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3 i Riemann Stein 2 3 n 2n n Cauchy Riemann Dolbeault 3 Dolbeault 2 Čech

4 ii TEX

5 iii i ii C n Cauchy Riemann Riemann Vitali

6 iv Schwarz Weierstraß Weierstraß O n UFD Noether Riemann Mittag-Leffler Weierstraß Γ Stirling Cousin Dolbeault Dolbeault Riemann

7 4 20. C n C C C }{{} n z = (z,...,z n ), z j = x j + iy j (x j, y j R) n C n C Euclid C n a r = (r,...,r n ) (R + ) n (a;r) = {z C n z j a j < r j, j =,..., n} (.) C n a a (a;r) = {z C n z j a j r j, j =,..., n} (.2) C n 2 C n n d(z,w) = z j w j 2, z w = max z j w j. (.3) j n j= C n Euclid R 2n.2 D C n R +

8 D f z 0 D n a,..., a n C ε(z,z 0 ) = f (z) f (z 0 ) a j (z j z 0 j) j= ε(z z 0 ) lim z z 0 z z 0 = 0 (.4) a,..., a n f f (z 0 ),..., f (z 0 ) z z n f D f D f (z) z f z j D f g f ± gf g D f /g D {g = 0} D \ {g = 0} {() = 0}.2. f D () f (2) f j f (z,...,z n ) z j z j.2 (), (2) f C C Jordan γ j ( j =,..., n) D j D D n D f D z D D n Cauchy f (z,...,z n ) = f (ζ,z 2,...,z n )dζ 2πi γ ζ z = f (ζ,ζ 2,z 3,...,z n )dζ 2 dζ (2πi) 2 γ 2 (ζ z )(ζ 2 z 2 ) γ = = (2πi) n γ γ n f (ζ,...,ζ n )dζ dζ n (ζ z ) (ζ n z n ) (.5) D \ {g = 0} 9.3 {g = 0} {z D g(z) = 0} f /g D \ {g = 0} γ C Jordan Jordan C \ γ γ γ 2 γ 2 γ

9 (Cauchy ). z D D n f (z,...,z n ) = (2πi) n γ γ n f (ζ,...,ζ n )dζ dζ n (ζ z ) (ζ n z n ). (.6) (D D n ) γ γ n n n n = 2 D D 2 /0 (D D 2 ) = ( D D 2 ) (D D 2 ) = (γ D 2 ) (D γ 2 ) γ γ 2 f.2 (.6) k + +k n f z k... zk n n (z,...,z n ) = k! k n! (2πi) n γ γ n f (ζ,...,ζ n )dζ dζ n (ζ z ) k + (ζ n z n ) k n+ (.7) f D z,..., z n Cauchy.4 (Cauchy ). f (a;r) (a;r) f (z) M k + +k n f z k (a) zk n Mk! k n! n r k. (.8) rk n n (.7).5 (Liouville ). C n C n f M z (.8) ( f / z j )(z) M/r j r j + ( f / z j )(z) = 0 f f Hartogs 3

10 (Weierstraß). D f, f 2,..., f ν,... f ν f f k + +k n f z k zk n n k+ +kn f ν = lim ν z k zk n n (.9) a D (a;r) D r (R + ) n γ j = (a j ;r j ) C Cauchy (.6) f ν ν f ν f f (a;r) f (z,...,z n ) = (2πi) n γ γ n f (ζ,...,ζ n )dζ dζ n (ζ z ) (ζ n z n ) f (.7).3 a = (a,...,a n ) C n a ν ν n (z a ) ν (z n a n ) ν n, a ν ν n C (.0) ν,..., ν n = 0 a.7 (Abel). r = (r,...,r n ) (R + ) n {a ν ν n r ν rν n n ν,..., ν n N} (.0) (a;r) (.0) (a;r) f (z) k + +k n f z k zk n n (z) = () (.) ν,..., ν n a ν ν n r ν rν n n M ε > 0 (.0) (a;r ε) (.0) r ε (r ε,...,r n ε) (R + ) n

11 .3 5 (a;r) z (a;r ε) a ν ν n (z a ) ν (z n a n ) ν n = a ν ν n r ν rν n n z a r ν ( M ε ) ν ) ( εrn νn r z n a n r n (.0) (.0) (a;r ε) Weierstraß.6 ν n.7 r (R + ) n a C n 2 a (.) z = a k + +k n f z k zk n n (a) = k! k n!a k k n (.2).8. f D C n a D (a;r) D a (a;r) f a ν ν n = f (ζ,...,ζ n )dζ dζ n (2πi) n γ γ n (ζ z ) ν + (ζ n z n ) ν (.3) n+ γ j = (a j ;r j ) Cauchy.3 z (a;r) f (z) = (2πi) n γ γ n z j a j / ζ j a j < ζ j z j = f (ζ,...,ζ n )dζ dζ n (ζ z ) (ζ n z n ) (ζ j a j ) (z j a j ) = ζ j a j = ζ j a j ( ) z j a ν j = ν=0 ζ j a j ν=0 z j a j ζ j a j (z j a j ) v (ζ j a j ) ν+ (ζ z ) (ζ n z n ) = ν,..., ν n = 0 (z a ) ν (z n a n ) ν n (ζ a ) ν + (ζ n a n ) ν n+.

12 ζ γ γ n (.3) a ν ν n f (z) = a ν ν n (z a ) ν (z n a n ) ν n ν,...,ν n =0 f a D Taylor.9 (). f D C n a 0 = (0,...,0) ν = (ν,...,ν n ) N n \ {0} ν + +ν n f z ν (a) = 0 (.4) zν n n f D D 2 f, g D f, g D { A = z D ν + +ν n f z ν zν n n } (z) = 0 ( ν N n \ {0}) D D A = D f A z f Taylor A (.2) 0 f z A A D D D A 2 D.4 Cauchy Riemann. (z,...,z n ) C n, z j = x j + iy j (x j, y j R) (x,...,x n,y,...,y n ) R 2n C n R 2n C n dx j, dy j dz j = dx j + idy j, dz j = dx j idy j (.5)

13 .4 Cauchy Riemann 7, x j y j = ( i z j 2 x j y j ), = ( + i ) z j 2 x j y j (.6) z j dz k = δ jk, z j dz k = 0, z j dz k = 0, z j dz k = δ jk (.7) f (z) C d f = n j= (.5), (.6) d f = n j= f x j dx j + f z j dz j + n j= n j= f y j dy j (.8) f z j dz j (.9) f, f (.6), f z j z j z j z j f. z j f f f f z j.0. z = x + iy x zn = nz n x z = nzn, y zn = nz n y z = inzn (.6) z zn = 2 ( x i ) z n = ( nz n i 2 nz n ) = nz n y 2 z n z.. D C n f D C () f f 0, j =,..., n z j (2) f. f z j z j

14 () f (z) = u(z) + iv(z) f z j 0, j =,..., n {( u v ) ( v + i + u )} 0, 2 x j y j x j y j j =,..., n u = v, x j y j v = u, j =,..., n (.20) x j y j u, v z j Cauchy Riemann f.2 (2) f x j = f z j, f y j = i f z j f = ( f + ) f = ( i ) f = f z j 2 x j i y j 2 x j y j z j C f. () f z j 0, j =,..., n (.2) Cauchy Riemann (.20) (.20) f u 2 u + 2 u = 0, x j x k y j y k 2 u 2 u = 0, j, k =,..., n (.22) x j y k x k y j.2. D C n f f f f f f Cauchy Riemann (.2) f f / z j 0, j =,..., n f f f 0 f 0 f (a) 0 a r = f a f C θ(z) f (z) = re iθ(z) 0 f z j = ire iθ θ z j

15 .5 9 θ/ z j 0 θ θ f a.9 f D.5.3 (). D C n f f D f D a f a D a f f a D.9 f D D C n f D D D S sup{ f (z) z D} = sup{ f (z) z S} (.23) f S D.4. D C n D S 0 (D).4 S 0 (D) D Sirov.5. D = {z C z < } S 0 (D) = D a C e az e az D a a S 0 (D).6. D = (0;r,r 2 ) C 2 γ j = {z C z = r j }, D j = {z C z < r j }, j =, 2 D = (γ D 2 ) (D γ 2 ) S 0 (D) = γ γ 2.3

16 r = r 2 = f (z,z 2 ) D D θ R f (e iθ,z 2 ) z 2 (0;) z 2 (z,z 2 ) D f z f (z,z 2 ) = f (e iθ,z 2 ) θ z 2 f (e iθ,z 2 ) = f (e iθ,e iθ 2) θ 2 γ γ 2 D.5 S 0 (D).7. 2n D = {z C n z z n 2 < r}.5 S 0 (D) = D

17 D C n f,..., f m F = ( f,..., f m ) F : D C m F 2. (). a D F(a) = b m n det ( f,..., f m ) (a) 0 (2.) (z,...,z m ) a D = D D 2 DD C m D 2 C n m D 2 ϕ (z m+,...,z n ),..., ϕ m (z m+,...,z n ) D z = (z,...,z n ) z = ϕ (z m+,...,z n ) F(z) = b. z m = ϕ m (z m+,...,z n ) (2.2) 2.2 (). a D m = n det ( f,..., f n ) (a) 0 (2.3) (z,...,z n ) a D D D C n F D F D (F D ) : D D D

18 D C n F : D C n (2.3) R Rt t 3 0 C Cz z D C f : D C a C f (z 0 ) = a z 0 D f a f (z 0 ) = a, f (z 0 ) = 0,..., f (m ) (z 0 ) = 0, f (m) (z 0 ) 0 (2.4) z 0 m a f z 0 Taylor f (z) = a + a m (z z 0 ) m +, a m 0 f (z) = a + (z z 0 ) m g(z), g(z) D g(z 0 ) 0 (2.5) 2.3. D C f : D C z 0 f m a z 0 U (i) W = f (U) W a = f (z 0 ) (ii) f U\{z0 } : U \ {z 0 } W \ {a} m (iii) m = g : W U g W m ( ) g : W U m z w P(z,w) = z m + a (w)z m + + a m (w) w w = w 0 P(z,w 0 ) = 0 z,..., z m w 0 (z,...,z m ) m 2.3 f (z) a = (z z 0 ) m g(z), g(z 0 ) 0 z 0 = (z 0 ;ε) g(z) 0 c > 0 f (z) a > c W = (a;c) 2.3 L. Ahlfors, Complex Analysis, 3rd ed., McGraw Hill Chapter 8, 2 O. Forster, Lectures on Riemann Surfaces, Springer Verlag Chapter, 8

