4 4 No. pdf pdf II Fourier
No. No. 4 No. 4 4 38 f(z) z = z z n n sin x x dx = π, π n sin(mπ/n) x m + x n dx = m, n m < n e z, sin z, cos z, log z, z α 4 4 9
4 9 i = imaginary unit z = x + iy x, y R x real part y imaginary part x = Re(z), y = Im(z) z := x + y z absolute value z+w z + w z := x iy z complex conjugate C C C ( ) x y x, y R z = x + iy y x z θ z = r(cos θ + i sin θ) (r >, θ R) z polar form θ = arg(z) z argument arg z πn n z, w z w {z n } n= z lim z n z = z n = x n + iy n, z = x + iy n {z n } n= z {x n} n=, {y n } n= x, y. () z = i () z 3 = i..3 z, w C, z w z + w z w = z w z w + i Im( zw) z w z, w, z, w 3 z zw + w = ρ = cos ( π 3 ) + i sin ( π 3 ).4 ( a, b, ) c, d R ad bc > z C Im(z) > az + b Im > f(z) = az + b H := {z C cz + d cz + d Im(z) > } f(h) H f : H H.5 α C α < z C z < z α ᾱz < g(z) = z α := {z C z < } ᾱz g( ) g :
No. 4 6 e z cos z, sin z n= z n z n C S n := z + + z n {S n } n= n= z n n= z n absolute convergent. () n= z n () test by majorant series n= z n M n z n M n n =,,... n= M n n= z n (3) n= z n, n= z n n= (z n ± z n) = n= z n ± n= z n, n= αz n = α n= z n α C (4) n= z n, n= z n w n := n k= z kz n k Cauchy n= z n, n= z n n= w n ( n= z n) ( n= z n) = n= w n n= a nz n a n C, z C power series. n= a nz n (i) z C (ii) (iii) ρ > z < ρ z > ρ z ρ n= a nz n radius of convergence z = ρ circle of convergence (i), (iii) ρ =, ρ. n= a nz n ρ () / =, / = ρ = lim sup n n a n * Cauchy Hadamard () n a n an a n+ n ρ d Alembert e z cos z, sin z exponential function e z := n= zn n! trigonometric function cos z := zn n= ( )n (n)!, sin z := zn+ n= ( )n (n+)! z C e z () e iz = cos z + i sin z Euler () e z+w = e z e w (3) z = x + iy x = Re(z), y = Im(z) e z = e x (cos(y) + i sin(y)) e z (4) e z = e w z = w + nπi n Z (5) z C e z 4 4 6 * {b n } n= c n = sup b k {c n } n= k n {b n} n= lim sup b n n 3 lim c n n
. () (4) n= ( + ) n z n () n n 3 z n (5) n= ( n 3 n ) z n (3) n= n= ( + n + + ) z n (6) n n= n= z n! (n!) (n)! zn. (a) z = (b) z = (c) z = z =.3.(4).4 () cos z = eiz + e iz, sin z = eiz e iz i () e z z, w C cos(z + w) = cos z cos w sin z sin w, sin(z + w) = sin z cos w + cos z sin w, cos(z + π) = cos z, sin(z + π) = sin z, cos z + sin z =. (3) R sin x {nπ n Z} e z C sin z cos z.5 x cosh x, sinh x cosh x = ex + e x, sinh x = ex e x () x cos(ix) = cosh x, sin(ix) = i sinh x () z = x + iy cos(z), sin(z) (3) z cos z, sin z 4
No. 3 Cauchy Riemann 4 3 Cauchy Riemann α C, r > r (α) := {z C z α < r} D C D open set α D r > r (α) D D arcwise connected α, β D D () = α, () = β : [, ] D D C D D domain * Cauchy Riemann C D f : D C, z f(z) f(z) D z = x + iy f(z) = u(x, y) + iv(x, y) u(x, y) v(x, y) (x, y) f(z) D f(z) z = z D lim z z f(z) = f(z ) z = x + iy u(x, y), v(x, y) (x, y ) f(z) D D f(z) f(z ) f(z) D f(z) z = z D lim z z z z f (z ) z z z 3. (i) (ii) f(z) = u(x, y) + iv(x, y) D z = x + iy D f(z) z = z u(x, y), v(x, y) (x, y ) Cauchy Riemann (x, y ) u x = v y, u y = v x, f(z) = z n (n =,,...) f (z) = nz n f (z) = u x + iv x = i (u y + iv y ) u x := u x f(z) D f(z) (i) D (ii) f (z) D f(z) D holomorphic function * 4 4 3 * C D D C * (i) (ii) (ii) 5
D f D f = f 3. C f(z) = z z f(z) = z D 3. f(z) = z 3 f( + i) f( + i) = f ( + ti) t ( + i) ( + i) 3.3 D f () z D f(z) R f () f(z) f(z) : ()() 3.4 Cauchy Riemann () f = u + iv D v () u + v 3.4 f(z) = u(x, y) + iv(x, y) D u(x, y), v(x, y) D u x + u y =, v x + v y = Laplace 6
