Re iθ ) α = R α e 2παi ( [, 2π] ) 2π (Re iθ ) α f(re iθ ) ire iθ 2π dθ Rα+ f(re iθ ) dθ R α+ 2π M R 2 C 3 z α f(z) dz = C 3 ε R = 2πM R α (te (2π δ)i ) α f(te (2π δ)i ) e (2π δ)i dt (R ). R R = e (2π δ)αi e (2π δ)i t α f(te (2π δ)i ) dt = e 2παi e (+α)δi t α f(te δi ) dt. δ t [ε, R] t α f(te δi ) t α f(t) R z α f(z) dz e 2παi t α f(t) dt. C 3 ε C 4 C 4 z α f(z) dz = 2π δ δ ε (εe iθ ) α f(εe iθ ) iεe iθ dθ 88 2π (εe iθ ) α f(εe iθ ) iεe iθ dθ (δ ). ε

90 θ = (εe iθ ) α = ε α, θ = 2π (εe iθ ) α = ε α e 2παi ( [, 2π] ) 2π (εe iθ ) α f(εe iθ ) iεe iθ 2π dθ εα+ f(εe iθ ) dθ ε α+ 2π M ε = 2πM ε α (ε ). ( ) δ ε, R t α f(t) dt e 2παi t α f(t) dt = 2πi Res(z α f(z); c). c t α f(t) dt = 2πi e 2παi Res(z α f(z); c). c δ C z α f(z) dz R C 3 z α f(z) dz e 2παi ε t α f(t) dt, R ε t α f(t) dt [ε, R] 2πi c Res (f(z) log z; c) = f(z) log z dz = ( e 2παi ) C ε,r ( C ε,r ) R f(x)dx x α ( + x 2 dx = 2πi e 2παi = 2πi e 2παi Res ( e παi/2 2i ( ) z α + z 2 ; i = π ( e παi/2 e 3παi/2) e 2παi = e3παi /2 2i π 2 cos πα 2 ( z α + Res ). )) + z 2 ; i Mathematica, Maple ( Integrate[x^a/(+x^2), {x,-infinity,infinity}], integrate(x^a/(+x^2),x =-infinity..infinity) ) (Ahlfors p.74) x /3 + x 2 dx ( : π 3 ) 89

91 4.4.6 f(x) dx ( ) f ( ) f(x) dx = 2 f(x) dx = 2 2πi Im c> Res(f; c) = πi Im c> Res(f; c) f z (, ) 2πi c (log z) = (log z) + 2πi Res (f(z) log z; c) = f(z) log z dz = 2πi C ε,r R f(x)dx +. ( C ε,r ) Proposition () f(z) = deg P (z) deg Q(z) + 2, x [, ) P (x) f(x) dx = c C\[, ) Q(z), P (z), Q(z) C[z], P (z) Res (f(z) log z; c). log Im log (, 2π) f(z) log z 4.4. ε, δ R 4 f(z) log z dz = 2πi Res(f(z) log z; c). C j δ δ δ j= f(z) log z dz C f(z) log z dz C 3 ε c C\[, ) R R C +C 3 f(z) log z dz 2πi C 2 f(z) log z dz 2π ε f(t) log t dt, f(t) [log t + 2πi] dt. R ε f(t) dt. f(re iθ )(log R + iθ) ire iθ dθ, 9

92 2π f(re iθ )(log R + iθ) ire iθ 2π dθ R(log R + 2π) f(re iθ ) dθ R(log R + 2π) M R 2 2π C 4 ε 2πi f(t) dt = 2πi c C\[, ) 2πi [9] pp.6 63 f(x) dx = f(z) Log( z) dz = 2πi C Res(f(z) log z; c). c f (R ). Res (f(z) Log( z); c). C C ( ) (?) (principal value) Log ( Log( z) ) f(x) dx = c f Res (f(z); c) Log( c) I = dx x 3 + = 2 3π (log x) n g(x)(log x) n dx () Ω = {z C; Im z > }, g Ω x R g( x) = g(x) R θ [, π] g(re iθ )R log R. 2 g(x) log x dx + iπ g(x) dx = 2πi Res(g(z) Log z; c). Log z Im(Log z) (, 2π) a > I = 2 log x x 2 dx + iπ + a2 Im c> log x π log a x 2 dx = + a2 2a. x 2 dx = 2πi Res + a2 ( ) Log z z 2 + a 2 ; ia = π (log a + π ) a 2 i log x π log a x 2 dx = + a2 2a. 9

93 4.4.7 a > I = 2 log x (x 2 + a 2 ) 2 dx+iπ log x π(log a ) (x 2 + a 2 2 dx = ) 4a 3. ( ) Log z (x 2 + a 2 dx = 2πi Res ) 2 (z 2 + a 2 ) 2 ; ia = 2πi d ( ) (z ia) 2 Log z dz (z 2 + z 2 ) z=ia 2 = 2π a > I = 2 log x x 4 dx + iπ + a4 log x π(2 log a π/2) x 4 dx = + a4 4. 2a 3 x 4 dx = 2πi + a4 ( Res ( Log z z 4 + a 4 ; aeπi/4 = = π 2 2a 3 [ (2 log a π 2 ) + iπ ] log x a > I = x 6 + a 6 dx = π(log a 2π/ 3) 3a 5. 2πi ( ) Res() = π 3a 5 2 log a π + iπ Im c> ) ( )) Log z + Res z 4 + a 4 ; ae3πi/4 (Ahlfors p.74) log x dx ( : ) + x (log x) n () f f(x)(log x) n dx f(x) (log x) n dx (n = 4.4.6) x α log () 92

94 a < b Log z a z b z = a, b C \ [a, b] x (a, b) ( lim Log z + ε z Log z + ) z=x+iε z = 2πi z=x iε f [a, b] (C ) D b a f(x) dx = f(z) Log z a 2πi C z b dz. C [a, b] D C C f 4.4. I = dx x 4 + = ( Log + ) 2 + π I = π log sin θ dθ = π log 2. (Ahlfors p.74) log( + x 2 ) x +α dx ( < α < 2) ( π α sin απ ) 2 a b Γ a,b (Fresnel ( ) ) f(z) := exp( z 2 /2) C := C + C 2 + C 3, C := Γ,X, C 2 := Γ X,(+i)X, C 3 := Γ (+i)x, (X (, )) cos ( x 2) dx = sin ( x 2) dx = π ( ) f C = f(z) dz = C f(z) dz + C f(z) dz + C 2 f(z) dz. C 3 x 8 Augustin-Jean Fresnel ( ) Fresnel C(x) := sin ( t 2) dt x Fresnel 93 x cos ( t 2) dt, S(x) :=

95 f(z) dz = dz + C 3 C C z = x (x [, X]) C f(z) dz = x/ 2 = t dx = 2dt C f(z) dz = X/ 2 X C 2 e x2 /2 dx. e t2 2 dt 2 C 2 z = X + iy (y [, X]) e (X2 y 2 +2iXy)/2 z2 dz. 2 = y 2 + 2iXy X2, dz = i dy 2 e t2 = π 2. X f(z) dz = e (X2 y 2 +2iXy)/2 i dy. C 2 = e (X2 y 2 )/2, (X 2 y 2 ) = (X + y)(x y) X(X y) X f(z) dz e (X2 y 2 )/2 dy C 2 = X = 2 X e Xt/2 ( )dt = X ( ) e X2 /2 X e X(X y)/2 dy e Xt/2 dt = (X ). [ 2 X e Xt/2 ] X C 3 z = ( + i)t (t [, X]) z 2 = ( + i) 2 t 2 = 2it 2, dz = ( + i)dt C 3 f(z) dz = X X ( e it2 ( + i)dt = ( + i) cos(t 2 ) i sin ( t 2)) dt. ( X X ) ( + i) cos(t 2 )dt i sin(t 2 )dt = 2 X/ 2 π e t2 dt + f(z) dz C 2 2 (X ). X cos(t 2 ) dt i sin(t 2 ) dt = + i π 2 = i 2 π 2. cos ( t 2) dt = sin ( t 2) dt = π

96 4.4.3 a R e x2 cos(2ax)dx = e a2 π, e x2 +i2ax dx = e a2 π. e x2 sin(2ax)dx =. ( ) a = () ( e x2 +i2ax = e x2, e x2 dx < ) a > (a < a a ) f(z) := exp( z 2 ) a >, X > C := Γ X,X, C 2 := Γ X,X+ia, C 3 := Γ X+ia, X+ia, C 4 := Γ X+ia, X, C := C + C 2 + C 3 + C 4 f C = f(z) dz = C f(z) dz + C f(z) dz + C 2 f(z) dz + C 3 f(z) dz. C 4 f(z) dz = C 3 f(z) dz + C f(z) dz + C 2 f(z) dz. C 4 C z = x (x [ X, X]) C f(z) dz = X X e x2 dx. C 2 z = X + iy (y [, a]) (X + iy) 2 = (X 2 y 2 + 2iXy) a f(z) dz = exp [ (X 2 y 2 + 2iXy) ] i dy. C 2 X > a y [, a] Re [ (X 2 y 2 + 2iXy) ] = (X 2 y 2 ) = (X + y)(x y) X(X y) X(X a) a f(z) dz exp [ (X 2 y 2 + 2iXy) ] dy = C 2 a e X(X a) dy = ae X(X a) a e (X2 y 2) dy (X ). f(z) dz C 4 (X ). C 3 z = x + ia (x [ X, X]) exp( z 2 ) = exp [ (x + ia) 2] = exp [ x 2 + a 2 2aix ] = e a2 e x2 (cos(2ax) i sin(2ax)) 95

97 C 3 f(z) dz = e a2 ( X X X ) e x2 cos(2ax)dx i e x2 sin(2ax)dx. X e a2 ( X X e x2 cos(2ax)dx i X X ) X e x2 sin(2ax)dx = e x2 dx + f(z) dz + X C 2 π (X ). e x2 cos(2ax)dx = e a2 π. C 4 f(z) dz e x2 sin(2ax)dx =. e x2 e i2ax dx = e a2 π n, m N, m < n I = x m + x n dx = π/n sin(mπ/n). ( n ) dx + x 2 = π 2, dx + x 5 = 4π dx + x = ( r (, ) 9 dx + x 3 = 2π 3 3, dx + x 4 = π 2 2, 5 2 5, dx + x 6 = π 3, 5 + )π dx, + x 2 = ( 6 + 2)π. 2 x r + x dx = π sin(rπ) dx + x 8 = π 4 2 2, (r = m/n x n = u ) ( ) f(z) := zm ( + 2k)πi exp (k =,,..., n ) + zn n ω := exp πi n f ω2k+ (k =,,..., n ) ω n =, ω 2n = R (, ) C := Γ,R, C 2 : z = Re iθ (θ [, 2π/n]), C 3 := Γ Rω 2,, C := C + C 2 + C

