a(z) = k 0 1 z k = k 0 2 k z k = k 0 z k = (1 + z) n. k

Transcript

1 !" #$%% $&$'$ # %( $)%*&%' '+ &'&%

2 ! " " # $ " " % " & ' # () *+ (, *,-.$ / " " " * $ 0 * " # " $ * $ 0

3 # % " & ', # ' * # " & #! " # %& *%& $ % & ' " ( z D log! ) * (% % (+, ) " " -. // 0 ', % 0 ', %

4 ,, ( 0 * " 0 / ', % 0 0 /./!- ' % " & 0 % / % " &, % 0 / & ' " & - -&

5 = 0,,,... (" ) z & (z). = 0 + z + z +... = k 0 k z k. ' $ " % * ( ),,,... % " & (z) = k 0 z k = k 0 ( z < ) ( ),, 4, 8,... % " & (z) = k 0 ( z < ) ( ) ( n ) ( 0, n k z k = k 0 ),..., ( n n) % " & z k = z (z) k = z (z) = n k=0 ( ) n z k = ( + z) n. k

6 ( / " & $ " " ( " "" ) % & " ( ) (z) % & $ (z) = ( k 0 kz k 0 < z < r, r > 0 ) # " (' k & = k! (k) (0) (k) k ") z k & (z) # [z k ](z) = k $ " βk! z k [β k z k ](z) = β k k $ βk " z % / # f 0 = 0, f =, f k = f k + f k, k ' f0, f, f % " &,... f(z) = k 0 f k z k = 0 + z + k = 0 + z + k = z + k = z + z k 0 f k z k (f k + f k )z k f k z k + k f k z k + z k 0 = z + zf(z) + z f(z) / f k z k f k z k ' # f(z) = ϕ, ˆϕ ' ϕ, ˆϕ { ϕˆϕ z z z = = ϕ + ˆϕ =, z (α z)(ˆα z) = z ( ϕz)( ˆϕz)

7 ( / ϕ = ϕ ϕ ϕ = 0 { ϕ = + 5 ˆϕ = 5 $ (( $ ) " " $ A B ' A B A = B = z f(z) = ( ϕz)( ˆϕz) A ϕz + B ˆϕz. lim z ϕ lim z ˆϕ z ˆϕz z ϕz = = /ϕ ˆϕ/ϕ = ϕ ˆϕ = 5 /ˆϕ ϕ/ˆϕ = ˆϕ ϕ = 5 " & " f(z) = ( 5 ϕz ) ( ˆϕz = ϕ k z k ) ˆϕ k z k 5 k 0 k 0 = k 0 5 (ϕ k ˆϕ k )z k, ( f k = 5 ( ϕ k ˆϕ k), k 0. (' & " ) $ R(z) = P(z) Q(z) P(z) Q(z) z deg P < deg Q Q(z) = (z q ) d (z q ) d... (z q m ) dm q,..., q m " # Q(z) " ci $ R(z) = ( m di ) c ij. (z q i ) j i= j=

8 ( / ( ) ( z)z = 4 z + z + z ( ) z(+z ) = z(z i)(z+i) = z z i z+i = z z z + % " & # " "!"## $%% &'( (z) = k z k % & # k 0 k " z k )) k 0 r = sup{ρ : k 0 k z k z z < ρ} = inf{ z : k 0 k z k z } r = λ λ = lim sup k " /k k % z k k k 0 λ k z k k 0 k z k r > 0 # " k 0 ɛ > 0 ) k k ( ) ( ) k ( + ɛ r ( ) k ( ɛ r ) k k

9 ( / / ˆϕ + /ϕ # " & f(z) = k 0 f k z k = z z z = z ( ϕz)( ˆϕz), ϕ = ( + 5)/ $ ˆϕ = (! " 5) f(z) z = 0.68 z =.68 % " & ϕ ˆϕ r = /ϕ ( fk (( + ɛ)ϕ) k k # f k (( ɛ) ϕ) k k $& $%# $%$ &'( P(z) R(z) = $ deg P < deg Q $ & Q(z) Q(z) = (z q ) d (z q ) d... (z q m ) dm, qi " # qi R di " )! 0,,,... % " & (z) & $ " " q,..., q m " d,..., d m # k = m ( ) k A i (k), i= Ai (k) d i " (* " " " ) Ai = q i