No. 4 log z z α 4 3 5 7 No. 5 f(z) z D f(z) f(z ) = f (z )(z z ) + o( z z ) f (z ) = r(cos θ + i sin θ) f z f(z ) r θ r f (z ) z f f(z) = u(x, u) + iv(x, y) D u(x, y), v(x, y) D C Cauchy Riemann * z z f(z) x, y z z z := ( x i ) y z := ( x + i ) y Cauchy Riemann f z = f(z) z f (z ) = f z (z ) z z z =, =, z z = z z z = z (zn z m ) = nz n z m, mz n z m z, z z (zn z m ) = 4. () f(z), g(z) D D (a) (b) (f(z) ± g(z)) = f (z) ± g (z), (αf(z)) = αf (z) α C (f(z)g(z)) = f (z)g(z) + f(z)g (z). (c) f D /f D (/f(z)) = f (z)/f(z). () f(z) D g(z) D f(d) D g(f(z)) D (g(f(z))) = g (f(z))f (z) 4. (a) f(z) = c C (c) = (b) z C (z) = (c) z n n =,,... C (z n ) = nz n a n z n + + a z + a C (a n z n + + a ) = na n z n + + a (d) a n z n + + a z + a b m z n + + b z + b z (e) e z C (e z ) = e z e (+z) C (e (+z) ) = (z)e (+z) 4 4 3 * R D g(x, y) C D g x y g/ x, g/ y D R D g(x, y) C g D 7
4. f(z) = a n z n = a + a z + a z + + a n z n + f (z) = n= na n z n = a + a z + + na n z n + n= f (z) = f (z) 4.3 f(z) = a n z n a n = f (n) ()/n! n =,,... n= e z sin z, cos z C (e z ) = e z, (sin z) = cos z, (cos z) = sin z log z z α z e w = z w w = log z + i arg(z) arg(z) z π z log z := log z + i arg(z) log z z z ( π, π] Arg(z) π < Arg(z) π [, π) Log z := log z + i Arg(z) principal value Log D := C (, ] Log z D (Log z) = z * z, α z z α := e α log z *3 α z α log z z α i i = e i log i = e i(i( π +nπ)) = e ( π +nπ) n i i No. 4 4. () log ( + 3i ) () Log ( + 3i ) (3) ( + 3i ) i. 4. α, β, z = x + iy x, y R f(z) := αx + βxy + y f C α, β, : x = (z + z)/, y = (z z)/(i) f z z f x, y C f f z = No. No. 4 A. () C D () f D f z D * [, π) /z *3 e z e α log z α log z e z e exp(z) 8
(3) f D f(z) = u(x, y) + iv(x, y) z = x + iy f z = x + iy D u(x, y) v(x, y) (4) f D f D A. () z C Cauchy Riemann () C f(z) z = x + iy u = x y f(z) v = xy + C C (3) u = x + y C f(z) B. H := {z C Im(z) > } := {z C z < } f(z) := z i z + H f(h) g(z) := z + i i(z ) g( ) H f(g(z)) = z, g(f(z)) = z f : H f : H g f f = g f Cayley B. f(z) = ( z) m m =,,... z : z = z n (m ) 4. B.3 (B.3.) (B.3.) z z + z3 3 + 3 4 z5 5 + 3 4 5 6 z7 7 + = z + z z3 3 + z5 5 z7 7 + = n= ( ) n zn+ n + n= (n )!! (n)!! z n+ n + (n )!! = (n )(n 3) 3, (n)!! = (n)(n ) 4 (B.3.) arcsin z z = Taylor (B.3.) arctan z z = Taylor B.4 D f(z) D f( z) D : f(z) = u(x, y) + iv(x, y) f( z) u(x, y), v(x, y) n= B.5 Abel f(z) = a n z n z = s = n= f(z) s a n z n n= a n z n= a s = s n := n k= a k n k= a kz k = s + (s s )z+ +(s n s n )z n = ( z) ( n k= s kz k) +s n z n Abel f(z) = ( z) n= s nz n Abel f(z) = n= ( )n z n+ /(n + ) B.3 z = z < z < f(z) = arctan z z f(z) π/4 π/4 = n= ( )n /(n + ) x arctan x x = 9
Taylor arctan x = n k= ( )k x k+ /(k + ) + R n+ (x) R n+ (x) Abel f(z) := +z = z + z z < lim z f(z) = z = lim n na n = Abel Tauber
No. 5 5 7 C f(z) f(z)dz C curver [a, b] C : [a, b] C (a) initial point (b) terminal point : [a, b] C (a) = (b) closed curve : [a, b] C simple a t < t < b (t ) (t ) * 5. Jordan C(= R ) C Im() Im() := ([a, b]) Jordan Jordan : [a, b] C positive orientation t a b (t) Jordan (t) = e it t π : [a, b] C (t) := (a + b t) a t b (b) (a) : [a, b] C : [a, b] C (t) := (a + b t) : [a, b ] C, : [a, b ] C (b ) = (a ) + := : [a, b + (b a )] C (t) = { (t) (a t b ) (a + (t b )) (b t b + b a ) : [a, b] C smooth (t) t C (t) t d d dt (t) t dt (t) * : [a, b] C piecewise smooth a = t < t < < t n = b [t i, t i+ ] t i 4 5 7 * Jordan * t = a, t = b d (t) dt d (t) dt
[a, b] R f(t) [a, b] f(t) = u(t) + iv(t) b a f(t)dt := b a u(t)dt + i b a v(t)dt f(t) [a, b] b f(t)dt a : [a, b] C f(z) Im() := ([a, b]) f(z) (5.) f(z)dz := b a f((t)) d dt (t)dt = k i f(z)dz := k i= i f(z)dz 5. : [a, b] C ( ) () φ : [ã, b] [a, b] C dφ dt >, φ(ã) = a, φ( b) = b := φ : [ã, b] C f(z)dz = f(z)dz. () f(z)dz = f(z)dz. 5.3 () f(z), g(z) (f(z) ± g(z))dz = f(z)dz ± g(z)dz, αf(z)dz = α f(z)dz (α C). (), f(z), f(z)dz = f(z)dz + f(z)dz. (3) : [a, b] C f(z) f(z)dz f(z) dz. = k ( i : [a i, b i ] C ) k i= bi a i f((t)) d i dt (t) dt *3 (3) : [a, b] C (5.) (5.) Riemann [a, b] a = t < t < < t N = b (5.) lim N f((s i )) ((t i+ ) (t i )) i= ( := max i N t i+ t i, s i [t i, t i+ ] 5.4 f(z) (5.) (5.) (5.) *4 *3 : [a, b] C (t) = t b a f(t)dt b a f(t) dt *4 (5.) (5.) (5.)