19 2. 3 2_3 f ( ) z 0 f a c W 2_ W w µ(w) = #{z f (z) = w} γ 2 µ(w) = f 2πi γ f (z)dz f (z) w f γ 2 w W f γ µ(w) = µ(a) = m f (W) = U f U\{z0 } : U \ {z 0 } W \ {a} m 2_3 m = f U : U W g(w) = z f (z)dz 2πi D f (z) w i m U f (z) = w z (w),..., z m (w) 0 i g k (w) = z k f (z)dz m 2πi D f (z) w = z l (w) k l= w l σ l (z (w),...,z m (w)) g,..., g m w a l (w) P(z,w) = z m a (w)z m + + ( ) m a m (w)z + ( ) m a m (w) g : W U 2.4. D C f : D C z 0 D f (z 0 ) U f U\{z0 } : U \ {z 0 } W \ {a} 4.2

20 D C n F : D C n a 2.2 (2.3) F a F = ( f,..., f n ) a f,..., f n a Taylor C{z,...,z n } z,..., z n m = (z,...,z n ) F : D C m a i i m w i w i m = g(w) g(f(z)) (2.6) (i,...,i m ) N m C{w,...,w m } C{z,...,z n } F 2.6. D C n C n F = ( f,..., f n ) F F : C{w,...,w n } C{z,...,z n } F 2.2 F G f (z) C{z,...,z n } g(w) = (G f )(w) = f (G(w)) w C{w,...,w n } (F g)(z) = g(f(z)) = f (z) F G : C{z,...,z n } C{z,...,z n } G F F n m D C n C m F = ( f,..., f m ) det ( f,..., f m ) (z,...,z m ) (0) 0 Noether Noether 8.7 F m z, m w F m w m z m 2 w m2 z m w/m 2 w m z /m 2 z 2 w,..., w n z,..., z n (2.3) F

21 2.3 5 C n m C m ϕ j (z m+,...,z n,w,...,w m ), j =,..., m z = ϕ (z m+,...,z n,w,...,w m ) F(z) = w. z m = ϕ m (z m+,...,z n,w,...,w m ) ( f,..., f m,z m+,...,z n ) 0 D F D (z,...,z m,z m+,...,z n ) F : (z,...,z m,z m+,...,z n ) (z,...,z m) D, D 2 C n 2 F : D D 2 F D D 2 F D D 2 D = D2 F : D D F : D 2 D (0;,) = {z C 2 z <, z 2 < } D 2 = {z C 2 z 2 + z 2 2 < } Riemann F : D D 2 F (.6) ( ) (,...,,,..., =,..., z z n z z n x,,..., x n y y n ) 2 I n i 2 I n 2 I n i 2 I n F = ( f,..., f n ) f j (z,z) = u j (x,y) + iv j (x,y) ( j =,..., n) f j z k f j z k f j ( z k f = In j I n z k u ) j ii n x k ii n v j x k u j y k v j 2 I n i y k 2 I n 2 I n i 2 I n

22 det ( f,..., f n, f,..., f n ) (z,...,z n,z,...,z n ) = det (u,...,u n,v,...,v n ) (x,...,x n,y,...,y n ). (2.7) f,..., f n det (u,...,u n,v,...,v n ) (x,...,x n,y,...,y n ) = det ( f,..., f n ) 2 (z,...,z n ) > 0 F C C γ : [0,) C γ(t) = x(t) + iy(t) 0 γ (0) = dγ dt (0) = dx dt dy (0) + i (0) (2.8) dt γ (0) 0 γ t = 0 a 2 γ, γ 2 t = 0 t = 0 2_3 arg ( γ ) 2 (0) γ (0) mod 2π (2.9) 2.9. D C C f : D C a f ( ) D f (z) = u(x,y) + iv(x,y) det (u,v) (a) 0 c (2.0) (x,y) f a a t = z W γ, γ 2 γ γ 2 t = 0 f γ f γ 2 t = 0 f a 2_ D f f γ 2 f f γ 2 γ f γ 2_ D C C f : D C f i D f (z) 0

23 2.3 7 a γ ( f γ)(t) t = 0 ( f γ) (0) = f z (a)γ (0) + f z (a)γ (0) f γ, γ 2 arg ( ( f γ2 ) ) (0) ( f γ ) (0) f f = arg z (a)γ (0) = arg f z (a)γ 2 (0) ( γ ) (0) γ 2 (0) f a D f (z) = u(z) + iv(z) ( f / z)(a) = 0 f a (d f ) a (.6) (d f ) a = ( f / z)(a) (dz) a (u,v)/ (x,y) 0 (d f ) a 0 f a ( f / z)(a) 0 c = ( f / z)(a) ( f / z)(a) a γ, γ 2 arg ( ( f γ2 ) ) (0) ( f γ ) (0) f = arg z (a)γ 2 (0) + f z (a)γ 2 (0) f z (a)γ (0) + f = arg z (a)γ (0) ( γ 2 (0) + cγ 2 (0) ) γ (0) + cγ (0) arg(γ 2 (0)/γ (0)) γ, γ 2 γ (0) =, γ 2 (0) = eiθ (θ R) ( e arg(e iθ iθ ( + ce 2iθ ) ) ) = arg + c arg( + ce 2iθ ) = arg( + c) θ c = 0 ( f / z)(a) = 0 a f D, D 2 C F F 2.. z = (w i)/(w + i) H = {w C Imw > 0} B = {z C z < } z 2 = w i w + i 2 = 4Imw w + i 2

24 2_ 8 γ f γ 2 f H B γ w = i z + z w =, 0,, z =,, i, i 2_3 f γ i 0 i D C f z z 2 D f (z ) f (z 2 ) f f f (D) C f : D f (D) 2.2. D C n D D D Aut(D) 2.3. () D = B { Aut(B) = w = f (z) = e iθ z α αz } θ R, α B. Schwarz α B w = f α (z) = (z α)/( αz) w 2 = ( z 2 )( α 2 ) αz 2 > 0 f α B B z = f α (w) = (w + α)/( + αz) f α B B f Aut(B) f (α) = 0 f f α Aut(B) f f (0) = 0 Schwarz f f α (w) = e iθ w f (z) = e iθ f α (z) = e iθ z α αz. (2) D = C Aut(C) = {w = az + b a, b C, a 0}. (3) D = H { az + b Aut(H) = cz + d 2.3 (2), (3) α } a, b, c, d R, ad bc > 0.

25 (). D 0 C n f 0 D 0 D /0 D C n f f D 0 D f 0 f f 0 D f f 0 D 2, D 3,..., D m f 2, f 3,..., f m D j D j+ /0 f j+ f j D j+ f 2, f 3,..., f m f 0 f 0, f,..., f m f 0 D 0 f 0 D = D 0 D D m f D 0 D m /0 f 0 D0 D m f m D0 D m f f D = D 0 D D m 3.2. z C \ {0} 2 D 0 = {re iθ 0 < θ < π} f 0 (z) = r e iθ/2 f 0 { D = re iθ π 2 < θ < 3π }, f (z) = r e iθ/2, 2 D 2 = {re iθ π < θ < 2π}, f 2 (z) = r e iθ/2, { D 3 = re iθ 3π 2 < θ < 5π }, f 3 (z) = r e iθ/2, 2 D 4 = {re iθ 2π < θ < 3π}, f 4 (z) = r e iθ/2.

26 D 4 D 0 f 4 f 0 f 4 = f 0 f (). () f 0, f,..., f m f 0 f 0 / z k, f / z k,..., f m / z k f 0 / z k (2) D j D j+ /0 D 0, D,..., D m D 0 l f 0,..., f l0 D 0, D,..., D m f i0, f i,..., f im (i =, 2,..., l) U j ( j = 0,,..., m) C l ( f j,..., f l j ) U j F 0 U 0 F 0,..., F m U 0,..., U m F 0 ( f 0 (z),..., f l0 (z)), F ( f (z),..., f l (z)),..., F m ( f m (z),..., f lm (z)) F 0 ( f 0 (z),..., f l0 (z)) D 0, D..., D m F 0 ( f 0 (z),..., f l0 (z)) 0 F m ( f m (z),..., f lm (z)) z = e w w = logz z = re iθ θ π < θ < π, 0 < θ < 2π{z = re iθ r > 0, θ ()} log z + iθ logz logz logz 3.3 () logz (logz) = /z. 3.3 (2) z = 0 logz 3.5. α C z α = e α logz z α z = 0 z α z = 0 (ze 2πi ) α = z α e 2απi z(z α ) = αz α () α R \ Q mα (m Z \ {0}) e 2mαπi. z α (2) α Q α = q/pp, q e 2m(q/p)πi m p z α = z q/p p z = 0 z q/p (3) α R (), (2) z α = e α logz = e Re(α logz) =

27 3.2 2 e α Re(logz) = e α log z = z α 0, α > 0, lim z 0 zα =, α = 0,, α < f (z) = a ν,...,ν n (z a ) ν (z n a n ) ν n (3.) ν,..., ν n (a;r) C n f (a;r) z (a;r) f z z f (a;r) f (a;r) (a;r ) a Taylor (.2) (3.).8 (3.) (a;r ) 3.7 (Vivanti). 0 f (z) = a n z n n=0 (a n 0) z = f 3.8 (H. Cartan Thullen). D C n K D ρ = K, D = inf{ z z z K,z D} a D D f f (a) sup f (z) (3.2) z K D (a;ρ) f D a f a ν ν n (z a ) ν (z n a n ) ν n (3.3) n (ρ,...,ρ) ρ

28 (.2) a ν ν n = ν + +ν n f ν! ν n! z ν (a) zν n n ν! ν n! sup ν + +ν n f z K z ν (z) zν n. (3.4) n z K (z ;ρ ε) D K ρ ε = {z C n z,k ρ ε} D, M ε = sup f (z) Cauchy.4 (3.4) z K ρ ε M (ρ ε) ν (ρ ε) ν n (3.3) (a;ρ ε) ε 0 f (a;ρ) D f (3.2) a ˆK K 3.3 D C n f f D D f D f D C n D K ˆK D D U U D 3.8 D K ˆK, D K, D ˆK =

29 (Hadamard ). f (z) = ν=0 a ν z n ν, lim n ν+ /n ν > ν f 3.. f (z) = z ν! f ν=0 lim (ν +)!/ν! = Hadamard 3.0 ν p f (re iq/p ) = (re iq/p ) ν! + r ν! (r ) ν=0 ν=p 3.2. (a;r) C n a = 0 f (z) = z ν! ν=0 f ( z r ) + f ( z2 (0;r) r 2 ) + + f ( zn r n ) 3.3. B = {z C n z z n 2 < } K B r = sup z z n 2 (< ) z K ˆK {z B z z n 2 r 2 } = B r B z B \ B r B f f (z 0 ) > sup f (z) z K 3.4 (Hartogs ). K (0;r) C n D K = (0;r) \ K n > f D K (0;r) D K R.E. Greene & S.G. Krantz, Function Theory of One Complex Variable, 3rd ed., American Mathematical Society Theorem 9.2. Ostrovski f (z) = a n z n {n ν } lim n ν+ /n ν > n=0 ν n ν ϕ ν (z) = a n z n f (z) f (z) n=0 {ϕ ν } ν