98 C (C 2 Rω 2 ) f f ω ( ) f(z) dz + C f(z) dz + C 2 f(z) dz = 2πi Res(f; ω). C 3 ω f Res(f; ω) = zm ( + z n ) = zm z=ω nz n = ωm z=ω nω C z = x (x [, R]) C 2 C f(z) dz = R C f(z) dz = f(x) dx = 2π/n R n = ωm x m + x n dx. f(re iθ ) ire iθ dθ nω n = ωm n. M, R f(z) M/ z 2 ( z R ) R > R 2π/n f(z) dz f(re iθ ) R dθ M 2π/n C R 2 R C 3 z = tω 2 (t [, R]) ( ) C 3 f(z) dz = R f(tω 2 ) ω 2 dt = ( ω 2m ) R R dθ = 2πM nr (R ). t m ω 2(m ) R + t n ω 2n ω 2 dt = ω 2m x m + x n dx. x m + x n dx + f(z) dz = 2πi ωm C 2 n. R Γ ( ω 2m ) x m + x n dx = 2πiωm n x m 2πiωm dx = + xn n( ω 2m ) = π 2i n(ω m ω m ) = π n sin(mπ/n). ( ) ( q Γ q ) ( q = B p p p, q ) x q = p p + x p dx = π sin (πq/p). B(α, β) = Γ (α)γ (β)/γ (α + β) Γ () = x p = u (4.26) x q + x p dx = u q/p p + u du. 97

99 B(p, q) = x = t/( + t) (4.27) B(p, q) = x (, ) C x p ( x) q dx t p dt ( + t) p+q x q + x p dx = ( q p B p, q ) p Γ (x)γ ( x) = π sin(πx). Γ (z) = e γz z k= e z/k + z/k (γ Euler ) Γ (z)γ ( z) = e γz e γz sin πz = z 2 k= π Γ (z)γ ( z) = πz2 e z/k + z/k k= k= e z/k z/k = z 2 k= (z/k) 2. ( (z/k) 2 ) ( = πz (z/k) 2 ). k= 4.5 [3] 4.5. (, infinity, point at infinity) ( C) z ( ) 98

100 (+ ), + + = ( ) (+ ) + ( ) = lim f f : I R (I R), a I, A R lim x a f(x) = A lim f(x) = + def. x a lim f(x) = A x + def. ε > δ > x I x a < δ = f(x) A < ε. U R δ > x I x a < δ = f(x) > U. def. ε > R R x I x > R = f(x) A < ε. lim f(x) = + x + + lim f(z) = A z a lim f(z) = z a lim f(z) = A z f f : Ω C (Ω C), a Ω, A C def. ε > δ > z Ω z a < δ = f(z) A < ε. def. U R δ > z Ω z a < δ = f(z) > U. def. ε > R R z Ω z > R = f(z) A < ε. lim f(z) = z 4.5. lim x x lim x x = + lim x + x = +, lim z lim x z = 99 x =

101 4.5.2 () +, n N lim z zn =. (Cf. lim x + xn = +, lim x xn = ± (n +, n )) ( ) P (z) lim P (z) =. n N lim =. (Cf. n lim zn x +, lim = lim x xn x x n ) exp z (Cf. lim lim z z x + ex = +, lim sin z =. lim =. z sin z lim Log z lim Log z =, z z C\(,) z lim x ex =.) z = +. n lim xn x + cos z =, lim z π/2 lim z C\(,) z x n = tan z =. z π/2 Log z =. log z Re log z = log z ( log log) log z =, lim log z =. lim z z C\{} z C \ {} z z α α R z α = z α ( ) (i) α > lim z zα =, lim z α =. z z (ii) α < lim z zα =, lim z α =. z z ( ) ( 2 ) ( lim = + ) ( C) C Ĉ P : Ĉ = P = P (C) := C { }. ( ( P (C) ) ) 2 sin, cos ( tan ) ( ) e iθ = cos θ + i sin θ ( ) cos θ sin θ ( ) sin θ cos θ

102 lim a C a + = + a =. b C \ {} b = b =. =. a a C \ {} =. b C b =. +,, Ĉ = C { } f(z) = az + b cz + d (a, b, c, d ad bc ) C f : Ĉ Ĉ f(z) = z + 2 { 3z + 4 z 4 f : C\ 4 } 3 3 C (z z 2 = f(z ) f(z 2 )) f(z) = 3 z ( z + 2 3z + 4 = 3(z + 2) = 3z ) + 2/z lim f(z) = lim z z 3 + 4/z = 3, lim f(z) = z 4/3 z 4/3 z + 2 3z + 4 { (z C \ 4 } ) 3 f(z) := (z = 4 3 ) (z = ) 3 f : Ĉ Ĉ f ( lim f(z) = f 4 ), lim f(z) = z 4/3 3 f( ) z Ĉ ( lim ) f f f : Ĉ Ĉ

103 S := {(x, x 2, x 3 ) R 3 ; x 2 + x x 2 3 = }, N := (,, ) x x 2 (x 3 = ) H C (x, x 2, ) H x + ix 2 C P S \ {N} N P H P P P φ: S \ {N} P P H = C N (stereographic projection) P = (x, y, ) = x + iy ( 2 ) x = x x 3, y = x 2 x 3, x + iy = x + ix 2 x 3. φ: S \{N} H z = φ(x, x 2, x 3 ) x, x 2, x 3 2 z 2 = + x 3, x 3 = z 2 x 3 z 2 +, x = z + z z 2 +, x i(z z) 2 = z 2 +. φ() = {z C; z > }, φ( ) = {z C; z = }, φ() = {z C; z < }, φ( ) = φ: S \ {N} C φ: S Ĉ = C { } φ(n) := ( φ ) φ Ĉ = C { } S (Riemann sphere) P N φ(p ) Ĉ φ N 4.5. S ξ 2 + η 2 + (ζ /2) 2 = (/2) 2 () Ĉ a C {D(a; ε)} ε> a 4.5. X a X X U a 2 (i) U U a (U a ) (ii) ( V : V a ) U U s.t. U V. C Ĉ = C { } ( ) 2 x 2 +x 2 2 +x 2 3 = z 2 = x 2 +y 2 = x2 + x 2 2 ( x 3 ) 2 = x2 3 ( x 3 ) 2 = + x 3. x 3 x 3 x 3 = z 2 z 2 +. x 2 3 = z 2 + x = x( x 3 ) = z + z 2 2 z 2 + = z + z z 2 +. x 2 = y( x 3 ) = z z 2 i(z z) 2i z 2 = + z

104 (a) a C (C ) {D(a; ε)} ε> a (b) {U R } R> U R := {z C; z > R} { }. ( = + U R = {z Ĉ; R < z + } ) ( a ) a Ĉ a (C a ) Ĉ z () 4.5. f(z) = z z = f() = =, lim f(z) = z z lim z f(z) = f() f z = Ĉ = C { } d : (4.28) d(z, z ) := φ (z) φ (z ) = 2 z z ( + z 2 ) ( + z 2 ) (z, z C ). S R 3 Ĉ () z = f(z) ( )( ) z = ( ) w g(w) := f w = ( w ) 3

105 : c C c C f (isolated singularity) R > s.t. f {z C; < z c < R} {a n } n Z s.t. a n f(z) = (z c) n + a n (z c) n n= ( < z c < R). f c (c ) Laurent f c Laurent f c 3 (i) c (removable singulariry,, regular point) n N a n = (a) (b) (a) ε (, R) s.t. f < z c < ε (b) lim z c z c f(z) C ( lim f(z) = a ) z c z c (ii) c (pole) k N s.t. a k n > k a n = ( k c (order), c f k ) lim f(z) = z c z c (iii) c ( ) (essential singularity) k N, n > k s.t. a n lim f(z) C lim f(z) = z c z c z c z c (Ĉ ) f c (residue) Res(f; c) Res(f; c) = Res z=c f(z) dz := a c D(c; R) C f(z) dz = 2πi Res(f; c) C 4

106 4.5.2 ( ) f (isolated singularity) R (, + ) s.t. {z C; z > R} f f ( ) g(w) := f ( < w < w R ) g g f (i) f (removable singularity,, regular point) g ( ) (ii) f (pole) g g k f k (iii) f (essential singularity) g 4.5. ( Laurent ) f R (, + ) s.t. f R < z <!{a n } n Z s.t. (4.29) f(z) = a n z n + a + a n z n (R < z < + ) (4.3) a n = 2πi ζ =r f(ζ) dζ (n Z) ζn+ (r R < r < ) () (3) () f n N a n =. (2) f k N a k n > k a n =. (3) f k N n > k s.t. a n. f A(; R, + ) = {z C; R < z < + } Laurent (4.29) {a n } n Z (c =, R = R, R 2 = + ) g(w) = f ( w ) = a n w n + a + (4.29) f ( ) Laurent a n w n ( < w < R ) a n z n a n (4.3) f ( ) 5

107 4.5.2 ( lim ) f (),(2),(3) () f lim f(z) C ( a ) z C z (2) f lim f(z) =. z C z (3) f lim f(z) C = (Ĉ ) z C z z = z w w (Cor. 4.3., p.55) Riemann ( ) Ĉ = C { } Riemann U R = {z Ĉ; R < z } w = z f(z) = z z = ( ) g(w) := f w = lim z z ( ) = w w f(z) = z z = ( ) g(w) := f = w w = w = () w = lim z z = ( ) f(z) := z z + f ( ) = w w w = w + + w w w = f z = ( z lim f(z) = ) f( ) =. g(z) := Log z z + z = g( ) = n N n P (z) = a z n + + a n (a ) ( ) P = a w w n + + a w + a, a w = n z = P n 6

108 4.5.4 z = sin z, exp z ( ) f : C C {a n } n s.t. f(z) = a n z n (z C). a n a n ( ) a n f = w w n + a (w C) ( ) w f w = w = ( ) w f ( ) n= C U R = {z C; R < z < + } { } = {z Ĉ; R < z + } ( Riemann z < ) z Riemann w r (R, + ) C z = re iθ (θ [, 2π]) Riemann C w C w = r eiθ (θ [, 2π]) C z = Riemann (,, ) w C w = ( ) 7