No. 5 5. n α C α r > (z α)n dz 5. +i 3 (t) = t+it t (t) = t+it t 3 := 3 3 3 (t) = t t 3 (t) = + it t i =,, 3 i Re(z) dz 5.3 R > (t) = t R t R, (t) = Re it = R(cos t + i sin t) t π = *5 () z n dz (n =,,,...) () 5.4 5.4 [a, b] ε > δ > t i+ t i < δ t [t i, t i+ ] d d (t) dt dt (s i) < ε d dt ((t i+) (t i )) d dt (s ti+ i)(t i+ t i ) = d ti+ t i dt (t)dt d t i dt (s i)dt ti+ d d (t) dt dt (s i) dt < ε(t i+ t i ) f((t)) [a, b] M (5.3) N N f((s i )) ((t i+ ) (t i )) f((s i )) d dt (s i)(t i+ t i ) i= N i= f((s i )) ((t i+) (t i )) d N dt (s i)(t i+ t i ) M ε(t i+ t i ) = M(b a)ε i= Riemann b d f((t)) a dt (t)dt = lim N i= f((s i)) d dt (s i)(t i+ t i ) (5.3) ε N lim i= f((s i)) ((t i+ ) (t i )) = b a t i i= d f((t)) dt (t)dt z dz *5 () := f(z) := z n C Cauchy f(z) dz = Cauchy () πr / i z 3
No. 6 Cauchy 5 4 5 Cauchy Cauchy C U c U c r > r (c) U r (c) := {z C z c < r} c r C A closed set A C A C X open neighborhoodx X C z C X adherent pointz U X U X z X z X z X z X z z X X C X X X closure X X X C X C z X X z r > r (z) X X int(x) int(x) C X := X int(x) X boundary C X bounded R > X R () Cauchy 6. Cauchy Cauchy s integral theorem D f D U D f(z)dz = * D D D D,..., k D f(z)dz = k i= i f(z)dz i D 6. f () f U f (z) Cauchy Green f (z) f (z) D f(z)dz = * () f D D 4 5 4 * f(z)dz * Goursat 4
Cauchy α C, r > 5. n *3 { (6.4) (z α) n πi (n = ) dz = ( ) z α =r (6.4) Cauchy 7 Green Green Gauss Stokes Cauchy Green : [a, b] R (R (t) = (x(t), y(t)) t [a, b] z = x + iy C (x, y) R D R : [a, b] D (t) = (x(t), y(t)) t [a, b] P (x, y), Q(x, y) D P (x, y)dx + Q(x, y)dy *4 (6.5) P (x, y)dx + Q(x, y)dy := b a { P (x(t), y(t)) dx } (t) + Q(x(t), y(t))dy dt dt (t) dt P dx + Qdy line integral 6.3 Green D R P (x, y), Q(x, y) D C D P (x, y)dx + Q(x, y)dy = D ( Q x P ) dxdy y D D C R R : [a, b] C = R f(z) = u(x, y) + iv(x, y) (6.6) f(z)dz = (u(x, y)dx v(x, y)dy) + i (v(x, y)dx + u(x, y)dy). R dz = dx+idy fdz = (u + iv)(dx + idy) *3 (z α) n dz z α =r *4 P (x, y)dx + Q(x, y)dy dx dy 5
Cauchy Green Cauchy 6. f(z)dz = (u(x, y)dx v(x, y)dy) + i (v(x, y)dx + u(x, y)dy) D D D ( = v D x u ) ( u dxdy + i y D x v ) dxdy y = (6.6) Green 6.3 Cauchy Riemann Cauchy Riemann Green Cauchy 6.() No. 6 6. X C X X 6. () α n () (3) (z α)n dz ( dz z = z(z ) z(z ) = sin(z ) e ez dz z = z z ) () ( 6.3 (6.4) z + ) n dz π z = z z cos n (n )!! θdθ = π (n)!! (n )!! = (n )(n 3), (n)!! = (n)(n ) 6.4 a, b : [, π] C (t) = a cos t + ib sin t π d () 6.() dz := (t)dt z (t) dt π () () a cos t + b sin dt t 6.5 D D zdz = i area(d) area(d) = dxdy D Green D 6
No. 7 Cauchy 5 5 8 Cauchy Cauchy Cauchy Cauchy Cauchy Cauchy f(z) Cauchy f(z)dz = f(z)dz = 7. Cauchy Cauchy s integral formula D f D U z D f(z) = f(ζ) πi D ζ z dζ D D Cauchy f f Cauchy Cauchy 7. Cauchy Cauchy Cauchy f () f U f (z) () f D D Cauchy D 7.3 Cauchy r (c) c r > f r (c) z r (c) f(z) = f(ζ) dζ πi ζ c =r ζ z 7.3 z = c c + re iθ θ [, π] (7.7) f(c) = π π f(c + re iθ )dθ f mean-value property 4 5 7
K C f n (z) n =,,... f(z) K {f n } n= f K converge uniformly lim sup f n (z) f(z) = n z K 7.4 : [a, b] C ([a, b]) {f n } n= f f lim f n (z)dz = f(z)dz n u n (z) f(z) = n= u n(z) K f n (z) := u (z) + + u n (z) {f n (z)} n= f(z) = n= u n(z) ([a, b]) u n (z)dz = u n (z)dz n= n= 7.5test by majorant series Weierstrass M-M-test C K n= u n(z) M n sup z K u n (z) M n n =,,... n= M n n= u n(z) K f(z) = n= a nz n ρ > < r < ρ f(z) = n= a nz n r () = {z C z r} M- r = ρ f(z) = n= a nz n ρ () f(z) = n= zn = z sup z < f(z) n k= zk = sup z < k=n+ zk z = sup n+ z < z = ρ () D f n (z) n =,,... f(z) D D = K * {f n } n= f K {f n } n= f converge uniformly on compact subsets f(z) = n= a nz n f(z) ρ K r (ρ) = z K K r * r < K r () f(z) r () K D {f n } n= f f : [a, b] D D ([a, b]) = 7.4 f n (z)dz = f(z)dz lim n No. 7 7., 7. Cauchy * C * = K K K 8
7. z + = i ( z + i ) z i z dz + 7. x +dx Cauchy x = tan θ 7 R > (t) = t R t R, (t) = Re it = R(cos t + i sin t) t π = () z + = i ( z + i ) z i () lim R z + dz = 7.3 f(z) = n= 7.4 z dz + x dx + z n n =,,... = {z C z < } zn D {f n } n= f f 9