30 r = (r,...,r n ) r j < r j r = (r,...,r n) K (0;r ) f (0;r) \ (0;r ) r j < r j < r j r (0;r ) f (z,...,z n ) = (2πi) n ζ =r ζ n =r n f (ζ,...,ζ n )dζ dζ n (ζ z ) (ζ n z n ) r < z < r Cauchy f = f z f f (0;r ) \ K f f f (0;r) 3.5. n 2 f a a / f \ {a} 3.4Hartogs 3.6. D C C D C D {z m } m =, 2,... z m D D ζ m {a m } m =, 2,... a m < () f (z) D (2) {ζ m } m =, 2,... D f (z) = m= a m z ζ m (3) lim z ζ m, z z m ζ m f (z) = z m ζ m z m ζ m f D (), (2), (3) m= (2) (2 ) D U C D U C z m ζ m U z m ζ m ζ m C C D U ζ C D U C (ζ ;2ε) U ε > 0 (ζ ;ε) C z m z m D ζ m (ζ ;2ε) (ζ ;2ε) ζ z m ζ m U z m ζ m ζ m D U C

31 () K D K, D = d K z ζ m d f (z) K (2) ζ D ζ ε (ζ ;ε) D z m z m ζ m ε ζ ζ m ζ z m + z m ζ m ε + ε 2ε. (3) m N a n < a m N z z m ζ m, z ζ m n>n f (z) = an z ζ n ζ n =ζ m a n z ζ m a m z ζ m z ζ m ζ n ζ m n>n,ζ n ζ m ( a m n>n a n z ζ n a n z ζ n ) a n n N n N,ζ n ζ m a n z ζ n. a n z ζ n z m ζ m z z ζ m z ζ n z ζ m 3.4 D C n a D {(U, f ) U a f U }. (3.5) {(U, f )} (U, f ) (V,g) a W U V f W = g W (3.6) 7_section_3 U f a W g V 8_ z n

32 {(U, f )}/ O a a O a (U, f ) f a O 0 C{z,...,z n } O = O a D a D O f O a O f (U, f ) N(U, f ) = { f b b U} O N(U, f ) f O O Hausdorff p : O D f a a p O. Forster, Lectures on Riemann Surfaces, Springer Verlag Chapter, 6

33 a C n f a (z)a a γ : [0,] C n f a γ f (z,t) (i) t [0,] f (z,t) γ(t) (ii) f (z,0) = f a (z). (iii) t [0,] ε s t < ε s f (z,s) f (z,t) f (z,t) f a γ γ() f (z,) f a γ 4.2. () f a γ (2) f a γ f b f b γ f a () γ 2 f (z,t), g(z,t) J = {t [0,] s t f (z,s) = g(z,s)} 0 J J /0. J [0,] J = [0,]

34 J t J f (z,t) = g(z,t) (iii) s t < ε s f (z,s) f (z,t) g(z,s) g(z,t) Taylor f (z,s) = g(z,s) J t J ε > 0 t ε < s < t s J f (z,s) = g(z,s) (iii) ε > 0 f (z,s) f (z,t) g(z,s) g(z,t) f (z,t) = g(z,t) t J (2) f a γ f (z,t) f (z, t) f b γ 4.3. f D a D f a f a D a f (z,t) f γ(t) 4.4. D 0, D,..., D m C n f 0, f,..., f m D j D j+ /0, f j+ f j f j f j+ n D j D j+ C j+ γ : [0,] D i 0 = t 0 t t 2 i= t m t m+ =, γ([t j,t j+ ]) D j, t j C j f 0 γ(0) γ f (z,) f m γ() 4.5. f (z,t) γ(t) r = (r,...,r n ) (r j > 0) f (z,t) t [0,] (γ(t);r) n = f (z,t) r(t) t t [0,] f 0 (z,t) (γ(t);r(t)) r(t) t t [0,] ε > 0 U t = {s [0,] s t < ε} s (i) f (z,s) f (z,t) ( (ii) γ(s) γ(t); r(t) ) 2

35 _5 γ(s) γ(t) (γ(t); r(t) 2 ) (γ(t);r(t)) 4_7 s γ t [0,] a (γ(t i );r(t i )) min r(t i)/2 r i k 0 t γ U t [0,] t,..., t k U t U tk = [0,] t [0,] U ti (i), (ii) f (z,t) (γ(t);r(t i )/2) b 4.6. a D C n f a a a D γ f a f a D 4_ γ, γ a b 2 H : [0,] [0,] D, (t,s) H(t,s) γ γ D γ γ γ(s) γ(t) (γ(t); r(t) 2 ) (i) H(t,0) = γ 0 (t), H(t,) = γ (t). (γ(t);r(t)) (ii) H(0,s) = a, H(,s) = b. 4_7 s γ a b 0 t 2 γ 4.8. γ, γ, γ (0) γ γ. () γ γ = γ γ. (2) γ γ, γ γ = γ γ. (3) γ γ = γ γ. (0), (), (2)

36 γ γ [γ] 4.9 (). D C n a D f a f a D a b 2 γ, γ f a b f b, f b b γ γ γ 4.7 (i), (ii) H : [0,] [0,] D H s H(,s) a b γ s γ s f a b f γs f γs s s [0,] s f γs s γ s f γs (z,t) 4.5 r f γs (z,t), 0 t (γ s (t);r) H(t,s) = γ s (t) δ > 0 s s < δ s γ s (t) γ s (t) < r 3, 0 t ( γ s (t) γ s (t); r ). 3 s γ s f γs (z,t) f γs (z,t) f γs (z,t) J = {t [0,] t t f γs (z,t ) f γs (z,t ) } J t J t [0,] U ( (i) γ s (U) γ s (t); r ) (, γ s (U) γ s (t); r ). 3 3 (ii) t U f γs (z,t ) f γs (z,t) f γs (z,t ) 4_9 f γs (z,t) γ s (U) γ s (t) γ s (t) γ s (U) 4_0 δ c a γ b

37 4.2 3 t J f γs (z,t) f γs (z,t) (γ s (t );r) (γ s (t);2r/3) (γ s (t);r/3) γ s (t ) f γs (t ) f γs (t ) J t J t t < t (i) f γs (z,t ) f γs (z,t ) (ii) f γs (z,t) f γs (z,t ) f γs (z,t) f γs (z,t ) t t f γs (t) f γs (t) t J. 0 J J /0 [0,] J = [0,] s s < δ f γs (z,) f γs (z,) b 4.2 4_ γ a b δ b c γ δ a c γ s (U) δγ γ s (t) γ(2t), 0 t δγ = 2, (4.) δ(2t ), γ 2 t s (t) γ s (U) 4_0 δ c 4_4_after a γ b γ a 4.. γ γ, δ δ δγ δ γ γ 0 b [γ], [δ] [δγ] 4_ a D a γ 0 a j γ e a e a a g j [γ] = a i g i

38 γ S D γ 0 γ : S D B DB 4.3. () a b 2 γ, γ γ γ γ γ e a. (2) a γ [γ] π (D,a) π (D,a) 4_9 [δ][γ] = [δγ] (i) ([γ][γ ])[γ ] = [γ]([γ ][γ γ s (U) ]). (ii) e a γ s (t) (iii) [γ] [γ ] 4_9 γ s (t) γ γ s (U) π (D,a) a D D s (U) 2 a, b a 4_0 b γ 0 γ s (t) δ π (D,a) [γ] [γ 0 γγ0 ] π c (D,b) a γ s (t) γ γ s (U) b 4_0 4_4_after δ c γ 0 b a γ a γ b 4_4_after 4_ π (C) = {}. γ a g i γ 0 a j π (C \ {0},) = Z[γ]. γ π (C \ {a,...,a n },) = ([g ],..., [g n ] ) a 4_5 i b g j a j g j a i g i D (i) a D π (D,a) = {}. 3

39 (ii) D (iii) D 2 a, b D C n a D f a f a D () γ a 0 f a γ a f γ f 0 (ii) D z D f a z D f f a 4.9 D f a z γ [γ] z f [γ] f a D f f f z f 4.7. () π (D,a) (2) D 2 a, b a D f A [γ] π (D,a) A A ρ [γ] ρ [γ] : A f j ( f j γ ) A (4.2) ρ [γ] [γ] ρ : π (D,a) S(A) A, [γ] ρ [γ] (4.3) π (D,a) ρ f 4_8_after D γ 2 γ 5_8 h(d) z h

40 z /m C \ {0} m a = m z A = {e k(2π/m)i m z 0 k < m} γ π (C \ {0},) = Z[γ] ρ(z[γ]) = Z/mZ ρ [γ] : e k(2π/m)i m z e (k+)(2π/m)i m z. D C n f D z D f (z) 0 D g f (z) = e g(z) 4.9 (Cauchy). D C n g,..., g n g k / z j = g j / z k D f g j = f / z j n g,..., g n ω = g j (z)dz j ω = 0 D f ω = f j= C n g,..., g n g k / z j = g j / z k f g j = f / z j n n = n g n Hartogs f n (z) = g n (z) = ν=0 c ν (z,...,z n )zn ν ν=0 ν + c ν(z,...,z n )z ν+ n f n f n / z n = g n h j = g j f n / z j h n 0 h k / z j = h j / z k h j z n = g j z n 2 f n z n z j = g j z n g n z j = 0 h,..., h n z n f 0 f 0 / z j = h j f = f n + f (). X, X Hausdorff π : X X (X,π,X) x X U

41 (i) π (U) X U j (ii) π U j π U : U j j U π (X,π,X) a X, a X π(a ) = a a D γ a γ a X γ π γ = γ (X,π,X) X π 4.9 D O p : O D, f a a O O O }{{} n n O n = {(ϕ,...,ϕ n ) ϕ j O, p(ϕ ) = = p(ϕ n )} π : O n D { X 0 = (ϕ,...,ϕ n ) O n ϕ k = ϕ } j z j z k π X0 : X 0 D π 0 ψ : D X 0, z ((g ) z,...,(g n ) z ) O π 0 ψ = id D X = ψ(d) π 0 X : X D η : O X 0, η 4.20 (( ) ( ) ) f f f a,..., z a z n a X η (X) 4.22 η X : X X π 0 X η X : X D D π 0 X η X X D D f f

42

43 f : D D 2 f (a ) = a 2 f : π (D,a ) π (D 2,a 2 ) [γ] [ f γ] f f C n D, D 2 g : D D 2 D D 2 f f g 4.9 f g 5.. z /2 D 2 = C \ {0} 2 D = {z C Rez > 0} D D 2 z /2 D 2 z z 5.2 X Hausdorff Y 2 x, y Y d(x,y) C(X,Y ) X Y Y X X Y Y X C(X,Y ) Y X 5.2 (). K X, U Y U(K,U) = { f C(X,Y ) f (K) U} (5.)