109 Res(f; ) C f(z) dz = 2πi Res(f; ) z = w dz = w 2 dw, C C C f(z) dz = ( Res(f; ) = Res ( w 2 f w = w 2 C f ( ) ( ( w w 2 )dw = 2πi Res ( w 2 f w ) ) ; n Z a n w n w = n Z ) ) ;. a n w n 2 w = a ( ) f ( Res(f; ) = Res f(z) dz := Res ( z= w 2 f w f (residue) ) ; ) = a () Res(f; ) a Res(g; ) ( n N a n = ) Res(f; ) ( a ) ( ) Res(f; c) Res f(z) dz f f(z)dz ( [8] ) f(z) = z z = Res z= f(z) dz = Res w 2 w= f(z) = n Z a n z n a = w dw dw = Res w= w =. z=c ( Laurent ) Laurent ( ) f(z) = z4 + z z 2 z

110 ( ) ( ) z 4 + z z 2 z + 2 z 2 + z + 9, 7z 9 + : (4.3) f(z) = (z2 z + 2)(z 2 + z + 9) + 7z 9 z 2 z + 2 7z 9 z 2 z + 2 = ( z 7 9 ) z ( z 2 z + 2 ) = z g z 2 g w < ( ± 7i 2 4 ) Taylor : {b n } n= s.t. g(w) = n= = z 2 + z z 9 z 2 z + 2. ( ), g(w) := z b n w n ( w < 2 ). 7 9w w + 2w 2. 2 f(z) = z 2 + z z g ( z ) = z 2 + z b z + b z 2 + ( 2 < z < ). f Laurent (Laurent ) = z 2 + z, Res(f; ) = a = g() = 7. ( ) 7z 9 Res(f; ) = b = g() = lim z z z 2 z + 2 (4.3) (b, b 2,... b ) f(z) = Q(z) P (z) Q(z) P (z) q(z), r(z) f Laurent q(z) q(), lim zr(z) z ( 2) f(z) = ( z 4 + z ) z 4 + ( z 2 z + 2 ) = z 2 z z 4 z 2 z + 2 z 2 h w < 2 h(w) := + w2 + 9w 4 w + 2w 2 ( ) f(z) = z 2 h. z h(w) = + w2 + 9w 4 w + 2w 2 = c + c w + c 2 w 2 + c 3 w 3 + c 4 w 4 + ( w < 2 ) 9

111 Taylor {c n } + w 2 + 9w 4 = ( w + 2w 2 )(c + c w + c 2 w 2 + ) = c, = c + c, = 2c c + c 2, = 2c c 2 + c 3, 9 = 2c 2 c 3 + c 4, = 2c n c n+ + c n+2 (n 3). c =, c =, c 2 = 9, c 3 = 7, c 4 = 2, c 5 = 6, () h(w) = + w + 9w 2 + 7w 3 2w 4 6w 5 + ( w < 2 ). ( ) f(z) = z 2 h = z 2 c n z z n n= ( = z 2 + z + 9 z z 3 2 z 4 6 ) z 5 + = z 2 + z z 2 z z 3 + c 6 z 4 + c 7 z 5 + ( 2 < z < ). f 2 < z < Laurent (Laurent ) z 2 + z, 7. g 4 (c 3 ) 2 ( ) 2 2 ( ) 4.6 f (rational function) f(z) z P (z), Q(z) C[z] s.t. f(z) = Q(z) P (z) (P (z) z C) P (z) Q(z) ( 22 ) P (z) Q(z) 22 P (z) = z, Q(z) = z 2 f(z) = z2 C \ {} f(z) = z z C

112 4.6. ( ) D Ĉ = C { } f D (meromorphic) ( ) E D s.t. D \ E Ĉ f : D \ E C c E c f E = ( ) 4.6. ( Riemann ) Ĉ f(z) = Q(z) P (z) (P (z), Q(z) ) P (z) c,..., c N f C \ {c,..., c N } f c j (j =, 2,..., N) z = g(w) := f ( ) w g(w) = w w w = f(z) z = (: R := max{ c,..., c N } + R > R < z < f f lim f(z) z z f f ) 4.6. E c E c f R > s.t. f {z C; < z c < R} D(c; R) E = {c} ( ) c C, R >, f A(c;, R) = {z C; < z c < R} c f f D(c; R) = {z C; z c < R} c f c f ( R > s.t. f A(c;, R) = {z C; < z c < R} ) (a) ( ), (b) ( ) f D(c; ε) (c) ( ) f D(c; ε) c f c f

113 4.6. (Picard) f < z c < R c f < z c < R f(z) ω ω Ĉ 2 Ĉ \ f(a(c;, R)) f(z) = exp z Ĉ \ f(a(c;, R)) = {, } 4.6. ( ) (Mittag-Leffler ) (partial fraction decomposition) 23 f(z) = Q(z) P (z), (P (z), Q(z) ) f(z) C P (z) c,..., c r, c j (j) f j (z) = a(j) k j (z c j ) k + + a j z c j R := max{ c,, c r } + f R < z < {a n } n Z s.t. f(z) = a n z n + a + a n z n f Laurent f (z) := a n z n. f Laurent n := deg Q(z), m := deg P (z), N := n m (R < z < ). f(z) lim z z N = a N, n > N a n = N f (z) = a n z n N (N = ). g(z) := f(z) r f j (z) f (z) (z C \ {c j } r j= ) j=

114 g Ĉ C g C lim g(z) = a z z C R >, ( z C : z R ) g(z) a + g(z) max{m, a + }, M := max z R g(z). Liouville g : C C z C g(z) = C. z C = a f(z) = f (z) + N f j (z) + a (z C \ {c j } r j= ). j= f(z) f(z) ( ) ( ) f(z) f(z) s.t. ( ) f(z) = N a n z n + {a n } N n=, r N {}, {c j } r j=, {m j } r j=, {a jk } j r k m j m r j a jk (z c j ) k (z C \ {c j } r j= ), N = a N, a jmj. n= j= k= {a n }, r, {c j }, {m j }, {a jk } N a n z n = n= N b n z n (z C) = (a, a,..., a N ) = (b, b,..., b N ). n= N (a n b n )z n = b a (z C) z a n b n = (n =, 2,..., N). = b a =. (a, a,..., a N ) = (b, b,..., b N ). 3

115 a,..., a m lim m z k= a k (z c) k = (i) a m (ii) c c lim m z c k= lim m z c k= a m k (z c) k = N a n z n n= a k (z c) k =, k= a k (c c) k ( ). (i), (ii) 4.6. f(z) := z4 + 3z (z ) 2 (z + 2). f Laurent f(z) ( f(z) ) 2 f(z) f z 4 + 3z Res(f; 2) = lim (z + 2)f(z) = lim z 2 z 2 (z ) 2 = ( 2)4 + 3 ( 2) ( 3) 2 = 6 6 =. 9 f 2 Laurent f 2 (z) := z + 2 f(z) 2 f 2 f Laurent f(z) = b 2 (z ) 2 + b z + b n (z ) n b 2 = lim(z ) 2 z 4 + 3z f(z) = lim = + 3 =, z z z ( ) d 2 b = Res(f; ) = lim (z ) 2 f(z) = lim z (2 )! dz z ( d = lim z 3 2x 2 + 4z ) z dz z + 2 = = 2. d dz ( = lim 3z 2 4z + 4 z ( z 4 ) + 3z z (z + ) 2 f Laurent f (z) := (z ) z f(z) 2 < z < (C 2, ) {a n } n Z s.t. ( ) f(z) = a n z n + a + a n z n 4 (2 < z < ). )

116 f(z) z 4 + 3z (z ) 2 (z + 2) = z 3 3z + 2 z, 3z 2 + z ( ) f(z) = z + 3z2 + z (z ) 2 (z + 2). ( ) ( ) lim z 3z 2 + z (z ) 2 =, lim (z + 2) z a + (a )z + a n z n n=2 a n z n = (z ). a =, a =, a n = (n 2) f Laurent f (z) := z Res(f; ) = lim z 3z 2 + z z (z ) 2 (z + 2) = 3. f(z) = f 2 (z) + f (z) + f (z) + a = z (z ) z + z f(z) f Ĉ () f Ĉ f E Ĉ Ĉ \ E f E {c n } n N s.t. n N c n E, j k c j c k. Ĉ {c n } n N c c (i.e. c E) c (i.e. c E, c f ) R > s.t. f < z c < R lim c n = c 24 n E c j (j =, 2,..., N) Laurent f j N f f j C j= f(z) = N f j (z) + C j= (z Ĉ). Laurent f j z f C (Bolzano- Wierstrass ) Ĉ 24 c n c N N s.t. n N c n c < R. c N c N+ E D(c; R) 2 E 5

117 4.6.2 f(z) = Q(z) P (z), (P (z), Q(z) ) f(z) C c,..., c N (N = ) R > {c,, c n } D(; R) ( N R := max{ c,..., c N } + ) N f(z) dz = Res(f; c j ). 2πi z =R Res(f; ) ( ) f Ĉ = C { } : α=c,...,c N, j= Res(f; α) =. ( c,..., c N f C ) ( f Res(f; ) Res(f; ) f C Res(f; ) ) (trivial) f(z) = z4 + 3z (z ) 2 (z + 2) Res(f; ) = 2, Res(f; 2) =, Res(f; ) = 3 Res(f; ) + Res(f; 2) + Res(f; ) = ( 3) = a, b, c, d 4 (4.32) bcd (a b)(a c)(a d) + acd (b a)(b c)(b d) abd + (c a)(c b)(c d) + abc (d a)(d b)(d c) = ( ) a, b, c, d f(z) := abcd (z a)(z b)(z c)(z d) z 6

118 f a, b, c, d, a f Res(f; b) = Res(f; a) = lim z a (z a)f(z) = lim z a = bcd (a b)(a c)(a d). abcd (z b)(z c)(z d) z = abcd (a b)(a c)(a d) a acd (b a)(b c)(b d), Res(f; c) = abd (c a)(c b)(c d), Res(f; d) = abc (d a)(d b)(d c). abcd Res(f; ) = lim zf(z) = lim z z (z a)(z b)(z c)(z d) = abcd ( a)( b)( c)( d) =. f(z) z 5 lim z z4 f(z) = f Laurent f(z) = n Z a n z n ( n Z : n 4) a n =. Res(f; ) = a =. Res(f; a) + Res(f; b) + Res(f; c) + Res(f; d) + Res(f; ) + Res(f; ) = (4.32) = Res(f; a) + Res(f; b) + Res(f; c) + Res(f; d) = Res(f; ) Res(f; ) = = = (4.32). () Proposition 4.6. f lim z Res(f; ). lim zf(z) = Res(f; ) =. z zf(z) A A = Remark 4.6. c f Res(f; c) = f Res(f; ) = f(z) = lim z f(z) = z f Res(f; ) =. Proposition Res(f; ) = 7