No. 8 Cauchy Cauchy Part 5 8 Cauchy Liouville Cauchy D z Cauchy Liouville {a n } n= c C n= a n(z c) n c z c = n= a n(z c) n ρ z c < ρ z C z c > ρ z C ρ ρ ρ > 8. f(z) D c D r > r (c) D z r (c) (8.8) f(z) = f(ζ) πi ζ c =r ζ z dζ f(z) r (c) r f(z) = a n (z c) n, n= ( a n = πi ζ c =r ) f(ζ) dζ. (ζ c) n+ f(z) r (c) f(z) r (c) f(z) D Cauchy c D r (c) D c r (c) (8.8) 8. D f c D Taylor f D 8.3 f(z) = n= a n(z c) n z = c 4.3 a n = f (n) (c) n! f(z) = f (n) (c) n= n! (z c) n Taylor Taylor a n = f (n) (c) n! f(z) z = c Taylor f (n) (c) n! f(z) z = c Taylor 8.3 8.4 Cauchy Cauchy Cauchy f(z) f (z) 8. f(z) f(z) f (z) f(z) f (z) 4 5 8
8.5 D f D U z D f (n) (z) = n! f(ζ) dζ (n =,,...) πi (ζ z) n+ * D D D 8.5 D r (c) c r > f r (c) z r (c) (8.9) f (n) (z) = n! f(ζ) dζ πi ζ c =r (ζ z) n+ (n =,,...) Cauchy Liouville (8.9) Cauchy Cauchy Liouville 8.6 Cauchy f(z) R (α) := {z C z α < R} M R (α) z f(z) M f (n) (α) n!m R n (n =,,...) C entire function Liouville M z C f(z) M 8.7 Liouville Liouville Cauchy z C z R Cauchy n = f (z) M R R z C f (z) = f Liouville fundamental theorem of algebra 8.8 d ( ) P (z) = a d z d + + a d P (z) = a d (z α ) (z α d ) α i C d ( ) P (z) = a d z d + + a... P (z) C f(z) := P (z) f(z) Liouville P (z) * Cauchy Goursat
No. 8 8. R C 8. D f z D R C f(x) 8.3 () z = i z () z = i z (3) z = i z(z ) (4) Log z z = (5) e z / z = 8.4 Cauchy (8.9) () z = sin z z dz () + () z = 8.5 () sin z, cos z C () f(z) = C + z 8.6 e z dz (3) (z i) 3 z = Log z (z ) dz f(z) C z C Im(f(z)) > f e if(z) Cayley z i B. z + i 8.7 f(z) C k C > (i) z C f(z) C( + z k ) (ii) f(z) k (i) = (ii) 8. f(z) = n= anzn R > (8.9) a n = f(z) πi z =R z dz n > k n+ an = 8.8 f(z) := {z C z < } z f(z) z f (n) () < (n + )!e < r < (8.9) f (n) () = n! πi z =r f(z) dz zn+
No. 9 Cauchy Cauchy Part 6 4 Cauchy Cauchy 9. identity theorem f(z), g(z) D z D z k D, z k z k =,,... lim k z k = z f(z k ) = g(z k ) k =,,... f(z) g(z) D f(z) g(z) C f(z) g(z) 9. f D D f z D f(z ) = r > r (z ) D f r (z ) z 9.4 9.3 f(z) D z D k f (k) (z ) = f D : [a, b] C f(z) (a) f(z) (b) analytic continuation 3 9.4maximum modulus principle f(z) D z D f(z) f(z) 9.5 D f(z) D D max f(z) = max f(z) z D z D 9.6 D {f n (z)} n= D {f n(z)} n= D f(z) f(z) D 4 6 4 3
9.6 8. * 9.7 9.6 k k {f n (k) (z)} n= k f (k) (z) D 9. R C f(x), g(x) (, ) f(x) g(x) 9. e z+w = e z e w z, w C z, w () w R e z+w e z e w z C z z C w R e z+w = e z e w () z C e z+w e z e w w z C w C e z+w = e z e w 9.3 () f(z) x f(x) e x () f ( ) f(z) n f = n n (3) f ( ) f(z) n f n = n 9.4 9.3 9.5 f(z) R () < r < R M(r) := sup z =r f(z) f(z) M(r) (, R) 9.6 () f(z) D z D f(z) z D f(z) f(z) () f(z) D D z D f(z) D f(z) f(z) D 9.7 9.6 9.7 R {f n (x)} n= f(x) () f n (x) n =,,... C f(x) C * () f n (x) n =,,... f(x) C {f (x)} n= f (x) * Cauchy Morera Morera 9.6 * f n(x) n =,,...f(x) 7.4 4
No. 5 No. 9 * 3 A.3 () : [a, b] C () : [a, b] C f(z) f(z)dz (3) Cauchy (4) Cauchy A.4 n c C, r > A.5 z c =r α C A.6 () z = z dz z 4 () Cauchy z = ( z = z z+ (z c) n dz z dz ) dz z α B.6 B.7 π e eiθ dθ r (c) c r > f r (c) () f (n) (c) = n! πr n π () m =,,... B.8 f(c + re iθ )e inθ dθ n =,,,... π f(c + re iθ )e imθ dθ f C R > z < R f B.9 Schwartz Schwartz. f(z) := {z C z < } f() =, f(z) < z D f(z) z z D f () f () = z D {} f(z ) = z θ R f(z) = e iθ z z f() = g(z) f(z) = zg(z) < r < r z = r g(z) = r z r g(z) r f(z) z r g(z) B. f : *4 *3 No. No. 4 *4 biholomorphic 5
() f() = θ R f(z) = e iθ z z () f() = α θ R f(z) = e iθ z α z ᾱz () f f Schwartz () f f(α) = α h(z) = z α ᾱz h(z).5 f h () 6
No. Cauchy Cauchy Morera 6 8 6 5 6 Cauchy Morera Cauchy Cauchy Morera Cauchy. (i) D f(z) D 3 D F (z) f(z) = F (z) F f primitive function (ii) D f(z)dz = (iii) D, f(z)dz = f(z)dz (ii) (ii) D f(z)dz = (iii). D f(z) D D f(z) F (z) D : [a, b] C f(z)dz = F ((b)) F ((a)) r (α) := {z C a α < r} f(z) 8. f(z) r (α) f(z) = n= a n(z α) n r (α) F (z) = a n n= n+ (z α)n+ f(z) D D D D simply connected, -connectedd D * C 4 6 * C = R X X X c X c X c : [a, b] X (a) = (b) = c H : [a, b] [, ] X H(a, s) = H(b, s) = c s [, ] H(t, ) = (t) t [a, b] H(t, ) = c c C D 7
Jordan Cauchy. (ii) (ii) Cauchy.3 Cauchy D f(z) D D f(z)dz =.4 D D f(z) f(z) D F (z) f(z) = F (z). D f D = {} f(z) = /z f(z)dz = πi.(ii).(i) /z F (z) = log z log z {}.(i) Morera Morera Cauchy Cauchy Morera.5 Morera U f(z) U U f(z)dz = f(z) U Morera z U r (z ) U r > f(z) r (z ) U r (z ) z r (z ) [z, z] z z F (z) := [z,z] f(ζ)dζ h z + h r (z ) F (z + h) := [z,z+h] f(ζ)dζ = [z,z] f(ζ)dζ + [z,z+h] f(ζ)dζ = F (z) + f(ζ)dζ. (iii) = (i) [z,z+h] F (z+h) F (z) lim h h = lim h h f(ζ)dζ [z,z+h] f(z) F (z) r (z ) F (z) = f(z) F (z) r (z ) 8. F (z) r (z ) f(z) = F (z) r (z ). D f(z) D F (z) F (z) f(z) F F D. D R C f(z) D D R D f(z) D II 3.4 8