44 K,..., K l X U,..., U l Y U(K,U ) U(K l,u l ) 5.3 (). g C(X,Y ), ε > 0 { } U(g,K,ε) = f C(X,Y ) supd(g(x), f (x)) < ε x K (5.2) g 5.4. ϕ : C(X,Y ) X Y, ( f,x) f (x) C(X,Y ) X Hausdorff ϕ : C(X,Y ) X Y, ( f,x) f (x) C(X,Y ) O U Y, K X U = U(K,U) O f U { f } K ϕ (U) C(X,Y ) f V V K ϕ (U) V U 5.5 (Ascoli Arzelà ). X Hausdorff Y C(X,Y ) F C(X,Y ) F (i) x X F(x) = { f (x) f F} Y (ii) x X F F x ε > 0 x U X f F f (U) f (x) ε 5.6. Hausdorff X, Y F C(X,Y ) F F X σ C(X,Y ) F F f, f 2... f ν, f ν2,... X

45 (Montel ). D C n F D K K M > 0 f F sup f (x) < M F C(D,C) x K Ascoli Arzelà 5.5 (i), (ii) (i) F (ii) a D (a;2r) D r M > 0 z (a;2r), f F f (z) M Cauchy.4 z (a;r), f F f (z) z j M r j f (z) f (a) = d f (a +t(z a))dt 0 dt = n M r j = nm r j j= 0 n f (a +t(z a))(z j a j )dt z j j= F 5.8 (H. Cartan). D C n 0 D, ϕ = (ϕ,...,ϕ n ) Aut(D), ϕ(0) = 0 ϕ = id D ϕ ϕ (0) (0) z z n..... ϕ n ϕ n (0) (0) z z n (5.3) ϕ id D ϕ = (ϕ,...,ϕ n ) 0 j ϕ j (z) = z j + p (l) j (z) + p (l+) j (z) +, p (l) j (z) 0 p (k) j (z) z k l 2 ϕ m = ϕ ϕ : D D }{{} m (ϕ m ) j = z j + mp (l) j (z) +

46 {(ϕ m ) j } m=,2,... Montel 5.7 D Weierstraß.6p (l) j (z) 0 p (l) j (z) 5.3 Riemann 5.9 (Riemann ). D C C = {w C w < } z 0 D f : D (i) f : D (ii) f (z 0 ) = 0, f (z 0 ) > 0. D f f 5. D Liouville.5D C Step. g : D D g(z) < g(z 0 ) = 0, g (z 0 ) > 0 Step 2. F = {g : D U g(z) < ( z D), g(z 0 ) = 0, g (z 0 ) > 0} (5.4) Step f g F ϕ = g f :, ϕ(0) = 0 Schwarz ϕ (0) ϕ (0) = c = c ϕ(w) = cw (5.5) c = ϕ (0) ϕ (w) = g ( f (w)) f ( f (w)) ϕ (0) = g (z 0 ) f (z 0 ). (5.6) g (z 0 ) f (z 0 ) g(z) = f (z) s = sup g (z 0 ) f f F f (z 0 ) = s g F f f (z 0 ) = s f F

47 5.3 Riemann 4 Step 3. Step 2 f f (D) = f 4_8_after D = D Step. D C z C \ D (z z ) /2 C \ {z } 2 D C \ {z } 2 h(z) γ 2 h(z) h(z) 2 = z z h(z) h(z ) = h(z 2 ) h(z ) 2 = h(z 2 ) 2 z z = z 2 z h(d) h(d) = /0 h(z ) = h(z 2 ) h(z ) 2 = h(z 2 ) 2 z = z 2 h(z ) = h(z ), h(z ) = 0 z = z D 5_8 γ h(d) z h h 0 h(d) D 6_4_ z 0 D h h(d) r h(z 0 ) ε h(d) h(d) + h(z 0 ) C r ζ 0 f ε h(z) + h(z ρ 0 ) > ε, z D. D ρ λ r ρ g(z) = ε h(z) + h(z 0 ) g : D g(z ) = g(z 2 ) h(z ) = h(z 2 ), z = z 2 ϕ : ϕ(w) = e iθ w g(z 0) g(z 0 )w ϕ g : D (ϕ g)(z 0 ) = 0 (ϕ g) (z) = e iθ g(z 0 ) 2 (4 g(z 0 )g(z)) 2 g (z) (ϕ g) (z 0 ) = e iθ g (z 0 ) g(z 0 ) 2 0 θ (ϕ g) (z 0 ) > 0 θ ϕ g F

48 Step 2. g F g (z 0 ) > 0 (z 0 ;ρ) D g(z 0 + ρz) 0 0 Schwarz d dz f (z 0 + ρz). z=0 ρ f (z 0 ) f (z 0 ) /ρ s = sup f (z 0 ) F f, f 2, f 3,... lim f f F n n(z 0 ) = s { f n } Montel 5.7 { f nν } D f Weierstraß.6{ f n ν } f f (z 0 ) = s > 0 f f nν (D) f nν (z) < f (z). z D f (z) =.3 f (z) f (z) <. f f F f 5.0 (Hurwitz ). D C { f n } n =, 2,... f () D C Jordan γ 0 γ D f a γ f a n f n (z) D a (2) { f n } f 5.0 () a = 0 inf f (z) > 0 n z γ D D C C γ ζ C \ γ n(γ,ζ ) = 2πi γ dz z ζ L. Ahlfors, Complex Analysis, 3rd ed., McGraw Hill Chapter 4, 2, Lemma γ ζ n(γ,ζ ) ζ C \ γ γ Jordan ζ γ D Cauchy n(γ,ζ ) = ζ ζ ζ n(γ,ζ ) = 0 ζ C \ D 4.2 γ e a H(t,s) γ s (t) = H(t,s) γ 0 = γ 0 < s < γ s C H(t,s) n(γ,ζ ) = 0 n(γ s,ζ ) s n(γ,ζ ) = 0 ζ D D D (2) γ D () (2)

49 5.3 Riemann 43 inf f (z) > sup z γ f n (z) f (z) z γ z γ f (z) > f n (z) f (z) (5.7) f (z) f n (z) D g t (z) = f (z) +t( f n (z) f (z)) g 0 (z) = f (z), g (z) = f n (z) (5.7) g t (z) γ D g t dg t 2πi γ g t t t = 0 t = (2) 2 z, z 2 D f (z ) = f (z 2 ) = a z, z 2 D, D 2 () n f n D, D 2 a f n 5.9 Step 3. f (D) = f (D) w 0 \ f (D) g : D \ {0}, g(z) = f (z) w 0 w 0 f (z) g(z) /2 D 2 h(z) h(z) h(z) f (z) = e iθ h(z) h(z 0) h(z 0 )h(z) D z 0 D 0 e iθ f (z 0 ) > 0 f F f (z) = e iθ h(z 0 ) 2 ( h(z 0 )h(z)) 2 h (z) f (z 0 ) = e iθ h (z 0 ) h(z 0 ) 2. 2h(z)h (z) = ( w 0 2 ) f (z) ( w 0 f (z)) 2 2h(z 0 )h (z 0 ) = ( w 0 2 ) f (z 0 ) = ( h(z 0 ) 4 ) f (z 0 ). f (z 0 ) = e iθ + h(z 0) 2 2h(z 0 ) f (z 0 ) > f (z 0 ) γ (5.7) f f n D Rouché

50 h(z 0 ) < f f, g 2 g f Schwarz D D f f (z) /2 D 2 (5.8) D Riemann D 5.. D C Riemann P P \ D D (5.8) D Riemann D 5.4 Vitali 5.2 (Vitali ). D C n A D D f A 0 D 0 D { f ν } A D n = A D.9 A D { f ν } D K D δ > 0, { f ν } { f νk }, { f µk } k z k K f νk (z k ) f µk (z k ) δ Montel 5.7 { f νk }, { f µk } f νk f, f µk g {z k } K K {z k } z 0 K f (z 0 ) g(z 0 ) δ f, g D A f, g

51 5.4 Vitali 45 A D f = g 5.3. f (z) = a n z n n f n (z) = a k z k { z = } k=0 k n=0 {z f n (z) = 0} { z = } f n ζ ζ U n U f n (z) 0 U { z < } U 0 U 0 f n f Hurwitz 5.0 U 0 f (z) 0 ϕ n (z) = n f n (z) U () {ϕ n } (2) z U 0 ϕ n (z) Vitali 5.2 ϕ n (z) U z 0 > U ε > 0 n ϕ n (z 0 ) < + ε f n (z 0 ) < ( + ε) n a n z n 0 = f n (z 0 ) f n (z 0 ) ( + ε) n + ( + ε) n 2( + ε) n. a n 2( + ε) n / z 0 n a n /n 2 /n ( + ε)/ z 0 lim a n /n + ε n z 0. z 0 > ε lim n a n /n < (), (2) () lim n a n /n = a n /n K > a n < K n+ R U (0;R) f n (z) = a 0 + a z + + a n z n K + K 2 R + + K n+ R n (n + )K n+ R n ϕ n (z) (n + ) /n K +/n R (2) z U 0 n f (z) f n (z)/ f (z)

52

53 C Jordan γ D Jordan C \ γ 2 γ p. 2 D P \ D = γ D { } D P 5. D 6.2 (Carathéodory ). D, D 2 C Jordan f : D D 2 f D D 2 f : D D 2 f D D 2 γ C f (z) Jordan γ D C ζ 0 D f D \ {ζ 0 } f (ζ ) M (ζ D \ {ζ 0 }) f (z) M (z D) D R D (ζ 0 ;R). g(z) = (z ζ 0 )/R D 0 < g(z) <. D g(z) /m (m N) D

54 g m (z) 6_4_2 f m (z) = g m (z) f (z) D D \ {ζ 0 } z D \ {ζ 0 } ζ 0 g m (z) 0, f (z) f m (z) 0 f m (ζ 0 ) = 0 f m D f m (ζ ) M (ζ D) f.3 z D f m (z) M. m f (z) f (z) M (z D) C rn a n rn ζ 0 b n α n λ rn D 6.4. Riemann P Jordan D f (z) D \ { } f (ζ ) M 6_4_3 (ζ D \ { }) f (z) M (z D) ζ 0 f 6.5 (Lindelöf). D C C Jordan γ ζ 0 γ γ ζ 0 γ, γ 2 f (z) D 6_7 D\{ζ 0 } a C ζ γ lim f (ζ ) a m, ζ ζ 0, ζ γ lim f (ζ ) a m ζ ζ 0, ζ γ 2 (6.) lim f (z) a m. (6.2) z ζ 0, z D ζ 0 γ2 f a f a = 0 D (ζ 0 ;R) D \ {ζ 0 } f (z) K K D \ {ζ 0 } (ζ 0 ;R) \ {ζ 0 } ( ) R w = ilog z ζ 0 log z, z 2 D ilog(r/(z ζ 0 )) = ilog(r/(z 2 ζ 0 )) e R/(z ζ 0 ) = R/(z 2 ζ 0 ) z = z 2 D D γ, γ 2 γ, γ 2 5 ( ) ( ) R ilog = i log R z ζ 0 z ζ 0 + iarg R = ilog R z ζ 0 z ζ 0 arg R z ζ 0 z ζ < R R/(z ζ 0 ) > D γ, γ 2