119 ad bc a, b, c, d (i) (c ) a φ(z) := c az + b cz + d (z = d/c) (z = ) (z d/c, ) (ii) (c = ) φ(z) := (z = ) az + b cz + d (z ) φ: Ĉ Ĉ (linear fractional transformation) d/c (a) c z d c cz + d, az + b ad c az + b lim z d/c cz + d = = φ( d/c), z d/c + b = ad bc c lim az + b z cz + d = lim a + b/z z c + d/z = a c = φ( ). z (b) c = ad bc a, d, a d ( a lim z d z + b ) = = φ( ). d z c = d/c = 2 ( d/c ) ad bc ad bc = φ ad bc az + b (z) =, φ(z) φ() = (cz + d) 2 cz + d b d φ(z) = a c ad bc c cz + d (ad bc)2 = cz + d, Ĉ Ĉ ( ) 8

120 φ z Ĉ lim z z z z φ(z) = φ(z ) z C cz + d z = cz + d = ( ) a b A = GL(2; C) φ c d A (z) := az + b cz + d A φ A GL(2; C) () A, B GL(2; C) φ A φ B = φ AB. ( ) (2) I = φ I = id (Ĉ ). (3) A GL(2; C) φ A = (φ A ). ( ) ( ) a b p q () A =, B = GL(2; C) c d r s ( ) ( ) ( ) a b p q ap + br aq + bs AB = =. c d r s cp + dr cq + ds z C, rz + s, cφ B (z) + d ( ) pz + q a pz + q φ A φ B (z) = φ A (φ B (z)) = φ A = rz + s + b a(pz + q) + b(rz + s) rz + s c pz + q = rz + s + d c(pz + q) + d(rz + s) (ap + br)z + (aq + bs) = (cp + dr)z + (cq + ds) = φ AB(z). φ A φ B (z) = φ AB (z). Ĉ Ĉ 25 φ A φ B = φ AB. (2) c = { (z = ) φ I (z) = z (z ). φ I Ĉ (3) A GL(2; C) A GL(2; C) φ A φ A = φ AA = φ I = id, φ A φ A = φ A A = φ I = id. φ A = (φ A ). 25 (X, d), (X, d ) Ω X f : X X, g : X X x Ω f(x) = g(x) x X f(x) = g(x). 9

121 4.6. φ: Ĉ Ĉ T b (z) := z + b z + b = z + b z + ( ) b M a (z) = az (a C \ {}) az = a z + z + ( ) a 2 a = re iθ (r >, θ R) w = az = re iθz ζ = e iθ z, w = rζ ζ = e iθ z θ w = rζ (r > ) ( < r < ) (r = ) R(z) = z z = z + z + ( ) ( ) (a) c az + b cz + d a/c, b ad/c ad az + b cz + d = a b c c cz + d = a c c ad bc cz + d T d/c, R, M (ad bc)/c 2, T a/c = ad bc c 2 z + d c + a c. (b) c = az + b cz + d = az + b = a d d z + b d M a/d, T b/d 2

122 Ĉ Ĉ C () β 2 ac a, c R, β C azz + βz + βz + c = ( p.9.4) a = a Ĉ Ĉ T d M a (a ) w = R(z) = z z = w ww a w ( ) ( ) + β + β w w w + c =. a + βw + βw + cww =. c := a, β := β, a := c a ww + β w + β w + c =. Ĉ ( {z Ĉ; azz + βz + βz + c = } w = z {w Ĉ; a ww + β w + β w + c = } ( ) ) () α, β, γ Ĉ 3 φ(α) =, φ(β) =, φ(γ) = φ ( ) α, β, γ C φ(z) := α γ α β z β z γ, β = φ(z) = α γ z γ, 2

123 γ = α = φ(z) = z β α β, φ(z) = z β z γ () φ, φ 2 α, β, γ,, φ := φ 2 φ φ() =, φ() =, φ( ) = φ( ) = a, b C s.t. φ(z) = az + b (z C). φ() = b =. φ() = a =. φ = id. φ = φ φ(z) α γ α β z β z γ z, α, β, γ (cross ratio) (z, α, β, γ) : (z, α, β, γ) = α γ α β z β z γ α, β, γ Ĉ 3 α, β, γ Ĉ 3 φ(α) = α, φ(β) = β, φ(γ) = γ φ φ αβγ φ := φ α β γ φ αβγ φ 4.6., 2, 3 2, 3, ( ) 3 α, β, γ C,, φ α,β,γ ( ) 2 4, 3 φ 2,3, φ,2,3 ) ( ) 2 4 ( 3 φ α,β,γ (z) = α γ α β z β z γ φ,2,3 (z) = 3 2 z 2 2(z 2) = = 2z 4 z 3 z 3 z 3, φ 2,3, (z) = z 3 z = z 3 (z ) = z + 3 z. 3 = ( ) 3 ( )( ) 3 φ φ = ( ) ( ) = 3 2 ( ) φ(z) = 5 2 z z = 5z 3z 7. 22

124 . c f Res(f; c) () f(z) = sin z z, c = (2) f(z) = z sin z, c = sin z z (3) f(z) =, c = π (4) f(z) = z sin z, c = π. () Laurent (2) c f,, Res(f; c) (a) f(z) = cos z, c = (b) f(z) = z3 z, c = (c) f(z) = z sin z, c = π. (: Laurent ) 2. () (2) R R 3. f(z) := sinh ( z 3) z ( cos z) z 2 =R dz z dz z =R z 2 4 () sinh z Taylor (Maclaurin ) (2) Q(z) := sinh ( z 3) z Laurent (3) P (z) := cos z (4) f (sinh z Taylor sinh z sin z ) 4. f(z) := 5z2 4z + 3 (z 2 + )(z 2) () z < (2) < z < 2 (3) 2 < z < 23

125 5 N cot πz = lim N n= N z n = z + 2z z 2 n 2. ( z = n (n Z) z n ) Euler sin πz = lim N πz ( z ) ( + z ) ( z ) ( + z ) ( z ) ( + z ) 2 2 N N = πz ) ( z2 n ( ) Weierstrass M-test ( ) f n (z) = f n(z) n n f n (z) dz = f n (z) dz C n n C N () ( ) ( ) ( ) ( ) () 5..2, 5..5, 5..7, 24

126 5.. 3,4 ( ) 5.. ( ) lim a n = A def. ε >, N N, n N n N = a n a < ε. n 5..2 ( ) {a n } n N (Cauchy ) def. ε >, N N, n N, m N n, m N = a n a m < ε (R, C ) = ( ) 5..4 ( ) a n := lim n k= n a k (, ) lim a n =. n lim n a n = a n a n ( ) lim a n ( n ) a n 5..6 ( ) M n n (i.e. M R, n N, M k M) 5..7 ( ) k= a n def. a n 5..8 ( ) R C n a n s n := a k n > m s n s m = n k=m+ k= a k n k=m+ a k = S n S m. 25

127 S n := n k= a k {S n } n N ( 5..3 ){s n } n N {s n } n N ( 5..3) a n 5..9 ( ) a n φ: N N a φ(n) = () 5.. ( ) (i) n N (ii) a n b n b n a n a n, a n. b n (i) b n a n n N b n ( ) n N n a k k= n b k k= b k. ( ) M n M R, n N a k M a n k= k= 5.. () () α > { (α > ) n α = (α ). (2) r C r n = n= r ( r < ) ( r ). 26

128 n n () k α + dx k= x α, n k= n+ k α dx x α dx x α (2) n r n+ r k (r ) = r n + (r = ) k= ( ) f : Ω C, n N f n : Ω C () {f n } n N f Ω (pointwise convergence, pointwise convergent, converges pointwise) x Ω lim f n(x) = f(x). n i.e. x Ω, ε >, N N, n N n N = f n (x) f(x) < ε. ( ε >, x Ω, N N, n N n N = f n (x) f(x) < ε.) (2) {f n } n N Ω f (uniform convergence, uniformly convergent, converges uniformly) lim f n (x) f(x) =. sup n x Ω i.e. ε >, N N, n N, x Ω n N = f n (x) f(x) < ε. ( ε >, N N, x Ω, n N n N = f n (x) f(x) < ε.) 5..3 () {f n } f f n f (x n ) f n (x) := (x n ) nx ( n x n ), (x > ) f(x) := (x < ) nx (x = ) {f n } f f n f x = 27

129 5..4 ( ) {f n } f n 2 x ( x ( n ) f n (x) := n 2 x 2 ) ( n n x 2 n ) (x < or x > 2 n ), f(x) := (x R) {f n } f 2 f n (x); dx = 2 2 n n =, 2 lim n 2 f(x) dx = f n (x) dx = = 2 lim f n(x) dx. n 5..5 ( ) () (2) lim () ε >, N N, n N n N = sup f(x) f n (x) ε. sup f(x) f N (x) ε x Ω 3 x Ω 3. a Ω f N a δ >, y Ω y a < δ = f N (y) f N (a) ε 3. y a < δ y f(y) f(a) = f(y) f N (y) + f N (y) f N (a) + f N (a) f(a) f(y) f N (y) + f N (y) f N (a) + f N (a) f(a) ε 3 + ε 3 + ε 3 = ε. (2) b a f n (x) dx b a b f(x) dx = b (f n (x) f(x))dx f n (x) f(x) dx a b a sup x [a,b] a f n (x) f(x) dx = sup f n (x) f(x) x [a,b] = (b a) sup f n (x) f(x) (n ). x [a,b] b a dx 28

130 C C {f n } f f n (z) dz f(z) dz sup f n (z) f(z) dz. C C z C 5..6 ( ) I R C {f n } f {f n} I g f C f = g. a I n f n (x) = f n (a) + x a f n(t) dt (x I) C f(x) = f(a) + x a g(t) dt (x I). f (x) = g(x). C Ω {f n (z)} n N f(z) {g n (z)} n N g(z) f f = g. ( 5..26) 5.. ( ) 5..6 {f n} I g a I ( V : a ) s.t. {f n} V g f n (x) = f n (a) + f(x) = f(a) + x a x a f n(t) dt (x V ) g(t) dt (x V ) f (x) = g(x) (x V ) I I uniform convergence on every compact set uniform convergence on compacta Ω {f n } n N Ω f f Ω ( ) F = f F ) z a f n(ζ)dζ = f n (z) f n (a). 29 C f(z) dz = F (b) F (a) (a, b C