No. ( ) < κ < {α n} n= α n κ π < ( ) Arg(α n ) < κ π n =,,... (i) α n (ii) n= Re(α n ) n= w = f(z) = z z w () w = u + iv u, v R z = x + iy x, y R u, v x, y () Re z = a a R Im z = b b R f π 3 a, b a > b > dt a + b cos t () z = e it = cos t + i sin t t [, π] z = π a + b cos t dt = z = a + b ( z + z ) iz dz = z = i(bz + az + b) dz () bz + az + b = α = a a b, β = a + a b α, β b b b > α < < β < b < < β < < α (3) i(bz + az + b) z α z β π i(bz dz dt + az + b) a + b cos t 4 () S > R > ε > e iz z dz + e iz z dz + 3 e iz z dz + 4 e iz z R sin x dz + i ε x dx = (t) = R + it t [, S], (t) = t + is t [ R, R], 3 (t) = R+i(S t) t [, S] 4 (t) = εe i(π t) = ε(cos(π t)+i sin(π t)) t [, π] e iz ε 4 z e iz () lim dz = iπ (3) z dz R, e iz z dz R S, e iz 3 z dz R sin x (4) S R dx x 4 6 5 9
5 f(z) D No. (a) f D (i) f D (ii) f (z) D (b) z = x + iy f(z) = u(x, y) + iv(x, y) u(x, y), v(x, y) x, y D C D Cauchy Riemann u x = v y, u y = v x, ( f ) z = (c) D D D D D D f(z)dz = D (d) D (c) z D f(z) = f(ζ) dζ πi D ζ z (e) f D z D r > r (z ) D r (z ) f(z) = n= a nz n. r. (f) D D D D F z D F (z) = f(z) 6 f(z) U f(z) α,..., α n m,..., m n U g(z) U (.) f(z) = (z α ) m (z α ) m (z α n ) mn g(z) f n = () (.) f (z) f(z) = m + + m n + g (z) z α z α n g(z) () () f (z) πi f(z) dz = m + + m n (3) (.) (a) f α f m U h(z) h(α) f(z) = (z α) m h(z) (b) (.) f (z) 6 () argument principle dz f(z) f (3)(a) B.8 7 D f(z) D f(z) z D R f(z) R f(z) z D ir f(z) ir f(z) z D f( z) = f(z) ir := {iy y R} f( z) D Cauchy Riemann B.4 f(z) D R 3
α n = a n + ib n a n, b n R ( π κ) < Arg(α n ) < π κ a n > tan (Arg(α n )) = b n an b n < ( tan ( π κ)) a n (i) = (ii) a n = a n α n n= α n n= a n (ii) = (i) α n a n + b n < ( + tan ( π κ)) a n ( ( + tan π κ)) n= a n n= α n () z = (x + iy) = (x y ) + i(xy) u = x y, v = xy () x = a u = a y, v = ay (i) a = u = y, v = w (ii) a u = a v 4a y = b u = x b, v = bx (i) b = u = x, v = w (ii) b u = v 4b b ( ) 3 ()() (3) i(bz +az+b) = i a b z β z α β α Cauchy Cauchy z = i(bz +az+b) dz = i a b z = π a+b cos t dt = z β dz π a b i a b z = z α dz = (πi) i = π a b a b 4 () C = [ε, R], C = [ R, ε] C C t [ε, R] R t C C eiz z C,,, 3, C, 4 Cauchy C e iz z dz + sin z = e iz z dz + e iz z dz + 3 e iz z dz + C e iz z dz + 4 e iz z dz = eiz e iz i e iz C z dz + e iz C z dz = ( R e ix ε x ) e ix x dx = i R sin x ε x dx () () 4 (t) = εe i(π t) t π4 (t) = εeit t π ε e iz 4 z dz = e iz 4 z dz = π e iεeit εe iεe it dt = i π it eiεeitdt iπ e iz z = π )dt (eiεeit π e e iεeit iεe dt π sup it t π ε (3) e iz z dz S = e i(r+it) R+it dt S ei(r+it) R+it dt S e t R dt = R ( e S ) R 3 3 (t) = R + it t [, S] e iz 3 z dz = e iz 3 z dz S = e i(r+it) R+it dt S ei(r+it) R+it dt S e t R dt = R ( e S ) R (t) = t+is t [ R, R] e iz z dz = e iz z dz R = e i(t+is) R t+is dt R = R ei(t+is) t+is dt e S R dt R S e S R S R S (4) S R k =,, 3 e iz k z dz e iz 4 z dz + i sin x ε x dx = ε sin x x dx = π 5 (a) (b) 3. No. 4 (a) = (c) Cauchy 6. (c) = (a) Morera.5 (a) = (d) Cauchy 7. (d) = (e) 8. (e) = (a) 4. 4.3 (a) = (f).3 (f) = (a) F (a) = (d) = (e) F 3
f = F f (a) (f) 6 () f (z) = n i= m i(z α ) m (z α i ) m i (z α i ) mi (z α i+ ) m i+ (z α n ) m n g(z) + (z α ) m (z α n ) m n g (z) f (z) f(z) = m z α + + m n z α n + g (z) g(z) () U := {z U g(z) } U g (z) g(z) U Cauchy g (z) f πi g(z) dz = () (z) πi f(z) dz = n i= m i πi z α i dz + g (z) πi g(z) dz = m + + m n (3) (a) f K K = {z k } k= z k K f(z k ) = z := lim k z k f U f(z ) = z K f z f f f K f f α r > r (α) r (α) f(z) = n= a n(z α) n f 9.3 a n n n m r (α) f(z) = (z α) m (a m + a m+ (z α) + ) U h(z) = f(z) (z α) h(z) m U {α} z = α h(z) = a m + a m+ (z α) + h(α) = a m h(z) z = α h(z) U h(α) = a m (b) (a) f α,..., α n n (a) m,..., m n U g(z) f(z) = (z α ) m (z α n ) m n g(z) g(α ),..., g(α m ) f(z) α,..., α n g(z) 7 f( z) D Cauchy Riemann ( B.4 ) z D R f(z) = f( z) z D f(z) = f( z) * iz D z g(z) := if(iz) g(z) g( z) iz D ir z g(z) = g( z) iz D z g(z) = g( z) g( z) = if(i z) z D f( z) = f(i(iz)) = ig(iz) = ig(iz) = ( i) f(iiz) = f( z) = f(z) * Schwartz 7 3