55 ϕ(w) = f (z(w)) ϕ D λ > 0 Φ(w) = wϕ(w) w + iλ D w D ϕ(w) K, w < w + iλ Φ(w) = w ϕ(w) K. w + iλ Φ ε > 0 l > 0 γ γ 2 Imw > l w Φ(w) ϕ(w) m + ε λ > 0 γ Φ(w) w w + iλ K m. 6.4 w D Φ(w) m+ε. w D, Imw > l ϕ(w) = w + iλ w Φ(w) w + iλ w (m + ε). ε lim f (z) = lim z ζ 0, z D w, w D ϕ(w) m lim f (ζ ) = a, i =, 2 = lim f (z) = a. (6.3) ζ ζ 0, ζ γ j z ζ 0, z D 6.7 (Lindelöf). 6.5 lim f (ζ ) = a, ζ ζ 0, ζ γ lim f (ζ ) = b (6.4) ζ ζ 0, ζ γ 2 a = b lim f (z) = a. (6.5) z ζ 0, z D F(z) = ( f (z) a)( f (z) b) F D lim F(ζ ) = 0. ζ ζ 0,ζ γ j

56 lim F(z) = 0 (6.6) z ζ 0, z D 6_9_ (a a =, b b =, ε =.4,, 0.4 ) (w a)(w b) = ε (w a)(w b) = ε ε > a b 2 4 a b ε = a b 2 4 6_9_ (a =, b =, ε =.4,, 0.4 ) ε < a b 2 4 (w a)(w b) = ε 6_9_2 a b 2 ε a, b 2ε > δ γ f (γ ) 4, δ 2 ε r > 0 a γ (ζ 0 ;r) f (α) α, β α a b 2 a ε = b 4 γ 0 (i) α γ, β γ 2 f (α) δ f (β) δ 2 α, β a b 2 b ε < (ζ 0 ;r) D β γ 0 D 4 f (β) (ii) (ζ 0 ;r) D F(z) γ 2 < ε. 6_9_2 f (γ 2 ) 6 α γ D f (γ ) a f (α) D γ 0 a β D b b f (β) γ 2 6 f (γ 0 ) (w a)(w b) < ε f (α), f (β) δ, δ 2 f (γ 0 ) a = b D (6.5) 6.6 a b D f (γ 2 ) 6

57 4_8_after D 6.3 γ 5 γ D ζ 0 5_8 z ζ 0 f (z) ζ 0 w 0 f (ζ 0 ) = w 0 f h(d) γ C 6.2 D D 2 z D h 0 ζ 0 (ζ 0 ;ρ) = ρ h f ( ρ ) = D ρ D ρ D ρ D ρ 0 (ρ 0) ε > 0 h(d) ρ > 0 D ρ < ε D r < ρ (ζ 0 ;r) C r f λ r 6_4_ r C r ζ 0 f ρ D ρ λ r ρ λ r L r L r = f (z) r dθ. C r λ r C ρ 0 ρ L r dr = dr f (z) r dθ = f (z) r dr dθ 0 C r ρ ( ) /2 ( ) /2 f (z) 2 r dr dθ 2 r dr dθ = D ρ ρ ρ ρ ε πρ 2 = ρ πε. 4 Schwarz 0 < r < ρ L r πε r L r ε n 0 (n ) r n 0 (n ) 0 L rn πε n 0, L rn 0 C rn a n, b n C rn z a n, z b n

58 w = f (z) λ rn = f crn α n, β n λ rn L rn 6_4_2 C rn a n ζ 0 f α n D rn β n rn b n λ rn 6_4_3 α n β n α n = β n λ rn Jordan ζ 0 D rn f D rn D rn f (w) <, w D rn α n = β n λ rn 2 w f (w) 2 a n, b n f Lindelöf 6.7 ζ n λ rn L rn 0 α n β n L rn 0, α n β n 0 D α n, β n (α n β n ) (α n β n ) 0 6_7 (α n β n ) 0 n ν (ν =, 2,...) c > 0 γ (α nν β nν ) c D n ν {α nν }, {β nν } α, β D (α nν β nν ) c (αβ) c > 0 ζ 0 α β α nν β nν α β 0 D rn = λ rn (α n β n ) n 0 D rn 0 D rn w 0 w 0 D w 0 w 0. w 0 f (ζ 0 ) = w 0 f : D f f (ζ 0 ) = w 0, ζ 0 w 0 U V U w 0 V n D rn V f f ( rn ) D rn V U. f 2 ζ 0, ζ f (ζ 0 ) = f (ζ ) ζ 0 ζ f Jordan f Lindelöf ζ 0 = ζ γ2 D rn D D Jordan γ γ 5 D D λ rn p

59 _4_3 ζ 0 f ζ 6_7 f D w 0 D D γ {w n } w n = f (z n ) {z n } ζ 0 f ζ 0 f (ζ 0 ) = lim f (z n ) = lim wγ2 n = w 0 n n f : D f 6.2 Jordan D D Jordan D Riemann 5.9 f : D 6.2 f : D f D D f D, D 2 C Jordan γ, γ 2 γ, γ 2 3 z, z 2, z 3 γ, w, w 2, w 3 γ 2 f : D D 2 f (z j ) = w j ( j =, 2, 3)

60

61 55 7 6_9_ (a =, b =, ε =.4,, 0.4 ) (w a)(w b) = ε 6 ε > a b 2 4 a b ε = a b Schwarz 6_9_2 γ f (γ ) 7. (Schwarz ). D C {z Imz > 0} a D f (α) (a,b) R α f D D (a,b) (a,b) f D (a,b) D g(z) 6 D γ 0 { β f (z) z D (a,b), g(z) γ= 2 f (z) z D (a,b). f (γ 2 ) ε < b f (β) a b 2 4 (7.) D a D b g f (z) D = {z C z D} z = x + iy f (z) = u(x,y) + iv(x,y) g(z) = f (z) = u(x, y) iv(x, y) x, y Cauchy Riemann ( f / z)(z) = 0 6 z g(z) = ( ) z f (z) = z f (z) = 0 D {z z D}

62 D = D (a,b) D 6 2 C Jordan γ g(z)dz = 0 Morera γ g(z) γ γ g(z)dz = γ a γ γ 2 g(z)dz = g(z)dz + g(z)dz γ γ γ 2 γ 6 3 γ a b q γ 2 δ g(z)dz = (g(x) g(x + iδ))dx + i (g(q + iy) g(p + iy))dy. γ p 0 p, q δ 6 3 7_ a p δ γ q b b R a 6 2 p δ q b a ρ γ 7_ γ 2 δ γ 2 a R a ρ γ 2 b a C Laurent n= a n (z a) n = a n (z a) n a n + (z a) n (7.2) a p q b n=0 n= R 2 7_ /(z a) /ρ R ρ 0 ρ < R ρ < z a < R Laurent ρ < z a < R R 7 δ a ρ

63 Laurent a n (z a) n, n= n= b n (z a) n (7.3) ρ < z a < R a n = b n (n Z). Laurent f ρ < r < R f (z) dz = 2πi z =r (z a) m+ 2πi = 2πi = 2πi = a n. n= n= n= a n z =r (z a) n m dz 2π a n r n m e i(n m )θ ire iθ dθ 0 2π ir n m a n e i(n m)θ dθ 0 a n f 7.3. f (z) ρ < z a < R Laurent f (z) = n= ρ < r < R r a n (z a) n, ρ < z a < R. (7.4) a n = f (z) dz (7.5) 2πi z a =r (z a) n+ (7.4) f (z) Laurent a = 0 ρ < ρ < z < R < R z Cauchy f (z) = f (ζ ) 2πi ζ =R ζ z dζ f (ζ ) 2πi ζ =ρ ζ z dζ. f (ζ ) 2πi z =R ζ z dζ = 2πi z n f (ζ ) ζ =R ζ n+ dζ = a n z n. n=0 n=0 a n r Cauchy

64 ζ = ρ < z ζ z = z ζ /z = n= ζ n z n = n= z n ζ n+. z ζ = ρ f (ζ )/(ζ z) f (ζ ) 2πi ζ =ρ ζ z dζ = 2πi n= z n f (ζ ) dζ = ζ =ρ ζ n+ n= a n z n. 0 < z a < R f (z) a f 0 < z a < R f (z) = a n (z a) n a n + (z a) n (7.6) n=0 Laurent 2 Laurent () a n (n =, 2,...) 0 a f (z) = a 0 f (z) z < R (2) f a n=0 n= k f (z) = a n (z a) n a n + (z a) n, a k 0 (7.7) n= k ( ) k f (z) = (z a) k a n (z a) n+k + a n (z a) k n n=0 n= (7.8) = (z a) k a n k (z a) n n=0 g(z) = a n k (z a) n a n=0 g(a) = a k 0 / f (z) = (z a) k /g(z) a a k (3) f a f z > ρ f (/ζ ) ζ = 0 f

65 (Riemann ). f 0 < z a < R a Laurent (7.5) a n = f (ζ ) dζ 2πi z a =r (ζ a) n+ 0 < z a < R f (z) < M a n 2π z a =r f (ζ ) dζ rn+ 2π 2π 0 M r n dθ = M r n. n =, 2,... a n Mr n r 0 a n = f 0 < z a < R f a f (z) (z a) (7.9) f a k (7.8) f (z) = (z a) k g(z), g(z) g(a) 0, f (z) f (z) (z a) Riemann 7.4/ f (z) a a k / f (z) = (z a) k h(z), h(z) h(a) 0, f (z) = /(z a) k h(z) /h(z) a f a k 7.6. f (z), g(z) z a < R g(a) = 0 g(z) 0 f (z)/g(z) z = a z = a f (z)/g(z) z = a g(z) n

66 n 2 a C n 3.4 a O a Taylor O a C{z a,...,z n a n } a = 0 O 0 O n f O n 0 0 f (0) O n f, g O n f g = 0 f g 0 f, g f, g f g = 0 U ( f g) U 0 U f U 0 U f 0 U f 0 g U 0 g U 0 f, g O n f g h O n g = f h f g f g g f u O n = uv v O n 0 = u(0)v(0) u(0) 0 u O n u(0) 0 (7.0) f, g O n f = ugu f g f g f O n 0 f f f f, g,... O n O n f O n f = up ν pν k k u i j p i p j. (7.) 2 f = up ν pν k k = vqµ qµ l l k = l q,..., q l p j q j, ν j = µ j ( j =,..., k)