131 ( ) 5..7 (Weierstrass M test) M n {f n } n N Ω n N, z Ω f n (z) M n f n (z) Ω ( ) 5..8 z Ω f n (z) S(z) M n U z Ω n S(z) f k (z) = k= ε >, N N, n N k=m+ f k (z) k=m+ n N = U f k (z) n M k < ε. k= k=m+ M k = U n M k < ε. k= () f n (z) Ω a n (z c) n z C < ρ < z c n= ρ a n (z c) n D(c; ρ) = {z C; z c ρ} n= D(c; z c ) = {z C; z c < z c } z = c ( ) z c a n (z c) n lim a n(z c) n =. M R, n N a n (z c) n M. n z c ρ z ( ) a n (z c) n = z c n ( ) z c n ( ) ρ n a n(z c) n = a n (z c) n M. z c z c z c ρ ( ) ρ n z c < M ( 5..) z n= c 5..7 a n (z c) n D(c; ρ) n= 3 n=

132 5..2 ( ) (i) z C \ {c} (ii) z C a n (z c) n 3 n= (iii) r (, ) s.t. z c < r z z c > r z (i) r :=, (iii) r := z c < r z c > r r = z c < z z c < r z c > z c r = z C z c < r z c < r z C z c > r z z c > r (r [, ) r = ) D(c; r) = {z C; z c < r} ( ) D(c; ) =, D(c; ) = C 5..2 ( ) () (2) a n (z c) n = n Z a n (z c) n + n= a n (z c) n ζ := z c a n (z c) n z z c < ρ ρ a n z c ρ (z c) n 3

133 5..23 () a n (z c) n n N z, z 2 ( z c < z 2 c ) z c < ρ < ρ 2 < z 2 c ρ, ρ 2 A(c; ρ, ρ 2 ) = {z C; ρ z c ρ 2 } A(c; z c, z 2 c ) = {z C; z c < z c < z 2 c } ( ) a n (z c) n R R 2 n= s.t. A(c; R, R 2 ) = {z C; R < z c < R 2 } A(c; R, R 2 ) ( ) () (2) () Ω C {f n } n N Ω f : Ω C {f n } n N Ω f f f (z) = lim f n n(z) (z Ω). f Ω ( ) a Ω D(a; ε) Ω ε > z D(a; ε) n N Cauchy ( ) f n (z) = 2πi ζ a =ε f n (ζ) ζ z dζ, d := ε z a d > ζ a = ε ζ ζ z ζ a a z = ε a z = d. Ω {ζ C; ζ a = ε} {f n } f sup f n (ζ) ζ a =ε ζ z f(ζ) ζ z d sup f n (ζ) f(ζ) (n ). ζ a =ε 32

134 ( ) n f(z) = 2πi ζ a =ε f(ζ) ζ z dζ. z D(a; ε) f D(a; ε) ( ) k N f (k) (z) = k! f(ζ) dζ (z D(a; ε)). 2πi (ζ z) k+ ( ) ζ a =ε f n (k) (z) = k! f n (ζ) dζ. 2πi ζ a =ε (ζ z) k+ ( ) f (k) (z) f n (k) (z) k! 2π sup dζ f(ζ) f n (ζ) (n ). k+ ζ a =ε ζ a =ε ζ z ( ) ( d k ( lim n dz lim f (k) n n (z) = f (k) (z). ) k ( d = ) k lim n ) dz ( ) K := D(a; ε/2) z K ζ a = ε ζ ζ z = ζ a + a z ζ a a z ε ε 2 = ε 2 ζ z k+ ( ) 2 k+, ε ζ a =ε dζ ζ z k+ ( ) 2 k+ 2πε. ε ( ) sup f (k) (z) f n (k) 2 k+ (z) k! sup f(ζ) f n (ζ) z K ε ζ a =ε (n ). {f n (k) } K ( k = ) Ω {f n } Ω f f Ω k N f (k) (z) = lim f (k) n n (z), C C f n (z) dz = f(z) dz. lim n C C (C C Ω {f n } C f ) ( ) 33

135 5..27 (Riemann ) ( n z = exp (z log n) log n log n R ) ζ(z) := n z D := {z C; Re z > } D ζ Riemann α > Re z α n z = exp (z log n) = exp Re(z log n) = exp [(Re z) log n] = n Re z n α. M n := n α M n n z (n N, Re z α), M n = n α <. C.3.3 Re z > α ζ α > nz ζ Re z > Riemann s 2 : ζ(s) := n s. C Riemann Riemann (859 ) ζ(s) s = 2n (n N) Re s = 2 3 (859 ) < Re s < f(z) := z + 2z z 2 n 2 (z C \ Z) C \ Z f n 2z n 2 z < n z 2 /n 2 z 2 /n 2 > 2z z 2 n 2 = 2z n 2 ( z 2 /n 2 ) = 2 z n 2 z 2 /n 2 2 z n 2 ( z 2 /n 2 ). 2 Riemann (Riemann [4] ) 3 x π(x) π(x) x (x + ). Gauss log x 34

136 R > N N N 2R ( z C : z R) ( n N : n N) z 2 /n 2 (/2) 2 = z z 2 n 2 2R n = 8R 3 n 2 Weierstrass M-test z < R n=n 2z z 2 n 2 z R n N 2. ( z R, n N). {z C \ Z; z < R} f R > C \ Z f 5.2 φ φ(z) := π cot πz = π cos πz sin πz φ C z = n (n Z) P (z) := sin πz, Q(z) := π cos πz P (z) Q(z) C n Z P (n) =, P (n), Q(n), z C \ Z P (z). ( ) Q(z) Res(φ; n) = Res P (z) ; n = Q(n) π cos nπ P = (n) π cos nπ =. φ n Z z n φ(z) n= z n (z n z n ( n ) = ) n lim N N n= N z n = lim N = z + [ z + N 2z z 2 n 2 ( z n + ) ] ( N = lim z + n N z + ) 2z z 2 n 2 35

137 ( C \ Z ) φ(z) 5.2. z C \ Z (5.) π cot πz = z + ( z n + ) = z + n z + 2z z 2 n 2. cot z ( n ) f = Q P C c,..., c r Laurent ( f (z),..., f r (z), f (z) ) f a C s.t. r f(z) = f r (z) + f (z) + a (z C). j= f cot z cot ( cot x = tan ( π 2 x) ) : cot x (5.) z C \ Z f f(ζ) := π cot πζ ζ z N N z < N R := N ±R ± ir C 36

138 C f(ζ) dζ f C ζ = z, ζ = k (k Z) 4 Res(f; z) = lim(ζ z)f(ζ) = lim π cot πζ = π cot πz, ζ z ζ z ( ) Res(f; k) = Res π cot πζ; k = ζ z ζ z Res(π cot πz; k) = ζ=k k z = k z. 2πi C f(ζ) dζ = c C = π cot πz + N k= N = π cot πz z N Res(f; c) = k z k= c=z,,±,...,±n ( z k + ). z + k Res(f; c) N (5.) f(ζ) dζ = lim N C ζ = x + ir (x [ R, R]) cot πζ = cos πζ sin πζ = eiπζ + e iπζ 2 C 2i ieiπζ e iπζ e iπζ = + e iπζ + e 2iπζ e iπζ = i e iπζ e 2iπζ = i + e 2iπ(x+iR) + e2π(r ix) ie2π(r ix) = i e 2iπ(x+iR) e 2π(R ix) = + e 2π(R ix) cot πζ = e 2π(R ix) + e 2π(R ix) e 2π(R ix) + e 2π(R ix) = e2πr + e 2πR. R = N πR 2π 2 2 = 3π 9 e2πr 2 9 = 52. t t + t t > cot πζ = C cot πζ 2. C cot πζ () x, y R sinh(2y) + i sin(2x) cot(x + iy) = i cosh(2y) cos(2x) cosh(2y) ( y = ), cos(2x) ( x = nπ (n Z)), x = ±πr = ±π(n + /2) cos 2x = cos [±π(2n + )] = 2. y sinh(2y), cosh(2y) e 2 y /2 4 c f g Res(fg; c) = f(c) Res(g; c) 37

139 5 g(ζ) := 2πi C π cot πζ ζ (g( ζ) = g(ζ)) C g(ζ) dζ =. f(ζ) dζ = (f(ζ) g(ζ)) dζ = 2πi C 2πi = z cot πζ 2i ζ(ζ z) dζ C C π cot πζ ( ζ z ) ζ ζ R > z f(ζ) dζ 2πi z z cot πζ dζ 2 C 2 C ζ ζ z 2 C ζ ( ζ z ) dζ z C R(R z ) dζ z 8R R(R z ) = 8 z R z (N ). dζ 5.2. π cot πz = z + ( z n + ) z + n 6 : ( π π ) sin 2 = πz z 2 + ( (z n) 2 + ( ) ) (z + n) 2. π 2 sin 2 πz = z 2 + ( ) (z n) 2 + (z + n) 2. 5 C C, C 2 C = C + C 2. C g(ζ) dζ ζ = w g(ζ) dζ = g( w) ( dw) = g(w)dw. C C C 2 C 2 g(ζ) dζ =. C 6 38

140 5.2.2 ( ) ζ Riemann ζ(s) = πz cot πz Taylor b n z n n= n s ζ(2m) = b 2m 2 (m N). z Bernoulli ( e z + z 2 = + ( ) n B 2n (2n)! z2n {B 2n } n ) (5.) ζ(2m) = 22m B 2m π 2m (2m)! 2z 2 πz cot πz = + z 2 n 2 = 2 = 2 ζ(2m)z 2m. m= m= (m N). ( ) z 2 m = 2 n 2 ( ) n 2m z 2m m= b 2m = 2ζ(2m) (m N) ζ(2m) = b 2m 2 (m N). Bernoulli πz cot πz Taylor (5.2) πz cot πz = 2 2n B 2n π 2n z 2n. (2n)! ζ(2m) = 22m B 2m π 2m (2m)! (m N) (?) 39

141 6 6. (conformal mapping) f 6.. (( )) D f(z) D D D f 6..2 (univalent function) 6..3 ( ) D w = f(z) w = f(z ), f (z ) z U w V f U V g V g (w) = f (z) 6..4 () 6..5 D (z z f(z) f(z )) f D f(d) f f(d) 6.2 Cauchy Goursat 4