No. Laurent 7 Laurent Laurent 9 6 3 Cauchy D α f(z) D z α D f(z) c D isolated singularity r > f(z) r (c) {c} = {z C < z c < r} f(z) z = c c sin z z, z, sin z z = f(z) z = c c z = c * Laurent. Laurent c C f(z) R (c) {c} = {z C < z c < R} R (c) {c} (.) f(z) = n= a n (z c) n ( < z c < R) (.) r z c r < r < r < R a n f(z) Laurent (.) f(z) z = c Laurent f(z) n= a n(z c) n = + a (z c) + a z c f(z) Laurent principal part f(z) z = c Laurent () f(c) = a f(z) z = c Laurent z = c f(z) = n= a n(z c) n z c < R f(z) z = c Tayler z = c f(z) removable singularity () k Laurent a k (z c) k + a (z c) + a (z c) + a + a (z c) + (a k ) z = c f(z) k pole * 4 7 * sin( π z ) z = ± n =,,... z = n log z z = log z * f(z) z = c f(c) = f(z) z = c f(z) = a m(z c) m + a m+ (z c) m+ + m, a m z = c f(z) m 33
(3)..., a, a sin z z z = c f(z) essential singularity z = Laurent sin z z sin z z = = z zn+ n= ( )n (n+)! = z 3! + z4 5! z z z = sin ( ) z = n= ( )n (n+)!z = n+ n= ( )n ( n+)! zn z = sin ( ) z f(z) D f(z) D meromorphic function f(z). z = c f(z) z c f(z) () z = c f(z) lim f(z) z c () z = c f(z) lim f(z) = z c (3) z = c f(z) lim f(z) z c z = c f(z) () z = c f(z) Riemann. z = c f(z) (3) α C { } z n c lim n f(z n) = α Weierstrass.3 f(z). Laurent (3) z 3. () z z z = () z sin z z = (3) z cos z sin z Riemann Riemann. (z = ) D = R (c) {c} = {z C < z c < R} f(z) D M z D f(z) M z = c f(z) f(z) z = c Laurent f(z) = n= a nz n m < r < R a m = f(z)(z πi z c =r c) m dz. a m Mr m m > r a m =.3 Weierstrass Casorati Weierstrass Weierstrass. f(z) R (c) {c} z = c f(z) c {z n } n= () lim f(z n) = () n α C lim f(z n) = α n () f(z) z = c Riemann z = c f(z) () α C ε > r > z r(c) {c} f(z) α ε g(z) := z = c Riemann z = c f(z) α g(z) f(z) = α + z = c g(z) 34
No. 3 7 9 f(z) z = c ρ > ρ (c) := {z C z c < ρ } f(z) R > ρ f(z) R (c) R > f(z) R (c) ρ = f(z) z = c n= a n(z c) n n= a n(z c) n ρ ρ = ρ f(z) = z z = f(z) z = i i = f(z) z i < z i f(z) i f(z) f(z) z = i 8.3() f(z) z = α Laurent n= a n (z α) n z α a α residue Res(f, α) Res z=α f(z) Res z=α f(z)dz * 3. D f(z) D D D α,..., α n D f(z)dz = πi n Res(f, α j ) D D D f(z) z = α f(z) z = α z = α j= Res(f, α) = lim z α (z α)f(z) f(z) Laurent f(z) = a z α + a + a (z α) + a = Res(f, α) = a = lim z α (z α)f(z) f(z) z = α k f(z) z = α z = α k ( ) d k Res(f, α) = (k )! lim z α dz k (z α)k f(z) k = f(z) Laurent f(z) = a k + + a (z α) k z α +a +a (z α)+ (z α) k f(z) = a k + ( +a (z α) k +a ) (z α) k +a (z α) k+ + d k Res(f, α) = a = (k )! lim z α dz k (z α)k f(z) 4 7 9 * f(z) - f(z)dz Res z=αf(z)dz Res(f, α) 35
* [I] π R(cos θ, sin θ)dθ R(cos θ, sin θ) sin θ, cos θ sin θ, cos θ R(cos θ, sin θ) R(cos θ, sin θ)dθ z = e iθ ( ) z = z = e iθ cos θ = z + z ( ) sin θ = i z z π R(cos θ, sin θ)dθ = ( ( R z + ), ( z )) dz i z i z z z = z = π No. 3 a, b a > b > π a+b cos θ dθ z = e iθ π a + b cos θ dθ = i z = a + b ( z + z ) z dz = z = i(bz + az + b) dz. bz + az + b = α = a a b b, β = a+ a b b α β z = β lim z β (z β) i(bz +az+b) = lim z β ib(z α) = ib(β α) = i a b πi i = π a b a b *3 [II] Q(x) P (x) dx P (x), Q(x) deg Q + deg P P (x) lim R Q(x) R Q(x) P (x) P (x) dx R Q(x), R lim R P (x) dx + dx deg Q + deg P P (x) Q(x) P (x) dx *4 R R = R = R R lim R R R lim R R 3. Q(x) P (x) dx Q(x) P (x) Q(x) P (x) R Q(x) P (x) dx dx dx P (x), Q(x) deg Q + deg P P (x) P (x) H := {z = x + iy C y > } α,..., α n (3.) Q(x) n P (x) dx = πi ( ) Res P (z), α j * *3 No. 3 z = β i(bz +az+b) *4 c > M > x R x c x Q(x) P (x) M Q(x) P (x) M x [ ] R M c x dx = M R = M + M x c R c M c (R ) lim R Q(x) R c dx P (x) [ ] c c M R x dx = M = M x R c + M M R c (R ) lim c Q(x) R R dx P (x) j= 36
(t) = t t [ R, R] (t) = Re it = R(cos t + i sin t) t [, π] := A > max{ α,..., α n } A R A α,..., α n (3.) R R Q(x) P (x) dx + P (z) dz = P (z) dz = πi n j= ( ) Res P (z), α j R deg Q + deg P M > z C z A z M P (z) P (z) dz π = Q(Re it ) π P (Re it ) Rieit dt Q(Re it ) π P (Re it ) Rieit dt *5 (3.) R, (3.). M Mπ dt = (R ) R R n (+x ) dx *6 + x n+ i R > (t) = t t [ R, R] (t) = Re it = R(cos t + i sin t) t [, π] (3.3) R R ( ) ( + x dx + ) n+ ( + z dz = πi Res ) n+ ( + z ) n+, i R lim R (+z ) dz = n π ( + z dz ) n+ = π ( + (Re it ) ) n+ Rieit dt Rie it ( + (Re it ) ) n+ dt π R (R ) n+ dt = πr (R (R ) ) n+ (+z ) z = i n + n+ ( ) πi Res ( + z ) n+, i = πi ( d n n! lim (z i) n+ ) z i dz n ( + z ) n+ = πi ( ) d n n! lim (z + i) (n+) z i dzn = πi n! ( )n (n + )(n + ) (n)(i) (n+) = π (n)! (n)!!(n )!! (n )!! n = π (n!) ((n)!!) = π (n)!! ( = π ) (n )(n 3) (n)(n ) (3.3) R, (+x ) dx = π (n )!! n+ (n)!! sin z 3. () z 4 z = sin z () z = z = (z )(z + ) 3 3. [I] < a < π dθ [II] a cos θ + a (x + x + ) dx *5 P (z) dz P (z) dz M R dz = M R πr = Mπ (R ) R *6 I x = tan θ I = π/ π/ cosn θdθ = π cos n θdθ [I] I 6.3 [I] I = π (n )!! [II] I (n)!! 37