67 7.3 6 O n 7.7 O n (UFD) WeierstraßO n Noether 7.8 (Weierstraß). f O n k u O f (z,0,...,0) = z k u(z ) g O n k g(z) = q(z) f (z) + z j h j(z 2,...,z n ) (7.2) q O n h 0,..., h k O n j=0

68

69 Weierstraß O n O n [z n ] 0 C n O n [z n ] O n (z,...,z n,z n ) z (z,...,z n ) z O 0 = C n 2 n = 8.. h O n [z n ] z n k Weierstraßh h(z,z n ) = z k n + a (z )z k n + + a k (z ), a j (0) = 0 ( j =,..., k) (8.) h(0,z n ) = z k n f O n f (0) = 0 z n (regular) f (0,z n ) 0 Weierstraß 8.2 (Weierstraß). f (0) = 0 f O n z n f (0,z n ) = z k ng(z n ) g(0) 0 z n = 0 f (0,z n ) k k Weierstraß h O n [z n ] u O n f (z) = u(z)h(z,z n ) (8.2)

70 f a W f (z) (0;r) s n > 0 f (0,z n ) g V z n s n 0 inf f (0,z n ) > 0 f (z,z n ) z z n =s n 8_ z n ε > 0 s > 0 z n sup f (z,z n ) f (0,z n ) < ε z n = s n, z < s s n z sup f (z,z n ) f (0,z n ) < inf f (0,z n ) z n = s n, z < s z n =s n Rouché p. 43 z z s f (z,z n ) k ϕ (z ),..., ϕ k (z ) a l (z ) ϕ,..., ϕ k l h(z,z n ) = k l= (z n ϕ l (z )) = z k n a (z )z k n + + ( ) k a k (z ) (8.3) τ r (z ) = k l= ϕ l (z ) r = z r n 2πi z n =s n f (z,z n ) z n f (z dz n.,z n ) 8 z j ( j =,..., n ) 0 z a l (z ) τ (z ),..., τ k (z ) z h Weierstraßa l (0) ϕ (0) = 0,..., ϕ k (0) = 0 0 h (8.3) ζ = s n h(z,ζ ) 0 u(z,z n ) = f (z,ζ ) 2πi ζ =s n h(z,ζ ) dζ (8.4) ζ z n z < s, z n < s n z z n < s n h(z,z n ) f (z,z n )/h(z,z n ) Cauchy u(z,z n ) = f (z,z n )/h(z,z n ) f (z,z n ) = u(z,z n )h(z,z n ) (8.3) u(z,z n ) = f (z,z n )/h(z,z n )

71 8.2 Weierstraß 65 f = uh = ũ h z z n < s n f (z,z n ) h(z,z n ) h(z,z n ) k h(z,z n ), h(z,z n ) h(z,z n ) = h(z,z n ) u(z,z n ) = ũ(z,z n ) h = h, u = ũ. () f { f = 0} Weierstraß 9. (2) k = f (0) = 0 f ( f / z n )(0) 0 f z n regular of order Weierstraß f (z,z n ) = () (z n a (z )) f (z,z n ) = 0 z n = a (z ) (3) f (0) = 0 f 0 L f L 0 L z n z = C z f (C z) z n 8.2 Weierstraß 8.3 (Weierstraß). h(z,z n ) O n [z n ] k Weierstraß f O n f = g h + r (g O n, r O n [z n ], degr < k). (8.5) f, g (0;r) r s h(z,z n ) z n = s n, z s 0 f (z,z n )/h(z,z n ) g(z,z n ) = f (z,ζ ) 2πi ζ =s n h(z,ζ ) dζ. ζ z n (8.4) z n < s n, z < s f gh r r(z) = f (z) g(z)h(z) = f (z,ζ ) dζ h(z,z n ) 2πi ζ =s n ζ z n 2πi ζ =s n h(z,ζ ) = h(z,ζ ) h(z,z n ) f (z,ζ ) 2πi ζ =s n (ζ z n ) h(z,ζ ) dζ. f (z,z n ) ζ z n dζ

72 (h(z,ζ ) h(z,z n ))/(ζ z n ) z n k k z l nb l (z,ζ )b l (z,ζ ) l=0 r(z) = k 2πi l=0 z l n ζ =s n b l (z,ζ ) f (z,ζ ) h(z,ζ ) dζ l z f = gh + r = g h + r (g g)h = r r k z k r r = 0 g = g. 8.3 O n UFD Noether f O n g, g 2 O n f = g g 2 O n [z n ] 8.4. f O n [z n ] O n [z n ] O n [z n ] g, g 2 O n [z n ] f = g g Weierstraß h O n [z n ] () h O n h. (2) h h Weierstraß k Weierstraß h O n h = g g 2 z k n = h(0,z n ) = g (0,z n )g 2 (0,z n ) g, g 2 z n Weierstraß 8.2 g i = u i h i u i h i Weierstraß h = u u 2 h h 2 h h 2 Weierstraß 8.2 h = h h 2 () = (2) () = h = g g 2 g i O n [z n ], g i g, g 2 O n z k n = h(0,z n ) = g (0,z n )g 2 (0,z n ) g (0,z n ) = z l na (z n ), g 2 (0,z n ) = z m n a 2 (z n ) k = l + m, a i (0) 0 g O n g (0) 0 l = 0, m = k

73 8.3 O n UFD Noether 67 degg 2 k, degg 0 k degg = 0 g (z,z n ) = a(z ) g (0) = a(0) 0 a(z ) O n g O n [z n ] 8.6. O n UFD n n = 0 O 0 = C O n UFD Gauß O n [z n ] UFD 0 f O n z n Weierstraß 8.2Weierstraß h u f = uh h O n [z n ] O n [z n ] 8.5 (2) h = h h m h l O n [z n ] Weierstraß f = uh h m 8.5 () h l O n f = ũ h h l ũ h l h,..., h m, h,..., h l z n Weierstraß h h m = ũ u h h m h h m, h h m WeierstraßWeierstraß 8.2 ũ/u, h h m = h h m O n [z n ] UFD h l, h l O n 8.5 () O n [z n ] 2 O n [z n ] O n h l, h l Weierstraß h,..., h m h,..., h m 8.7. O n Noether O n n n = 0 O 0 = C OK. n O n Noether Hilbert O n [z n ] Noether 0 a O n 0 f a a f z n Weierstraß 8.2 f

74 Weierstraß h h a a O n [z n ] O n [z n ] O n [z n ] Noether g,..., g m O n [z n ] m a O n [z n ] = O n [z n ]g l l= a O n h, g,..., g m g a Weierstraß 8.3 g = g 0 h + r g g 0 h = r a O n [z n ]. g g 0 h g g 0 h = f g + + f m g m ( f,..., f m O n [z n ] O n ) g = g 0 h + f g + + f m g m (h,g,...,g m )

75 Riemann 9_2_ 9.. D C n X D y D y 2 x(x )(x 2)=0 x D U D U f ( 0) X U = {z U f (z) = 0} f X x O 2 x 9.2. C 2 2 x, y (x ) y (y ) y 2 x(x )(x 2)=0 9_2_ 9_2_2 y y 2 x(x y )(x 2)=0 y 2 x 3 = 0 O 2 x O 2 x O x 2 9_2_2 9_2_3 y y y 2 x 3 = 0 y 2 x 3 = 0 y y 2 x 3 x 2 = 0 O O x x O x 3 9_2_3 y 2 x 3 x 2 C y 2 = 0 y C y 2 n x 3 x 2 = 0 C n Cousin. O x O x

76 X x f x O n f = f f m x U X i = {z U f i (z) = 0} X U = X X m (9.) X x X Weierstraß 8.2 (3) (z,...,z n ) = (z,z n ) f i (z) = u i (z)h i (z,z n )u i h i Weierstraß X i x {h i (z,z n ) = 0} x U U = U U (U C n, U C) h i U δ i (z ) U 0 π : X i U U 0 Y i = π (U \ {δ i = 0}) U \ {δ i = 0} U Y i Y i X i U x X U X WeierstraßU U U X z = c (c U ) L c x U X L c = {z = c } 9_2_after z n L c x z 9_4 L c \X X c U U \X U 9.3. X D D \ X D D 9_7_before \ X D D \ X D \ X D 2 x, y D γ γ V ( f ) γ(t) U t U t \ X {U t } 0 t γ 0 = t 0 < t < < t N = U tk U tk (k =,..., N) Sing( f )

77 9. Riemann 7 U tk U k U k U k X x k x 0 = x, x N+ = y k = 0,,..., N x k, x k+ U k \ X U k \ X x k x k+ x = x 0 y = x N+ D \ X D \ X R n D X D \ X 9.4. D C n X X x f O x x U X U = { f = 0} U g g X U 0 O x f g Weierstraß 8.2 f k Weierstraß Weierstraß 8.3 g = f q + r, (r O n [z n ], degr < k) f X U g X U 0 r X U 0 {δ(z ) 0} z f (z,z n ) k 9.3 r X U 0. f g D C n f V ( f ) = {z D f (z) = 0} 9.5. f, g D C n {x D f x, g x O x } x D f x, g x O x V ( f ) V (g) x 9.4 V ( f ) V (g) x U n 2 f, g U x U f, g h O x h f, h g V ( f ) V (g) V (h) n 2 n 9.6 (Riemann ). D C n X D f D \ X x D U D f U\X D f f = f D\X

78 f X f D f x X U D f f U\X = f U\X x = 0 Weierstraß h X U = {h(z,z n ) = 0} 9_2_after z n Weierstraß 8.2s, s n > 0 z s z h(z,z n ) z n < s n k z n = s n f (z,z n ) = f (z,ζ )dζ x 2πi z (9.2) ζ =s n ζ z n f z < s, z n < s n z f (z,z n ) 7.4 z n s (9.2) 9_4 (z,z n ) U \ X f (z,z n ) X L c 9_7_before D V ( f ) X D C n n 2 f D \ X D f f = f Sing( D\X f ) 9.2 D C n D D C n 0 E D D \ E f z D z D A (nowhere dense) A A

79 U D U p, q f (z) = p(z)/q(z) (z U \ E) O x p, q p, q O x p, q 9.5 U x U p, q p, q x U p, q, U D f Sing( f ) D f E f 9_2_after z n 9.9. Sing( f ) U = {z U q(z) = 0} L c z U, q(z) 0 z q 0 z f p/q x z q(z) = 0 z f q f = p O z q p q p, q 9_4 Sing( f ) D Sing( f ) 2 X () z Sing( f ), p(z) 0 z f (2) z Sing( f ), p(z) = 0 z Sing( f ) n 2 V ( f ) = {z D f (z) = 0} { } = {z D p(z) = 0} (9.3) D Sing( f ) V ( f ) = { } 9_7_before D V ( f ) Sing( f )