142 6.2. () 2 III F () Cauchy Goursat ( ) Goursat F F F Cauchy Green Cartan H [] 6.3 Morera Cauchy Cauchy ( ) Morera Theorem 6.3. (Morera ) D C f : D C D γ f(z) dz = γ f D D a z a z D C C z F (z) := f(ζ) dζ C z F (z) C z F (z) = f(z) (z D) ε D(z; ε) D < h < ε h C C z+h C z + [z, z + h] F (z + h) F (z) h f(z) = h F (z + h) F (z) f(z) h h [z,z+h] [z,z+h] F f = F f(ζ) dζ f(z) h f(ζ) f(z) dζ 4 [z,z+h] dζ = h [z,z+h] (f(ζ) f(z)) dζ sup f(ζ) f(z) (h ). ζ [z,z+h]

143 6.4 Schwarz Theorem 6.4. (Schwarz (Schwarz Lemma)) D := {z C; z < }, f : D C f() =, f(z) (z D) f(z) z (z D), f () (i) z D \ {} f(z) = z (ii) f () = (iii) f D D a C s.t. a =, f(z) = az (z D). 6.5 Poisson 6.6 Schwarz (Schwarz reflection principle) Functions on One Complex Variables I, II, John B. Conway, Springer A C A := {z C; z A} A A A = A A 6.6. Ω C f : Ω C Ω := {z C; z Ω}, f : Ω C f(z) := f(z) (z Ω ) f c Ω c Ω. f Ω ε >, {a n } n s.t f(z) = a n (z c) n n= (z D (c; ε)). 42

144 z D(c; ε) z D (c; ε) f (z) = a n (z c) n. f (z) = f (z) = n= a n (z c) n = n= a n (z c) n. f D(c; ε) f Ω u Ω u (z) := u(z) Ω Ω C Ω, Ω = Ω (Ω ) Ω Ω R Ω R = Ω + := Ω H +, Ω := Ω H, H + := {z C; Im z > }, H := {z C; Im z < } n= Ω = Ω + Ω, Ω + Ω =. Ω Ω R. x Ω R Ω ε > s.t. D(x; ε) Ω. I := (x ε, x + ε) = {t R; x ε < t < x + ε} I Ω Ω C Ω, Ω = Ω (Ω ), f : Ω C f f(z) = f(z) (z Ω). I R f(x) R (x I). g Ω g(z) := f(z) f(z) (z Ω) g(x) = g in Ω. f(z) = f (z). (x I) ( ) Ω C Ω, Ω = Ω (Ω ) Ω + := Ω H +, Ω := Ω H, H + := {z C; Im z > }, H := {z C; Im z < }, σ := Ω R v : Ω σ = {z Ω; Im z } R v = in Ω +, v = on σ V (z) := { v(z) (z Ω + σ) v(z) (z Ω ) V = in Ω. 43

145 V = in Ω x σ V (x ) = D(x ; ε) Ω ε > P V (z) := 2π z x =ε P V z x < ε V P V V P V ( σ) V = v =, P V = V P V =. V P V V = P V. V (x ) = P V (x ) = (, ) f u, v f(x + iy) = u(x, y) + iv(x, y), x, y, u(x, y), v(x, y) R. f (x + iy) = f(x iy) = u(x, y) iv(x, y). u v u(x, y) v(x, y) u, v Ω + σ { u(x, y) ((x, y) Ω U(x, y) := + σ) u(x, y) ((x, y) Ω ), V (x, y) := { v(x, y) ((x, y) Ω + σ) v(x, y) ((x, y) Ω ) U Ω V V = on σ V Ω Morera C: z a = r C z C (, ) w a, z, w z a w a = r 2 z w z z lim z a z =, (z ) = z lim z z = a ( ) a, a : a =, = a. 44

146 Ĉ := C { } R(z) := z R: Ĉ Ĉ R = R. z a (z a) (z a) = r 2 A Ĉ C z C z = z R(A) = A a C, r > R: Ĉ Ĉ C: z a = r S(z) := a + r2 r2, T (z) := a + z a z a S C \ {a} T C \ {a} R(z) = S(z) = T (z) (z C \ {a}). Schwarz Ω σ Ω Ω C C := {z C; z a = r }, D := {z C; z a < r }, C 2 := {z C; z a 2 = r 2 }, D 2 := {z C; z a 2 < r 2 }, { } Ω D, σ := Ω C = C or a + r e iθ ; θ (α, β), β α 2π f : Ω C, f(ω) D 2, f(σ) C 2 { f(z) (z Ω) f(z) := R 2 (f(r (z))) (z R (Ω)) Ω := Ω σ R (Ω) C f Ω S : C\{a } C, T 2 : C \ {a 2 } C R (z) = S (z), R 2 (z) = T 2 (z) R 2 (f(r (z))) = T 2 (f(s (z))) f R (Ω) 45

147 6.7 Rouché Theorem 6.7. ( (argument principle)) C C f C f (z) dz = 2πi(N P ). f(z) f C N P C Mittag-Leffler Theorem 6.9. (Mittag-Leffler ) D C E D 6. Runge 46

148 7 7. Jordan D := {z C; z < } 7.. C Jordan Jordan C C Ω Ω D (Riemann ) Ω Jordan Ω C Ω D f f Ω D (Osgood-Carathéodory ) z Ω f(z ) =, f (z ) > f 7..2 D D f(z) = ε z a az, a <, ε =. f(z ) =, f (z ) > a = z, ε = f(z) = z z z z. 47

149 7..3 H := {z C; Im z > } Cayley f(z) = z i z + i z H z i < z + i f(z) <. z R f(z) =. Cayley unitary f(z) = az + b, ad bc >. cz + d 7.2 n(γ, c) = dz 2πi γ z c. 7.3 [5]

150 7.3. D C (i) D (ii) D γ D c D n(γ; c) = (γ c ). (iii) D ( ) γ D (iv) D γ D f γ f(z) dz =. (v) D f f F (D F F = f ) (vi) f D D D g f(z) = exp g(z) (vii) f D D D g f(z) = g(z) 2 (viii) D ( ) 7.4 Poisson (. [6] ) 7.5 Ascoli-Arzelra 7.5. (Ascoli-Arzela ) [, ] C([, ]; R) F (i), (ii) (i) ( ) M R s.t. f F, x [, ] f(x) M. (ii) ( ) ε >, δ > s.t. f F, x, x 2 [, ] x x 2 < δ = f(x ) f(x 2 ) < ε. F {f n } n N [, ] {f nn } k N [, ] Q {r n ; n N} { {m,n n N m,n = n (n N) fm,n (r ) } n N {m,n} n N {m,n } n N { f m,n (r ) } n N { fm,n (r 2 ) } n N {m,n} n N {m 2,n } n N { f m2,n (r 2 ) } n N k = 3, 4, {m k,n } n N {m k,n } n N { f mk,n (r k ) } n N { } fmn,n n N {f n} n N k N { f mn,n (r k ) } n N 49

151 ε > (ii) δ > N N min x r j < δ j N j {,, N} { f mn,n (r j ) } n N N N s.t. p, q N = f mp,p (r j ) f mq,q (r j ) < ε (j =, 2,, N). x [, ] x r j < δ j f mp,p (x) f mq,q (x) f mp,p (x) f mp,p (r j ) + f mp,p (r j ) f mq,q (r j ) + f mq,q (r j ) f mq,q (x) ε + ε + ε = 3ε. n k := m k,k R n ( [7] C ) Laplace Dirichlet 5

152 A A. X a X x 2 = a x X a A.., a a X = R A.. () : x R x 2 = x =. (2) a a a > x > x R x 2 = a x = ±x. (3) a a (4) a () x = x 2 = x > x 2 >, x < x 2 > x 2 = x =. (2) f(x) := x 2 a f : [, ) R f() = a <, f(a + ) = a 2 + a + > x (, a + ) s.t. f(x ) =. x >, x 2 = a. (x x )(x + x ) = x 2 x 2 = x 2 a R x 2 = a (x x )(x + x ) = x = ±x. (3) a <, x R x 2 > a x 2 a. (4) (), (2) A..2 a a a 5

153 A..2 X = C A..3 () : x C x 2 = x =. (2) a a a a a > x R x 2 = a x = ± a. (3) a a ai ai a < x C x 2 = a x = ± ai. (4) a a a C \ {} x C x C x 2 = a x = ±x. () x = x 2 =. z z z = z = x 2 = x 2 = x 2 = x =. x =. (2) x C ( x a ) ( x + a ) = x 2 ( a ) 2 = x 2 a. C x C x 2 = a x = ± a. (3) x C ( ) ( ) x ai x + ai = x 2 ( ai ) 2 = x 2 ( a)i 2 = x 2 a. C x C x 2 = a x = ± ai. (4) a = α + βi (a, β R) β = a = α R. a ± a a < ± ai β x = p + qi (p, q R) x 2 = (p + qi) 2 = p 2 q 2 + 2pqi x 2 = a p 2 q 2 = α 2pq = β. 52

154 β (p, q) = ± α + α 2 + β 2, β α + α 2 + β 2. 2 β 2 x 2 = a x = ± α + α 2 + β i β β α + α 2 + β 2. 2 a ( ) a a > a a a = α + βi (α, β R, β ) a = ± α + α 2 + β 2 + i β α + α 2 + β 2 ( ). 2 β 2 (a) a a = a 2 (b) a > a = ± a. A..3 a a a < (A.) a := ai a ( a a a < ) = i. = ±i. p 2 q 2 = α, 2pq = β p q = β 2p. p2 β2 4p 2 = α. 4p 4 4αp 2 β 2 =. p 2 2 p 2 = 2α± 4α 2 +4β 2 4 = α± α 2 +β 2 2 p p 2 p 2 = α+ α 2 +β 2 α+ α 2. p = ± 2 +β 2 2, q = β 2p = ±β 2(α+ α 2 +β 2 ) = ±β α+ α 2 +β 2 2(α 2 +β 2 α 2 ) = ± β β α+ α 2 +β