x m 3.3 m, n m < n I := dx + xn β = π xm f(z) = n + x n (t) = t t [, R], (t) = Re it t [, β], 3 (t) = (R t)e iβ t [, R] := 3 f(z) () f(z) α = e πi n z = α f(z) Res(f, e πi n ) = n e mπi n () (3) ( f(z)dz = e mπi n 3 ) e mπi n I = πi n e mπi f(z)dz lim R f(z)dz = π n I = n sin ( ) mπ n 38
No. 4 Riemann 7 3 7 6 [III] Fourier [IV] log z z α [IV] 4. [III] Fourier Q(x) P (x) eiλx dx λ > P (x), Q(x) deg P + deg Q Q(x) P (x) eiλx Q(x) Q(x) P (x) cos(λx), P (x) sin(λx) f(x) ˆf(ξ) := π f(x)e ixξ dx f(x) Fourier Fourier [III-] P (x) 4. λ > P (x), Q(x) deg Q + deg P * P (x) P (x) H := {z = x + iy C y > } α,..., α n (4.) Q(x) P (x) eiλx dx = πi n j= ( ) Res P (z) eiλz, α j R, R, S > (t) = R + t t [, R + R ], (t) = R + ti t [, S], 3 (t) = (R + is) t t [, R + R ], 4 (t) = ( R + is) t t [, S] 4 := 4 3 A > max{ α,..., α n } A R, R, S A α,..., α n (4.) R R Q(x) P (x) eiλx dx + 4 k= k P (z) eiλz dz = πi n j= ( ) Res P (z) eiλz, α j deg Q + deg P M > z C z A z M P (z) P (z) eiλz dz = S S 4 4 3 R P (z) eiλz dz 4 7 6 R Q(R + it) P (R + it) eiλ(r+it) idt M R + it e λt dt P (z) eiλz dz S M λr S Q(R + it) P (R + it) eiλr λt dt M e λt dt = M ( e λs ) M R λr λr 3 M R M R M (R t) + is e λs dt R S e λs dt R S dt = M λs (R + R ) * [II] deg Q + deg P deg Q + deg P 39
S R M λr + M λr Q(x) R P (x) eiλx dx πi ( ) n j= Res P (z) eiλz, α j R R R, R (4.). f(z) H := {z = x + iy C y } H α,..., α n A M > z C z A f(z) M Q(x) z 4. P (x) eiλx f(x)e iλx 4. λ > P (x), Q(x) α,..., α n 4. Q(x) () P (x) Q(x) n P (x) cos(λx)dx = πi ( ) Res P (z) eiλz, α j j= Q(x) () P (x) Q(x) n P (x) sin(λx)dx = π ( ) Res P (z) eiλz, α j ze iz x sin x +x dx x +x R j= x sin x +x dx = R R x sin x +x dx +z + z z = i zeiz +z z = i ( ) ze Res iz +z, i = lim z i (z i) zeiz +z = ie i = e 4. xe ix +x dx = πi e x sin x +x dx = π e [III-] P (x) P (x) x = d 4.3 λ > P (x), Q(x) deg Q + deg P P (x) x = d Q(d) P (x) H := {z = x + iy C y > } α,..., α n (4.3) { d ε } Q(x) ( ) Q(x) n ( ) lim ε P (x) eiλx dx + d+ε P (x) eiλx dx = πi Res P (z) eiλz, d +πi Res P (z) eiλz, α j ε > C ε (t) = d + εe i(π t) t [, π] [III-], 3, 4 [ R, R ] [ R, d ε] C ε, [d + ε, R ] { d ε R } Q(x) R Q(x) P (x) eiλx dx + d+ε P (x) eiλx dx + C ε P (z) eiλz dz + [III-] { d ε } Q(x) Q(x) (4.4) P (x) eiλx dx + d+ε P (x) eiλx dx C ε + C ε 4 k= k = πi P (z) eiλz dz = πi j= j= P (z) eiλz dz n ( ) Res P (z) eiλz, α j n j= ( ) Res P (z) eiλz, α j P (z) z = d P ( ) (z) = (z d)p (z) P (d) Res P (z) eiλz, d = lim z d (z d) P (z) eiλz = Q(d) P (d) eiλd π Q(d + εe it ) π P (z) eiλz dz = it) εe it P (d + εe it ) eiλ(d+εe εie it Q(d + εe it ) dt = i it) P (d + εe it ) eiλ(d+εe dt πi Q(d) P (d) eiλd = πires ( ) P (z) eiλz, d (ε ) P (z) eiλz z = d (4.4) ε (4.3) 4
Q(x) (4.3) Cauchy principal value, valeur principale p.v. P (x) eiλx dx * No. 4 eiz z { sin x x dx R sin x ε x dx = ε R ( eiz z z = Res e iz 4.3 lim ε { ε sin x x dx = π e ix x dx + e ix ε x dx } = πires sin x x dx + } R sin x ε x dx ) z, = lim z z eiz z = ( e iz z, ) = πi Riemann Riemann C z z 3 S S S (,, ) N z z N S Z C z Z S C S N z Z N N point at infiniy C S S Riemann Ĉ := C { } P (C) := C { } w = z w w = f(z) z > R f ( ) w < w < R z = f(z) w = f ( w ) *3 a, b, c, d C ad bc f(z) := az+b cz+d P (C) P (C) f f f C C 4. a >, b > cos(ax) cos(bx) x dx 4.3 eiaz e ibz z 4.3 = i(a b) z + h(z) h(z) 4. [IV] log z z α [IV] < α < x α ( z)α +x dx f(z) = +z ( z) α e (α ) Log( z) b * a < d < b φ(x) [a, b] x = d φ(x)dx = a d ε b lim φ(x)dx + lim φ(x)dx ε := ε = ε > ε ε a ε d+ε { d ε b } lim φ(x)dx + φ(x)dx Cauchy principal value, ε a d+ε valeur principale p.v. b a φ(x)dx ε dx = lim dx + lim dx x ε x ε ε x { ε } Cauchy p.v. dx = lim x ε x dx + ε x dx = *3 No. 3 - f ( ) ( w d ) f( ) w = w w dw f(z) z = Res(f(z), ) = Res( f( w ) w, ) 4