80 D C n X D f D \ X n 2 Y X x X \Y f D lim f (z) = (9.4) z x, z D \ X x X \Y x U inf f (z) M > 0 z U\X / f (z) U \ X / f (z) /M Riemann sup z U\X 9.6 / f (z) U f (z) D \Y f D 9.3 D C 9.8 D {a n } n= D D \ {a n} n= a n Laurent {a n } n= a n (a n ;ε) a n ε > 0 Res( f,a n ) = f (z)dz (9.5) 2πi z a n =ε f a n a n f Laurent /(z a n ) f D γ D C Jordan D γ 9.. f D a,..., a k f (z)dz = 2πi γ k Res( f,a i ). (9.6) i= Levi H. Grauert & R. Remmert, Coherent Analytic Sheaves, Springer Verlag Chapter 9, 5 9. f

81 9.4 Mittag-Leffler f D N P N P = f (z) 2πi γ f (z) dz = d arg f (z). (9.7) 2π γ 9.3. f D a,..., a N, b,..., b P g(z) D N P g(a i ) i= i= g(b i ) = 2πi γ g(z) f (z) dz. (9.8) f (z) 9.4 Mittag-Leffler. D C () D {a n } n= a n (2) a n /(z a n ) A (n) ν f n (z) = (z a ν= n ) ν. D \ {a n } n= f a n Laurent f n 9.4 (Mittag-Leffler ). D C dist(a n, D) = r n c n a n = r n c n D f n (z) C \ {a n } z c n > r n c n B ν Laurent (z c ν=0 n ) ν g n (i) f n (z) g n (z) < ε n (z C \ (c n ;2r n )), (ii) g n (z) C \ {c n } ε n > 0 ε n < n= ( f n (z) g n (z)) = ϕ(z) n= (2) f n f

82 ϕ D \ {a n } n= f n(z) g n (z) D \ {a n } a D \ {a n } n= a D \ {a n} n= /2 {a n } n= z c n 2r n /2 n N 2 {z z c n > 2r n } (n N). ( f n (z) g n (z)) f n (z) g n (z) n=n n=n n=n ( ε n z ). 2 ( f n g n ) /2 ϕ(z) n=n D \ {a n } n= ϕ(z) ( f k (z) g k (z)) = ( f n (z) g n (z)) a k g k (z) a k ϕ(z) a k Laurent f k D = C 0 {a n } n= {a n} n= a n (n ) f n (z) (0; a n ) f n Taylor g n ( f n (z) g n (z) < ε n (z 0; a )) n 2 n k D C ϕ(z) = ( f n (z) g n (z)) C \ {a n } n= ϕ(z) a n Laurent f n (z) n= 9.5. f D {a n } n= D f f a n f n D g g n (n =, 2,...) f (z) = g(z) + ( f n (z) g n (z)) (9.9) n= D D 2 D U U

83 9.5 Weierstraß 77 Mittag-Leffler 9.4 ϕ(z) = ( f n (z) g n (z)) f (z) ϕ(z) = g(z) n= sinz = ( z + ( ) n z nπ + ) = z nπ + 2z ( ) n z 2 n 2 π 2, n Z\{0} n= ( ) cosz = + ( ) n z (2n )π/2 +, (2n )π/2 n= cotz = ( z + z nπ + ) = z nπ + 2z z 2 n 2 π 2, n Z\{0} n= ( ) tanz = z (2n )π/2 +, (2n )π/2 n= cosec 2 z = (z nπ) 2, sec 2 z = n= n= (z (2n )π/2) Weierstraß. {a n } n= C C {a n } n= {a n} n= a = a n n k a k 9.7 (Weierstraß). () (2) {a n } n= f ( f (z) = z h e g(z) z ) e a z + ( ) ( ) z z λn n 2 an 2+ + λ n an (9.0) a a n 0 n g(z) C h a n = 0 n λ n ( ) R λn + R > 0 < a a n 0 n λ n = n

84 () a n = 0 n ( ϕ(z) = z ) e a z + ( ) ( ) z z λn n 2 an 2+ + λ n an (9.) a n= n E(z,λ) = ( z)e z+ 2 z2 + + λ zλ z < loge(z,λ) loge(z,λ) = k= k zk + z + 2 z2 + + λ zλ = k=λ+ k zk. loge(z,λ) k=λ+ z k = z λ z. z < /2 loge(z,λ) 2 z λ+ (9.) z < R n N R a n /2 N n N { ( log z ) e a z + ( ) ( ) } z z λn ( ) n 2 an 2+ + λ n an = z loge,λ n 2 z a n a n a n λ n + (9.2) λ n z < R 2 z λ n + a n=n n (9.2) ( z ) e a z + ( ) ( ) z z λn n 2 an 2+ + λ n an a n=n n z < R 0 N ( z ) e a z + ( ) ( ) z z λn n 2 an 2+ + λ n an a n n= (9.) ϕ(z) z < R a n < R a n R > 0 ϕ(z) (2) () 0 g e g(z) 9.8 (Weierstraß). sinz = z n Z\{0} ( z )e nπ z = z nπ n= ) ( z2 n 2 π 2.

85 9.5 Weierstraß f C f C 2 f {a n } n=, {b n} n= a n 0 f (z) = z k e g(z) b n 0 ( z a n ( z b n ) e a z ( ) ( ) z z λn + n an 2+ + λ n an ) e ( ) ( z z z b + n bn 2+ + µn bn ) µn (9.3)

86

87 Γ 0.. C {Rez > 0} Γ(z) = e t t z dt (0.) 0 t z = e (z )logt e (z )logt = e (Rez )logt = t Rez Γ(z) {Rez > 0} z F n (z) = e t t z (logt) n dt (0.2) 0 (0.2) 0 {a Rez}, {Rez b} (0.) Γ (n) (z) = F n (z) Γ(z) Rez > 0 [ ] Γ(z + ) = e t t z dt = e t t z + z e t t z dt = zγ(z) (0.3) Γ(z) = Γ(z + ) z (Rez > ) (0.4) Rez > Γ(z) = Γ(z + k) z(z + ) (z + k ) (Rez > k) (0.5) Γ Euler I IV 5.

88 Γ z = 0,, 2,... Γ(z) Rez > 0 Γ(z) = e t t z dt + e t t z dt 0 }{{} n=0 z C ( ) n = t n+z dt + e t t z dt n! 0 n=0 ( ) n = n!(z + n) + e t t z dt (0.6) (0.6) C \{0,, 2,...} Weierstraß.6.9 (0.6) C\{0,, 2,...} Res(Γ, n) = ( ) n /n! 0.2 (Gauß). n!n z Γ(z) = lim n z(z + ) (z + n). (0.7) Rez > 0 (0.) Γ(z) = lim n n 0 = lim n 0 = lim n z n = lim n n z n = lim n n z n! e t t z dt = lim n ( s) n n z s z ds { [s ] z ( s)n z { [ s z+ [ n 0 + n 0 ( t n) n t z dt s z ( s)n z(z + ) s z+n z(z + ) (z + n) n I n = e t t z dt 0 n 0 0 ] z ( s)n ds ] 0 + (n ) 0 ( t n) n t z dt = } 0 } s z+ z(z + ) ( s)n 2 ds n!n z = lim n z(z + ) (z + n). n 0 { ( e t t ) n } t z dt n 0 t n 0 e t ( t/n) n t 2 /2n f (t) = e t ( t n ) n Prym

89 0. Γ 83 ( f (t) = e t n) t n ( et n n) t n ( ) n = e t ( t n 0 t n f (t) 0 ) n ( t n ) = tet n ( t n) n 0 t ( 0 f (t) = e s s ) n s t 0 n n ds s et 0 n ds = et t2 2n n ( I n e t t n t 0 n) z dt a ( = e t t n n t 0 n) x ( dt + e t t n t a n) x dt a t x+ 0 2n dt + e t t x dt = ax+2 a 2n(x + 2) + e t t x dt. a ε > 0 a z e t t x dt < ε a I n < ax+2 2n(x + 2) + ε. n a x+2 /2n(x + 2) 0 lim n I n = 0. z Rez > 0 (0.5) Γ(z) = Γ(z + k) z(z + ) (z + k ) = z(z + ) (z + k ) lim n (z + k)(z + k + ) (z + k + n) n!n z+k n!n z = lim n z(z + ) (z + n) lim n k n (z + n + )(z + n + 2) (z + n + k) n!n z = lim n z(z + ) (z + n) 0.3 (Weierstraß). Γ(z) = zeγz n= ( ) γ Euler lim n k logn k= ( + z ) e n z. (0.8) n

90 Gauß 0.2 Γ(z) = lim z(z + ) (z + n) n z = lim n z z n n! n = lim n e zlogn ze ( nk= k ) z ( = lim e z nk= k logn) z n n k= n k= ( + z ) e k z k ( + z ) e k z. k n k= ( + z ) k k ( log + z ) e z ( k = log + z ) z k k k ( = z ) 2 ( z ) 3 + z/k 2 2 k 3 k z/k 2 z 2 k 0.4 (). Γ(z)Γ( z) = π sinπz. (0.9) Γ(z)Γ( z) = Γ(z)Γ( z) ( z) = = z n= ( z2 n 2 ze γz ) = sinπz π. n= ( + z n )e z n ( z)e γz z n= ( z ) e n z n Γ ( ) = 2 e t2 dt = π. (0.0) Γ(/2) 2 = π ( ) Γ = e t dt = 2 e s2 ds > t 0 (0.0)

91 0. Γ Γ(z) = C 0_6 e 2πiz e ζ ζ z dζ (z Z). (0.) C argζ = 0 ζ = ε argζ = 2π 0 I(z) = e ζ ζ z dζ z C C t I(z) = (e 2πiz )Γ(z) (Rez > 0) z d.9 (0.) { O R ε } I(z) = lim e t t z dt + e ζ ζ z dζ + e t (e 2πi t) z dt. ε 0 ζ =ε ε Γ(z), e 2πiz Γ(z) e ζ ζ z dζ e ζ ε x dζ τ ζ =ε ζ =ε 2π = e ε cosθ γ ε x dθ e ε ε x 2π 0 (ε 0) (Hankel ). Γ(z) = 2isinπz C e ζ ( ζ ) z dζ (z Z). (0.2) 0.8. Γ(z) = e ζ ( ζ ) z dζ. (0.3) 2πi C Hankel Γ( z) = e ζ ( ζ ) z Γ(z)Γ( z) dζ = e ζ ( ζ ) z dζ. 2isinπz C 2iπ C Γ( z) (0.3) z Z