10 ( / 0 # z f(z) = ( ϕz)( ˆϕz) = z (z )(z ). ϕ ˆϕ ( $ ϕˆϕ = ) ( fk = ϕ k + $ ˆϕ k, " " ' $ f 0 = 0 f " " = = 5 = 5

11 " " &, $ % " & " ) $ " # " ' = { 0,,,... i, i = 0,,,... } 0,,... + b 0, b,... = 0 + b 0, + b,... 0,,... b 0, b,... = 0 b 0, 0 b + b 0, 0 b + b + b 0,... }{{} ' " " k k + b k = k +b k k b k = j b k j j=0, ( $, +,,, ) = 0, 0, 0,... # =, 0, 0,... 0,,,... " # 0 X 0 + X + X +... $! k 0 kx k X k " " & $ " # # " " " # $ #, X ( X, +,,, ) ( ) ) ( % [X] $ " " )

12 ( / '' ( 0 *,,, ' * / $ " " $ " " X $ X $ X $," % X / % " X, 0, 0,... = X 0.," $ \ {0} $ X 0 =, 0, 0,... X 0 =, 0, 0,... X 0 X 0 = * " A = 0,,... X B = b 0, b,... $ " 0 0 ( # B = A ) $ # 0 0 B = b k " 0 b 0 = b 0 = 0 0 b + b 0 = 0 b = 0 b 0 0 b + b + b 0 = 0 b = 0 ( b + b 0 ) 0 b k + k j= jb k j b k = 0 k j= jb k j / $ $ " 0 = 0 0 b 0 = b0 ( X) " () {}}{ 0 ( X) =, {}}{, 0, 0,... () ( b0 = 0 = k b k = 0 ( b k + b k k b 0 ), 0 = $ = k = 0, k $ b k = ( b k ) = b k () ( X) =,,,... = k 0 X k (" % ")

13 ( / '' ( 0 *,,, ' # n 0 ( X) n (( X) ) n # " ( X) n = ( + X + X +...)( + X + X +...)...( + X + X +...) }{{ } n = ( ) n + k X k = ( ) n + k X k. k n k 0 k 0, " & ", " " % & " 0 F, G X $ " $ " G F 0 = G FG F = & k 0 kx " k F = DF (k + ) k+ X k k 0 k F k Xk k " " % # " " ' " ) " " $ & "% " $ ( " ) " F(Y ) = n Y n G(X) = + X # " n 0 F(G(X)) " F(G(X))! = F( + X)! = n ( + X) n n 0 = n 0 n ( k 0 ( ) n )X k k = k 0 ( ( ) n ) n X k. k n 0 ' X k " & $ " # n 0 ( n k ) n

14 ( / '' ( 0 *,,, ' # F0, F, F,... " $ " Fn = n k n 0 F n ( ) k n k $ n nk (n) k = 0. $ Fn k = (n) k n F0, F, F $,... Fn " $ $ ( ) D F n n 0 ( ) F n n 0 = n 0 DF n = n 0 * " $ F(G(X))! " $ G(X) 0 G(X) = b X + b X +... ( " G(0) = 0!)$ G 0 =, G, G, G 3,... " ( ) ( ) F(G(X)) = k 0 F n. k (G(X)) k, F(Y ) = k 0 k Y k. / F(G(X)) " # k 0 " k $ F " F G $ F(G(X)) = G(F(X)) = X F = k kx k $ 0 = 0 0 # G = n b nx n b0 = 0 b 0 $ F(G(X)) = X ' G = F [ ], G " # F(G(X)) = X k k ( j b j X j ) k [X ] b = b =

15 ( / '' ( 0 *,,, ' " [X n ] $ n > n b n + b n = k= n k= k k ( n k= j,...,j k, j +...+j k =n j,...,j k, j +...+j k =n k j,...,j k, j +...+j k =n b j... b jk = 0 b j... b jk = 0 b j... b jk ). (( " 0) % G = F [ ] ' Exp(X) = n 0 n! Xn =,,, 6,.... # F(X) = Exp(X) ( $ G(X) Ln( + X) $ Ln( + X) = ( ) n+ X n. n n ' " " r $ ( + X) r k 0 ( ) r X k, k ( r k) " ( r k ) = k! r (r ) (r )... (r k + ). & "! " % #

16 ( / '' ( 0 *,,, ' ( + X) r ( + X) s = ( k 0 = n 0 k=0 ( r + s n = }{{} n 0 ( ) )( r X k k j 0 ( n = ( + X) r+s. ( ) s )X j j ( )( ) ) r s X n k n k ) X n ( D Ln( + X) = D n ( ) n+ ) X n n = ( ) n+ X n n = ( ) n X n n 0 = ( + X).