155 (A.) A..4 a, b, c a x ax 2 + bx + c = x = b ± b 2 4ac. 2a A..5 a, b, c a b 2 4ac x b 2 4a < x ax 2 + bx + c = x = b ± b 2 4ac. 2a ax 2 + bx + c = x = b ± 4ac b 2 i. 2a (A.) ( ) A..5 b 2 4ac ax 2 + bx + c = x 2 + b a x + c a = ( x + b ) 2 b2 2a 4a 2 + c a = ( x + b ) 2 = b2 4ac 2a 4a 2. x + b b 2a = ± 2 4ac b 4a 2 = ± 2 4ac 2 a x = b ± b 2 4ac. 2a b 2 4ac < x + b 4ac b 2a = ± 2 4a 2 i = ± 4ac b 2 2 a i x = b ± 4ac b 2 i. 2a A.2 arg z, log z A.2. arg z z C z (z = x+iy, x, y R z = x 2 + y 2 ) arg z 54

156 ( ) f(x) dx F (x) = f(x) F (x) arg z O z θ z = z e iθ θ R z = re iθ, r, θ R θ z = arg z ( θ R z = z e iθ ) arg z z z θ θ [, 2π) arg z Arg z z C \ {} arg z Arg z z = z e iθ, θ [, 2π) θ Arg z arg z z = z e iθ = z e iθ 2, θ, θ 2 R e i(θ θ 2 ) = n Z s.t. θ θ 2 = 2nπ. θ 2 = Arg z θ = Arg z + 2nπ. θ R z = z e iθ n Z s.t. θ = Arg z + 2nπ {θ R; z = z e iθ } = {Arg z + 2nπ; n Z}. z arg z z = z e iθ θ R ( ) Arg z + 2nπ (n Z) A.2.2 log z log z z C exp w = z z = exp w = z w C exp w exp( w) = exp(w w) = exp = exp w z r := z, θ := Arg z r >, θ [, 2π), z = re iθ. u := Re w, v := Im w w = u + iv exp w = exp u exp(iv). exp u exp(iv) = r exp(iθ). exp u = r. z = exp(u) exp(iv) = r exp(iv). v z u = log r = log z (log log. v θ (mod 2)π. n Z s.t. v = θ + 2nπ. 55

157 (A.2) w = log r + i(θ + 2nπ) = log z + i (θ + 2nπ). log z = log z + i (Arg z + 2nπ) A.2.3 z α α Z C C \ {} α C \ Z z α := exp (α log z) log z C \ {} A.3 f(x) dx f(n) (A.3) n= σ (z) := π cot πz = π cos πz sin πz (sin πz ) n =, ±, ±2, ( ) π cos πz Res(σ ; n) = (sin πz) = (n Z). z=n f 2 f C N z = N + /2 ( R := N + /2 ±R ± ir 4 ) C N f(z)σ (z) dz = 2πi N k= N Res (f(z)σ (z); k) + 2πi N N k= N c f z < N + /2 Res (f(z)σ (z); k) = f(k) Res (σ ; k) = f(k) = f(k) f(k) = c f c <N+/2 Res (f(z)σ (z); c). Res (f(z)σ (z); c) + f(z)π cot πz dz. 2πi C N N f(n) = Res(f(z)σ (z); c) n= c f 56

158 A.3. n 2 = 2 n Z,n ( ) π cot πz n 2 = 2 Res z 2 ;. cot z = z z 3 45 z3 + π cot πz z 2 = z 3 π2 3z π4 z 45 + ( ) π cot πz Res z 2 ; = π2 3. n 2 = π2 6. A.3.2 a > S = n 2 + a 2 ( ) f(z) := z 2 + a 2 f ±ia n= n= Res(f; ia) = i 2a, f(n) = 2S + a 2. i Res(f; iia) = 2a. [ f(n) = π i 2a cot(iπa) + i ] 2a cot( iπa) = π coth πa. a S = 2 ( π a coth πa ) a 2. (A.4) σ 2 (z) := π sin πz n =, ±, ±2, Res(σ 2 ; n) = ( ) n (n Z) f 2 ( 2 ) f n= ( ) n f(n) = c f Res(f(z)σ 2 (z); c). 57

159 A.3.3 ( ) n n 2 = 2 n Z,n ( ) n n 2 = ( 2 Res π ) z 2 sin πz ; π z 2 sin πz = z 3 + π2 6π + 7π4 z = π2 2. A.3. 2 ( + z) n = n k= ( ) n z k k ( ) n = ( + z) n k 2πi C z k+ dz. A.4 cot z Maple series(cos(z)/sin(z),z=,2) A.4. {b n } f(z) = a n z n, a n= f(z) = b n z n n= ( = f(z) f(z) = n ) a k b n k n= k= z n a b =, a b + a b =, a b 2 + a b + a 2 b =,. n a b n + a k b n k = (n =,, 2,... ). k= 58

160 b = a, b n = a ( n k= a k b n k ) (n =, 2, ). A.4.2 f(z) = a n z n, g(z) = b n z n n= f(z)g(z) = n= ( n ) a k b n k n= k= z n A.4.3 cot z = cos z/ sin z 2 BASIC cot ( ) BASIC OPTION ARITHMETIC RATIONAL DECLARE EXTERNAL SUB INverse DECLARE EXTERNAL SUB conv LET maxn=2 OPTION BASE DIM c( TO maxn),s( TO maxn),is( TO maxn),cotangent( TO maxn) FOR n= TO maxn LET c(n)= LET s(n)= NEXT n FOR j= TO maxn/2 LET c(2*j)=(-)^j/fact(2*j) LET s(2*j)=(-)^j/fact(2*j+) NEXT j PRINT "z/sin(z)" CALL INverse(s,IS,maxn) CALL PRINTc(IS,maxn) PRINT "z cos(z)/sin(z)" CALL CONv(IS,c,cotangent,maxn) CALL printc(cotangent,maxn) END REM EXTERNAL SUB printc(a(),maxn) OPTION ARITHMETIC RATIONAL FOR n= TO maxn PRINT a(n) NEXT n END SUB REM

161 EXTERNAL SUB INverse(a(),b(),maxn) OPTION ARITHMETIC RATIONAL LET b()=/a() FOR TO maxn LET s= FOR k= TO n LET s=s+a(k)*b(n-k) NEXT k LET b(n)=-s/a() NEXT n END sub REM EXTERNAL SUB conv(a(),b(),c(),maxn) OPTION ARITHMETIC RATIONAL FOR n= TO maxn LET s= FOR j= TO n LET s=s+a(j)*b(n-j) NEXT j LET c(n)=s next n END sub (z/sin z z cot z ) z/sin(z) /6 7/36 3/52 27/648 73/ / / / / / z cos(z)/sin(z) -/3 -/45-2/945 6

162 -/4725-2/ / / / / / z sin z = z + 6 z z z z z + z cos z sin z = 3 z2 45 z z z z cos πz π sin πz = z 3 π2 z 45 π4 z π6 z π8 z π z 9. n 2 = π2 6, n 4 = π4 9, ( ) π cot πz n 2k = 2 Res z 2k ; n 6 = π6 945, n 8 = π8 945, n = π 93555, A.5 tan z tan z = (a + a w + a 2 w 2 + a 3 w 3 + ) z 6 z3 + 2 z5 54 z7 + 2 z z4 72 z6 + ( 2 w + 24 w2 72 w3 + a =, a 2 + a = 6, a 24 a 2 + a 2 = 2, = z 6 z2 + 2 z4 2 z z4 ) a 72 + a 24 a a 3 = 54, 54 z z6 + = 6 w + 2 w2 54 w3 + 6

163 a =, a = 6 + a 2 = 3, a 2 = 2 + a 2 a 24 = 2 5, a 2 = 54 + a 2 2 a 24 + a 72 = 7 35, tan z = z + 3 z z z7 + tan z sin z = 6 z3 + 8 z z7 + 62

164 B B. B p.85 63

165 C 28 C. 8, C.. arg z, log z arg z z C z (z = x+iy, x, y R z = x 2 + y 2 ) arg z ( ) f(x) dx F (x) = f(x) F (x) arg z O z θ z = z e iθ θ R z = re iθ, r, θ R θ z = arg z ( θ R z = z e iθ ) arg z z z θ θ [, 2π) arg z Arg z z C \ {} arg z Arg z z = z e iθ, θ [, 2π) θ Arg z arg z z = z e iθ = z e iθ 2, θ, θ 2 R e i(θ θ 2 ) = n Z s.t. θ θ 2 = 2nπ. θ 2 = Arg z θ = Arg z + 2nπ. θ R z = z e iθ n Z s.t. θ = Arg z + 2nπ {θ R; z = z e iθ } = {Arg z + 2nπ; n Z}. z arg z z = z e iθ θ R ( ) Arg z + 2nπ (n Z) 64

166 log z log z z C exp w = z z = exp w = z w C exp w exp( w) = exp(w w) = exp = exp w z r := z, θ := Arg z r >, θ [, 2π), z = re iθ. u := Re w, v := Im w w = u + iv exp w = exp u exp(iv). exp u exp(iv) = r exp(iθ). exp u = r. z = exp(u) exp(iv) = r exp(iv). v z u = log r = log z (log log. v θ (mod 2)π. n Z s.t. v = θ + 2nπ. w = log r + i(θ + 2nπ) = log z + i (θ + 2nπ). (C.) log z = log z + i (Arg z + 2nπ) z α α Z C C \ {} α C \ Z z α := exp (α log z) log z C \ {} < α < x α + x dx = π sin πα C.. R(z) = Q(z)/P (z) deg P (z) deg Q(z)+2, x (, ) P (x), z = R(z) < α < α x α R(x) dx = 2πi e 2πα Res(z α R(z); ζ). ζ ε > R > R(z) ε < z < R Γ := L ε,r + C R + L R,ε + C ε C R z = R, C ε z = ε z α R(z) dz = 2πi Res(z α R(z); ζ). Γ ζ 65

167 z L ε,r arg z = z α = x α ( ) R = L ε,r ε x α R(x) dx. z L ε,r arg z = 2π log z = log z + i arg z = log z + 2πi. z α = z α e 2πiα. C.2 9, C.2. C.2. {f n } Ω a Ω, r > s.t. D(a; r) = {z C; z a r} {f n } C.2.2 a n (z c) n n= Ω := D(c; R) (f n (z) := n k= a k(z c) k ) a Ω r := a c r < R R a c >. r := R a c 2 D(a; r) (c; ) C.2.3 Ω C.2.4 I R {f n } n N f n C (I) (i) c I s.t. {f n (c)} n N (ii) {f n} n N I 2 x I {f n (x)} n N f(x) f C (I), f (x) = lim n f n(x) f n C (I) A := lim n f n(c), ( ) lim f n(x) = lim f n n n(x). g(x) := lim n f n(x) (x I) f n (x) = f n (c) + x c f n(t) dt (x I). {f n} I g x lim n c f n(t) dt = 66 x c g(t) dt