92 Stirling Γ z = z = {a < argz < b} ( C \ {0}) 0.9. f (z) {a < argz < b} r a k f (z) z = n N zk/r k=0 lim z, a < argz < b zn/r ( f (z) n a k z k/r k=0 ) = 0 (0.4) f (z) k=0 a k z k/r (0.5) (0.4) f (z) = n k=0 a k z k/r + o ( ) z n/r (0.6) 0.0. F(τ) { ε < argτ < ε} τ = 0 F(τ) = a n τ n/r r. (0.7) n= K, b > 0 τ F(τ) < Ke b τ F Laplace argz < π/2 f f (z) = e zτ F(τ)dτ (0.8) 0 f (z) n= ( n ) a n Γ z n/r (0.9) r

93 0.2 Stirling 87 0 ( n e zτ a k τ )dτ k/r = k= n a k e zτ τ k/r dτ = k= 0 n ( ) k a k Γ z k/r r k= s n (z) ε > 0 lim z, argz < π/2 ε zn/r f (z) s n (z) = 0 C τ = 0 n F(τ) a k τ k/r C τ (n+)/r k= τ n F(τ) a k τ k/r Ce b τ. k= a > 0 z= x + iy ( a ) f (z) s n (z) C e τx τ (n+)/r dτ + e τx e bτ dτ. 0 a ( ) e τx τ (n+)/r n + dτ = 0 x (n+)/r Γ r ( ) z n/r z n/r z n/r f (z) s n (z) C + x (n+)/r x b e a(x b). z argz < π/2 ε z x 0 2 I k = e zτ τ k/r dτ zτ 0 τ e τ τ k/r z k/r dτ τ = zt, 0 t I k = e τ τ k/r z k/r dτ = Γ(k/r)z k/r 0

94 (Stirling ). Rez > 0 Γ(z) Γ(z) 2π z ( z e) z k=0 a k z k. (0.20) 0_6 a 0 =, a = /2,... Rez > 0 z Γ(z + ) ( z e) z = e t t z dt 0 e zlogz z = 0 e w(t) dt. w(t) = t zlogt (z zlogz) ( ( t z w(t) = z log + t z )) 0. z z argζ = 0 t τ = w/z s = (t z)/z τ = s log( + s) s = 0 t = z ζ = ε 0_6 argζ = 2π ζ = ε O τ = s2 2 s3 3 + s4 4 z argζ = 0 argζ = 2π R d z d t γ τ O R 0 2 t τ lim e t t z dt = 0 C : t = zλ, R d 0 < λ < τ Γ(z + ) zτ dt γ (z/e) z = e γ dτ γ dτ = z zτ ds e dτ dτ. γ + 0 < s < < s τ = s log( + s) s = f (τ), s = f 2 (τ) τ = z ( z ) z a k Γ(z) 2π e z k. k=0 (0.20)

95 0.2 Stirling 89 f (τ) = ( ) k b k τ k/2, f 2 (τ) = b k τ k/2 k= k= τ ds/dτ 0.0 ( z e ) z Γ(z + ) z = e zτ d f 0 ( = e zτ 0 k= dτ dτ + 0 e zτ d f 2 dτ dτ ( k 2 ( )k b k + k 2 b k ( = e zτ (2k + )b 2k+ τ )dτ k /2 0 k=0 (2k + )b 2k+ Γ(k + /2) k=0 ( b Γ(/2) z z (2k+)/2 + 3b ) 3Γ(3/2) z 3/2 +. )τ k/2 ) Γ(z+)/z Γ(z) b = 2, b 2 = 2/3, b 3 = 2/8,... 2 a 0 =, a = /2 dτ

96

97 Cousin Mittag-Leffler 9.4Weierstraß 9.7 Cousin.. D C n U = {U i } i I D U i ϕ i {ϕ i } i I 0 {ϕ i } i I Cousin i, j (ϕ i ϕ j ) Ui U j U i U j {ϕ i } i I Cousin i, j (ϕ i /ϕ j ) Ui U j U i U j (Cousin ). {ϕ i } i I D ϕ i I ϕ Ui ϕ i U i (Cousin ). {ϕ i } i I D ϕ i I ϕ Ui /ϕ i, ϕ i /ϕ Ui U i U i U j h i j {h i j } i, j I Cousin D C D D n (D j C) D j I

98 {ϕ i } i I i, j I h i j = (ϕ i ϕ j ) Ui U j (.) {h i j } i, j I {h i j } i, j I (h i j + h jk + h ki ) Ui U j U k = 0 ( i, j, k I) (.2) {h i j } i, j I C Z (U,O) {h i j } i, j I 0 { f i } i I h i j = ( f i f j ) Ui U j ( i, j I) (.3) {h i j } i, j I {h i j } i, j I = { f i } i I C 0 (U,O) Z (U,O) H (U,O) = Z (U,O)/ C 0 (U,O) (.4) O U Čech.2. {ϕ i } i I Cousin {ϕ i } i I Cousin h i j = ϕ i ϕ j {h i j } i, j I H (U,O) D ϕ ϕ i ϕ U i ϕ i ϕ Ui = f i { f i } i I 0 h i j = ϕ i ϕ j = (ϕ i ϕ) (ϕ j ϕ) = f i f j {h i j } i, j I = { f i } i I {h i j } i, j I = { f i } i I i, j U i U j ϕ i ϕ j = h i j = f i f j U i U j ϕ i f i = ϕ j f j ϕ = ϕ i f i U i D ϕ ϕ i ϕ Ui = f i U i ϕ Cousin

99 .2 Dolbeault 93.2 Dolbeault D C n D C C (D) D C C (D) dx,..., dx n, dy,..., dy n E (D) dz j = dx j + idy j, dz j = dx j idy j (.5) n n E (D) = C (D)dz j + C (D)dz j (.6) j= j= E (D) C (D) E (D) E (D) = C (D)dz j dz jp dz k dz kq (.7) E p,q (D) = j < < j p, k < < k q, 0 p,q n j < < j p, k < < k q C (D)dz j dz jp dz k dz kq (.8) (p,q) E (D) = 0 p,q n E p,q (D) (.9) α E p,q (D), β E p,q (D) α β E p+p,q+q (D) β α = ( ) (p+q)(p +q ) α β E p,0 (D) dz j dz jp p d : E (D) E (D) α = α jk dz j dz jp dz k dz kq (.0) dα = dα jk dz j dz jp dz k dz kq, (.) dα jk = n i= α jk z i dz i + n i= α jk z i dz i (.2)

100 α = i, j, k ( αjk z i ) dz i dz j dz k, α = i, j, k ( αjk d = + dd = 0 z i ) dz i dz j dz k. (.3) 0 = ( + )( + ) = 2 + ( + )+ 2 (.4) α E p,q (D) 2 α E p+2,q (D), ( + )α E p+,q+ (D), 2 α E p,q+2 (D) 2 = 0 E p,0 (D) E p, (D) 2 = 0, + = 0, 2 = 0 (.5) E p,2 (D) E p,q (D) (.6) α E p,q (D) α = 0 α α = β β α α H p,q (D) = { (p,q) }/{ (p,q) } (.7) Dolbeault.3 Dolbeault.4 (Grothendieck). D C C Jordan γ f D C f (z) = f (ζ ) 2πi γ ζ z dζ + f 2πi D ζ = f (ζ ) 2πi γ ζ z dζ + f 2πi D ζ dζ dζ ζ z dζ dζ ζ z. (.8) z D (z;r) D r γ r = (z;r), D r = D \ (z;r) Stokes γ f (ζ ) ζ z dζ ( ) f (ζ ) d ζ z dζ = f dζ dζ ζ ζ z. f (ζ ) γ r ζ z dζ = ( f (ζ ) d D r ) ζ z dζ f dζ dζ = D r ζ ζ z.

101 .3 Dolbeault 95 f (ζ )dζ r 2πi f (z) f dζ dζ γ ζ z ζ ζ z r 0 D f ζ dζ dζ ζ z = γ f (ζ )dζ ζ z 2πi f (z) D 2.5. D, γ, f D C g, h g h (z) = f (z), z (z) = f (z) (.9) z f w,..., w k f (z,w,...,w k ) f w C g, h w g, h w C g(z) = 2πi D f (ζ ) dζ dζ ζ z g g/ z = f f w C g w C z D (z;r) D r Stokes D r d ( ) f (ζ )log ζ z 2 dζ = f (ζ )log ζ z 2 dζ f (ζ )log ζ z 2 dζ (.20) γ γ r () = ( f (ζ )log ζ z 2) dζ D r = D r f ζ log ζ z 2 dζ dζ + D r f (ζ ) dζ dζ ζ z. r 0 (.20) 2 ζ = z + re iθ 2π lim f (ζ )log ζ z 2 dζ = lim r 0 γ r r 0 0 f (z + re iθ ) }{{} r 0 (logr 2 ) ( i)re iθ dθ = 0. z

102 f (ζ )log ζ z 2 dζ = γ / z = D D f (ζ ) dζ γ ζ z = D f ζ log ζ z 2 dζ dζ + D f ζ log ζ z 2 dζ dζ + 2πig(z) f dζ dζ ζ ζ z + 2πi g z. dζ dζ f (ζ ) ζ z.4 g/ z = f g/ z C g C.6 (Dolbeault ). (0;r) C n ω (0;r) (p,q) (q ) ω = 0 (0;r) (p,q ) η ω = η (0;r) η ω = η ω = α J,K dz J dz K J, K α J,K 0 J, K K ν α J,K 0 ω = 0 ν = 0 ν ν = 0 OK. ν ν ω ω = α dz ν + β α, β dz ν,..., dz n 0 = ω = α dz ν + β. k = ν +,..., n dz k α dz ν + β = 0 z k z k α z k = 0 (k = ν +,..., n). α/ z k α / z k (p,q )

103 .3 Dolbeault 97 α z ν+,..., z n α = α j,..., j p,k,...,k q (z)dz j dz jp dz k dz kq α j,..., j p,k,...,k q (z) z ν+,..., z n C.5 α j,..., j p,k,...,k q (z,...,z ν,z ν+,...,z n ) = z ν γ j,..., j p,k,...,k q (z,...,z ν,z ν+,...,z n ) γ j,..., j p,k,...,k q z ν+,..., z n C γ = γ j,..., j p,k,...,k q (z)dz j dz jp dz k dz kq γ = ν i= J, K z i γ J,K dz i dz J dz K = dz ν α + δ J = ( j,..., j p ), K = (k,...,k q ) δ dz ν,..., dz n ω = α dz ν + β = ( ) p+q γ + β ( ) p+q δ β ( ) p+q δ dz ν,..., dz n β ( ) p+q δ = ε ω = (( ) p+q γ + ε).7. C n () H p,0 ( ) p (2) H p,q ( ) = 0 (q > 0) () H p,0 ( ) = { (p,0) } = {ω E p,0 ( ) ω = 0} ω E p,0 ( ) α J ω = α J dz J ω = dz k dz J ω = 0 J, k z k J, k α J z k = 0 α J (2) q 2 2 ν = ν ν