ε ± δi, R ± δi δ > δ (e πi z) α, (e πi z) α z > () R e πi(α ) x α R e πi(α ) x α + f(z)dz + f(z)dz = πires(f(z), ) + x + x C(R) C(ε) Res(f(z), ) = () C(R) < α < C(R) f(z)dz π R α R (3) C(ε) C(ε) f(z)dz π ε α ε x α (4) ε, R + x dx = π sin(πα) 4.3 C S N z = x + iy C, Z = (X, Y, U) S N (X, Y, U) (x, y) (S X + Y + U = U ) No. No. 4 * 4 A.7 () D f(z) D : [a, b] D f(z)dz = () D f(z) D {a n } n= α := lim n a n n a n D α a n n =,,... n f(a n ) = f D (3) f(z) = z f(z) = z z = m= z m = n= zn n= zn f(z) z = Laurent B. f, g D fg D f g D No. 9 B. f, g C z C f(z) g(z) g () α C g(z) z = α f(z) g(z) 9. Riemann. () f(z) g(z) 3z B.3 f(z) := (z ) (z+) z = Laurent () f(z) 3z (z ) (z+) = A A, B, C, D *5 (z ) + B (z ) + C (z+) + () f(z) z = Laurent B. D (z+) A, B, C, D C *4 No. 5 No. 9 *5 f(z) := P (z), deg Q < deg P P (z) P (z) α j j k α j m j f(z) = k mj c jm j= m= (z α j ) m c jm C 4
B.4 () [II] Q(x) P (x) dx Imz < Q(x) P (x) 3. () [III] Q(x) P (x) eiλx dx Imz < Q(x) P (x) eiλx 4. B.5 λ > λ < Q(x) P (x) eiλx dx 43
4 7 3 ( ) 4 4 f(z) = z e z () z = f () z = f Laurent f(z) = (3) () a n z n n= a n z n a, a, a () a R f D f u, v D u + a v = f D () := {z C z < } F := {f(z) f(z) z f(z) } n f(z) F f (n) () ()() n= 3 a cos(ax) (x + ) dx 4 () P (z), deg Q + deg P P (z) := {z C z < } z = P (z) dz = () n P n := {P (z) P (z) n } R > P (z) P n max P (z) z R P (z) P n ()() () () π π P (Re it ) dt
4 7 3 ( ) z n () g(z) = g(z) g() = n! g(z) n= z = z z e z = z n= zn n! = = n= zn g(z) n! z = f () f(z) z = Laurent f(z) = n= a nz n () z = f(z) Laurent f() = = f(z) z = g() n= a nz n f(z) z = (e z )f(z) = z ( ) z! + z! + z3 (a 3! + + a z + a z + ) = z z, z, z 3 a =, a + a =, a 6 + a + a = a =, a =, a = (3) f(z) z = x + iy x, y R e z = e x (cos y + i sin y) e z = z = πni n Z z z = () z = f(z) f(z) {πni n Z, n } f(z) z = πi = π cf..(3) n! a n Bernoulli () z = x + iy x, y R u + a v = x y u x + a v x =, u y + a v = D f D D Cauchy Riemann y u x = v y, u y = v x u x a u u =, y y + a u x = u x = a u u = a y x a + a D u u = D = Cauchy Riemann D x y v x = v = D u, v D f y D () Cauchy f(z) F f (n) () n! g(z) := z n F g (n) () = n! f(z) F f (n) () n! () Cauchy Riemann cf. 3.3 f f
() Cauchy cf. 8.8 f Cauchy < r < r f (n) () = n! f(z) dz πi z =r zn+ f (n) () n! π f(re it ) πi (re it ) n+ ireit dt n! π f(re it ) π r n dθ n! r rn f (n) () n! 3 [I] a > I = cos(ax) cos(ax) (x dx + ) (x + ) R I = lim R cos(ax) (x + ) dx = R lim R R cos(ax) (x + ) dx R R > (t) = t R t R (t) = Re it t π := R i R eiaz eiaz (z + ) (z z = i z = i + ) z = i z = i (3.) e iaz (z + ) dz + (z + ) dz = e iaz e iaz ( ) e iaz (z dz = πi Res + ) (z + ), i (3.) e iaz (z + ) dz = R R e iax R (x + ) dx = R cos(ax) R (x + ) dx + i R sin(ax) (x + ) dx e iaz (z + ) dz (3.3) = π π e iareit ((Re it ) + ) ireit dt ar sin t e (R ) Rdt π π iar(cos t+i sin t) e ((Re it ) + ) ireit dt (R ) Rdt = πr (R (R ) ) t e it = e iar(cos t+i sin t) = e iar cos t e ar sin t = e ar sin t t π sin t a > e ar sin t e R sin t e R π t t π
e ar sin t t π a < e ar sin t t π e iaz (z z = i + ) (3.4) ( ) e iaz πi Res (z + ), i d = πi lim ((z i) e iaz ) z i! dz (z + ) = πi lim = πi = πi lim z i ( iae iaz (z + i) + e iaz ( )(z + i) 3) z i ( ) ) iae a (i) + e a (i) 3 = π e a ( a + (3.) (3.), (3.3), (3.4) R I = R lim cos(ax) (a + )π R R (x dx = + ) 4e a [II] a < cos(ax) = cos( ax) I := [I] I = ( a + )π 4e a = d ( e iaz (z + i) ) dz (a + )π e a cos(ax) (x + ) dx = cos( ax) (x dx + ) [I] lim e R iz (z + ) dz = 4 () R > A(, R) := {z C < z < R} P (z) A(, R) = {z C z R} Cauchy ( ) = A(,R) P (z) dz = z =R P (z) dz z = P (z) dz z = z = R deg Q + deg P M > z z P (z) M z *6 π P (z) dz Q(Re it ) π P (Re it ) Rieit dt M πm dt = R R z =R *6 deg P = n P (z) = a nz n + + a a n = b n z n + + b b n = lim z z P (z) = lim b n z n + + b z z a n z n + + a ) = b n a n r > z r z z P (z) b n a + z n P (z) { } K := {z C z r} K M b M := max n a +, M n z z z P (z) M P (z) M z 3
( ) z = P (z) dz = z =R P (z) dz πm R R R > R dz = z = P (z) () P (z) = z n + a n z n + + a a,..., a n Ca n = n n P (z) = a j a k z j z k π P (Re it ) dt π ( ) j= k= π π P (Re it ) dt = π π = n j= π M P := max P (z) z R ( ) π π n j= k= n a j a k R j+k e i(j k) dt n R j a j = R n + R j a j R n π P (Re it ) dt π j= (m ) e imt dt = (m = ) π M P dt = M P ( ) ( ) M P Rn M P R n M P = R n ( ) a = = a n = P (z) = z n max z R zn = R n M P = R n P (z) P n max P (z) z R Rn P (z) P n z n () P (z), () R = n z z n max z zn = n [, ] n T n(x) max x [,] n T n(x) = n T n(x) cos n cos cos(nθ) = T n (cos(θ)) Chebyshev 4 7 3 4 () P (z) = c(z α ) m (z α k ) m k c α,..., α k m,..., m k m k j P (z) = c jm (z α j ) m c jm C deg Q + deg P j= m= k j= c j = z = k P (z) dz = πi c j = j= 4