168 ( lim f n (c) + n x c ) f n(t) dt = A + {f n (x)} f(x) f(x) = A + x c g(t) dt. x c g(t) dt. f (x) = g(x) = lim n f n(x). C.2.5 ( ) C.3, 28 2 C.3. X a C, k N P (z) Q(z) z = a a P k (i.e., P (a) = P (a) = = P (k ) (a) =, P (k) (a) ), Q(a) f(z) := Q(z) P (z) a k C.3.2 C.3. I R {f n } n N I C (i) a I s.t. {f n (a)} n N (ii) I {f n} n N x I {f n (x)} n N f(x) f I C f (x) = lim f n n(x) (x I). ( ) lim f n(x) = lim f n n n(x). I def. x I, ε > s.t. D(x; ε) I = (x ε, x + ε) I I K x I ε K := D(x; ε) I I (x I D(x; ε) I ε > ε K = D(x; ε) = [x ε, x+ε] K x I 67

169 I = (a, b], x = b ε = (b a)/2 K = D(x; ε) I = [(a + b)/2, b] K ) K I x K ε x > s.t. D(x; ε x ) I x D(x; ε x ) K D(x; ε x ). K x,..., x r K s.t. K x K r D(x j ; ε xj ). D(x j ; ε xj ) I K = K I j= f n (x) = f n (a) + x a f n(t) dt K x > a [a, x], x < a [x, a] {f n} K C.3.2 ( 5.3) D C {f n } n N D D f f D k N lim f (k) n n = f (k) (D ). f {f n } f a D r > s.t. D(a; r) D {f n } n N r D(a; r) D D(a; r) D = D(a; r). Cauchy f n (z) = 2πi ζ a =r f n (ζ) ζ z dζ (z D(a; r)). K := {ζ C; ζ a = r} K {f n } f f n(ζ) ζ z f(ζ) ζ z K f(z) = 2πi ζ a =r f(ζ) ζ z dζ (z D(a; r)). f f D(a; r) k N f (k) (z) = k! f (k) (ζ) dζ (z D(a; r)). 2πi ζ a =r (ζ z) k+ f n f n (k) (z) = k! 2πi ζ a =r (z) dζ (z D(a; r)). (ζ z) k+ f (k) n f (k) (z) f n (k) (z) = k! 2πi ζ a =r f (k) (ζ) f n (k) (z) (ζ z) k+ dζ (z D(a; r)). z a r 2 ζ K ζ z ζ a z a r r 2 = r 2 68

170 f n (k) (z) f (k) (z) k! 2π k! 2π = k! 2π ζ a =r ζ a =r f n (ζ) f(ζ) ζ z k+ dζ sup f n (ζ) f(ζ) ζ a =r sup f n (ζ) f(ζ) ζ a =r k! 2k+ r k sup f n (ζ) f(ζ). ζ a =r ( ) 2 k+ 2πr r ( ) 2 k+ dζ r D(a; r/2) f (k) n f (k). C.3.3 ( 5.4 ( Weierstrass M )) D C {f n } n N D {M n } n N 2 (i) n N, z D, f n (z) M n. (ii) M n S(z) := f n (z) D k N, z D S (k) (z) = f n (k) (z). Weierstrass M n S n (z) := S(z) S D k N S (k) (z) = lim n S(k) n (z). n f j (z) D j= C.3.4 (Riemann ) ( n z = exp (z log n) log n log n R ) ζ(z) := D := {z C; Re z > } D ζ Riemann α > Re z α n z = exp (z log n) = exp Re(z log n) = exp [(Re z) log n] = n Re z n α. M n := n α M n n z (n N, Re z α), 69 n z M n = n α <.

171 C.3.3 Re z > α ζ α > ζ Re z > Riemann s : n z ζ(s) := n s. C Riemann Riemann (859 ) ζ(s) s = 2n (n N) Re s = 2 C.3.5 f(z) := z + 2z z 2 n 2 (z C \ Z) C \ Z f n 2z n 2 z < n z 2 /n 2 z 2 /n 2 > 2z z 2 n 2 = 2z n 2 ( z 2 /n 2 ) = 2 z n 2 z 2 /n 2 R > N N N 2R z < R n=n ( z C : z R) ( n N : n N) z 2 /n 2 (/2) 2 = z z 2 n 2 2R n = 8R 3 n 2 2 z n 2 ( z 2 /n 2 ). z R n N 2. ( z R, n N). 2z z 2 {z C \ Z; z < R} f n2 R > C \ Z f 7

172 C.4 2, 29 4 C.4. (analytic continuation) C.4. (, ) D ( C) f (analytic) D f C.4. ( vs ) ( ) (w = f(z) z w ) z w ( ) ( ) C.4. ( ) C.4.2 ( ) 2 ( ) C.4.3 ( [4] 2.38, (identity theorem)) D C f g D ( ) {z n } n N (i), (ii) (i) lim z n = a. n (ii) n N z n D, z n a, f(z n ) = g(z n ). D f g : z D f(z) = g(z). C.4.4 f g C.4.5 ( ) f g C.4.6 i.e., a f r > s.t. z D(a; r) \ {a} f(z). 7

173 C.4.7 () ( ) φ: I R (I R ) D ( I) f : D C (f φ x I φ(x) = f(x) ) g : D C φ f = g. ( ) ( ) C.4.8 (( ) ) D, D 2 D D 2 C f : D C f 2 : D 2 C z D f (z) = f 2 (z) f 2 f D 2 (analytic continuation) () C.4.9 ( ) () C.4.2 z log z ( ) Weierstarss h(z) D = D(c; R) a D h : h(z) = a n (z a) n. D = D(a; r) r r R a c r >. D D (D \ D ) h(z) (z D), f(z) := a n (z a) n (z D \ D) n= n= h D D D \ D Weierstrass h(z) D = D(c; R) (h(z), D) c C z = φ(t) (t [, ]) t [, ] φ(t) (h t (z), D t ) 72

174 t [, ], δ >, s [, ], t s < δ = D s D t h s (z) = h t (z) (z D s D t ). (h (z), D ) (h (z), D ) C Weierstrass 2 C C a b a C C b (h (z), D ), ( h (z), D ) b h (b), h (b) log x = log x x = Taylor ( ) n (x )n n n= = (x ) (x )2 2 h (z) := + (x )3 3 ( ) n (z )n n n= ( < x < ) D := D(; ) = {z C; z < }. x (, 2) h (x) = log x. (h (z), D ) Weierstrass h (z) = z h (z) = γ dζ ζ (γ D z ) γ D ( ) z γ z C \ {} C : z = φ(t) (t [, ]) C \ {} z (h (z), D ) C (h (z), D ) D := {z C; z z < z dζ z }, h (z) := ζ + dζ (z D ), h (z dζ ) = ζ ζ C (h (z) 2 z z D ) t (, ] h t (z) D t dζ z D t := {z C; z φ(t) < φ(t) }, h t (z) := C t ζ + dζ ζ C t C [, t] ( φ(t) ) z φ(t) C [, t] s φ(s) C (log x) = x log x = x Taylor dt t 73

175 z D t φ(t) z ( ) φ(t) t = D, h h t (z) φ(t) 2 /ζ D t ζ = φ(t) + ζ φ(t) = φ(t) + ζ φ(t) φ(t) = φ(t) ( ζ φ(t) ( ) n φ(t) n= ) n z φ(t) dζ ζ = φ(t) = n= z φ(t) n= ( ζ φ(t) ( ) n φ(t) ( ) n (z φ(t)) n+ n + φ(t) n+ (z D t ). h t (z) = φ(t) dζ ζ + n= ( ) φ (ε δ ε = φ(t) ) ) n dζ = φ(t) n= z φ(t) ( ) n (z φ(t)) n+ n + φ(t) n+ (z D t ). δ >, s [, ], s t δ = φ(t) φ(s) < φ(t). ( ) n φ(t) n (ζ φ(t))n dζ φ(s) D t. D s D t ( φ(s) ). z D t D s φ(s) ( dζ z h s (z) h t (z) = ζ + dζ φ(t) ) φ(s) ζ dζ z ζ + dζ φ(s) dζ z = φ(t) ζ φ(t) ζ + dζ φ(t) φ(s) ζ + dζ z ζ. /ζ D t h t (z) = h s (z). (h (z), D ) (h (z), D ) C (h (z), D ) Weierstrass z f(z) := z dζ ζ (z C \ {}) C \ {} z C z C C 2 74

176 [] E. T. I, II, III, (23), 976. [2], (946), [3] L., (97). [4],, (23). [5] Ahlfors, K.: Complex Analysis, McGraw Hill (953),,, (982). [6], (99). [7], (98). [8], (979). [9],, (23). [] Bak, J. and Newman, D. J.: Complex Analysis, Second Edition, Springer (999). [] 2, (965). [2], (98). [3] 3, (989). [4], (99). [5], (99), [6] 2 ( II ), pde/pde-23.pdf (997 ). [7],, (959), 75

177 Abel s continuity theorem, 22 absolutely convergent, 2 analytic, 7 analytic continuation, 72 annulus, 44 Cauchy-Riemann differential equations, 3 Cauchy-Riemann equations, 3 circle of convergence, 8 conformal mapping, 4 dominant series, 3 dominating series, 3 entire function, 37 isolated singularity, 43 Laurent series, 48 Liouville s theorem, 37 majorant, 3 majorant series, 3 meromorphic, multifunction, 56 pole, 5 power series, 7 principal part, 5 radius of convergence, 8 rational function, removable singularity, 5 residue, 62 Riemann sphere, 2 stereographic projection, 2 maximum principle, 35 maximum-modulus principle, 35 mean value property, 34 univalent function, 4 Weierstrass M-test, 6 Abel, 9 Abel, 22 (), 4 (annulus), 44 (Weierstrass ), 73, 72 ( ), 72, 7, 72, 5, 72 Cauchy, 37 Cauchy-Riemann, 3 Cauchy-Riemann, 3 Cauchy-Riemann, 3, 5 (isolated singularity), 43, 35 ( ), 3, 8, 8 ( ), 5 ( ), 2 ( ) (principal part of Laurent expansion), 5, 5, 5, 37, 34, 69, 2, 34, 7, 37, 4, 4 76

178 , 34, 7 Morera, 4, 3,,, 2 Riemann, 34, 69 Riemann, 34, 7, 37, 2, 62 (Laurent expansion), 48 Weierstrass M, 6 Weierstrass M, 6 Weierstrass, 73 77