17 " 0 f(z) = n 0, " & ( f z ) ( f : ) = 0,,,... " ( ) " % " & (%&) (z) = n 0 n z n ) % " & (%&) â(z) = n 0 n n! zn f n z n " (* )

18 ( / ((,* */,0 * (,* * *' / " %& * %& c n = n ± b n c(z) = (z) ± b(z) ĉ(z) = â(z) ± ˆb(z) n c n = k b n k c(z) = (z)b(z) k=0 n ( ) n c n = k b n k k k=0 c n = n (c 0 = 0) c(z) = z(z) ĉ(z) = (z) (0) c n = n+ c(z) = ĉ(z) = â(z)ˆb(z) â(z) dz ĉ(z) = D â(z) z 0 c n = n n c(z) = z D(z) ĉ(z) = z Dâ(z) c n = n n (z) (0) â(z) â(0) c(z) = dz ĉ(z) = dz z z # ' # fn+ f n+ f n = 0 $ f 0 = 0 $ f = ' " % " & f n f(z) f n+ f(z) z f n+ f(z) z z f(z) z f z z f = 0 z 0 f z fz fz = 0 f( z z ) = z f = z z z = z ( ϕz)( ˆϕz),

19 ( / ((,* */,0 * (,* * *' / f(z) = (ϕ n ˆϕ n ) z n, n 0 5 }{{} f n ' % " & f n ˆf(z) f n+ ˆf (z) f n+ ˆf (z) + ϕ = 5 ˆϕ = 5 ˆf ˆf ˆf = 0 ˆf = c e ϕz + c eˆϕz ˆf(0) = f 0 = 0 = c + c = 0 ˆf (0) = f = = c ϕ + c ˆϕ = { c = 5 c = 5 ˆf(z) = 5 ( e ϕz eˆϕz) = f n = [ z n n! ] ˆf(z) / ' " n 0(n + 4n + 5)/n! " " f(z) = = 5 (ϕ n ˆϕ n ). n 0 f(z) = {(z D) + 4z D +5}e z " ' " (n + 4n + 5) zn n! = z D(ze z ) + 4ze z + 5e z = z e z + ze z + 4ze z + 5e z = (z + 5z + 5)e z f() = e " f() 0 ( " π : [n] [n] ( % % )$ π(i) i i [n] d n $ π : [n] [n] " " π ( k ), n k ( ) n! = n k=0 ( ) n d n k = k n k=0 ( ) n d n k k

20 ( / ((,* */,0 * (,* * *' / 0 ' # ( ) % " &, n! zn n! n 0 z = n 0 ( n k=0 = e z ˆd(z) ˆd(z) = e z z = ( ( ) n ) z n d n k k n! ( ) k )( ) z k z m k! k 0 m 0 = ( n ( ) k ) z n k! n 0 [ z n k=0 ] d n = n! ˆd(z) = n![z n ] ˆd(z) ( = n!! +! 3! ( )n n! }{{} n e ) n! e. "," d = 0 d = π : [n] [n] $ n 3 ( ) π(n) = i $ π(i) = n $ π # [n ] \ {i} ( ) π(n) = i π (n) = j $ "! n j [n ], dn = (n )(d n +d n ) d n+ = n(d n +d n ) ' %& ˆd (z) = z ˆd (z) + z ˆd(z) = ( z) ˆd (z) = z ˆd(z) ˆd = ˆd dz = z z dz = ln ˆd(z) z = z dz = z ln( z) + C = ˆd(z) = e C e z z d = ˆd (0) = = C = 0

21 / " " " $ % " & C = (C, w) C w : C ) '! c n = w (n) = n σ C. $ cn < n " c n " # cn %& %& c(z) = n 0 " " ( ) c(z) = σ C c n z n, ĉ(z) = n 0 z w(σ), ĉ(z) = σ C c n n! zn, z w(σ) w(σ)!. $ C $ C #! C,..., C k! Φ C = (C, w) C $ Φ(C,..., C k ) Φ = (Φ e, Φ w ) Φ e C C... C k Φ w : k $ Φ e (σ) = (σ,..., σ k ) w(σ) = Φ w (w(σ ),..., w(σ k )) Φ " " % " & " " $ (%& " " & ϕ (%& ˆϕ ) $ c(z) = ϕ(c (z),..., c k (z)) ) ĉ(z) = ˆϕ(ĉ (z),..., ĉ k ))$ (z)) ) C Φ %& ) (" " %& ) ( %

22 ( / /'+ ',* /, ' / * %& %&!)$ ϕ " " %& ) (" " ˆϕ %& ) / C A + B 0 C C A B $ A B = $ & " w C (σ) = { w A (σ), σ A, w B (σ), σ B., %& %& $ A, B, C c(z) = σ C z w C(σ) = σ A " " %& ĉ(z) = â(z) + ˆb(z) C A B " C A B w C (α, β) = w A (α)+w B (β) z w A(σ) + σ B z w B(σ) = (z) + b(z) %& %& * " # # $ " % " & " " C A 0 + A + A +... i 0 A i C i=0 i$ A wc (σ) = w Ai (σ) σ A i c(z) = " i 0 i(z) n 0 [z n ] i 0 i (z) = i 0 i,n < n 0 w C (n) = w Ai (n) <. i 0 C A 0 A A... i 0 A i C i=0 A i $ wc (σ) = i 0 w Ai (σ i )

23 ( / /'+ ',* /, ' / # " C " $ σ = (σ0, σ, σ,...) w(σ i ) = 0 i 0 /! (w) Ai C i 0 c(z) = i (z) i 0 " [z n ] i 0 i(z) = k 0 ( i 0 +i +...+i k =n,i k 0 0i 0 i kik j [z j ] i (z) = ij = 0 i 0 j i 0 w A i (j) <. / %&, i 0 A i i 0 A i i (z) i 0 i (z) i 0 % (A A) (z ) " " " A ( (z)) ' µa z D (z) 0 / " A[B] (b(z)) 0 ( ) ( ) " P(A) exp j j (z j ) j * ' ( ) " M(A) exp j j (zj ) ) < ( ) ( " )

24 ( / /'+ ',* /, ' / C (A A) C A C c(z) = A {(α, α) α A}, w C (α, α) = w A (α) z w(α,α) = z w(α) = (z ) α A (α,α) C {ɛ} + A + A A +... = k 0 A k c(z) = k 0((z)) k = ( (z)). 0 = 0 C µa C n 0 {,,..., n} (A n [n]) $ A n = {α A w(α) = n} [n] = 0 & wc (α, i) = w A (α), cn = n n c(z) = z D(z) 0 " C A[B] C n 0 A n B n 0 & wc (α; β,..., β n ) = w B (β ) w B (β n ) c(z) = n $ (b(z)) n = (b(z)) n 0 b0 = 0 n = 0 n 0 C P(A) C {{α,..., α k } k 0, α,..., α k A} 0 & wc ({α,..., α n }) = w A (α ) w A (α n ) P(A) $ ( {ɛ} +{α}) α A }{{} w=0 c(z) = α A( + z w(α) ) = n 0( + z n ) n

25 ( / /'+ ',* /, ' / ln c(z) = n 0 n ln( + z n ), 0 = 0 = ( ) j n z nj j n 0 j = ( ) j n z nj j j n 0 = ( ) j (z j ) j j ( ( ) j c(z) = exp j j * C M(A) (z j )) 0 = 0 C {{α j,..., α j k k } k 0, α,..., α k A, j α! } 0 & wc ( {α j,..., α j k k } ) = j w A (α ) +...j k w A (α k ) M(A) α A{α} $ c(z) = α A( z w(α) ), w(α) 0 α A = n 0( z n ) n ( " ) =... ( ) = exp j j (zj ), 0 = 0 m [m] = {,..., m} C w=0 m {}}{ ( {ɛ} + i= w= {}}{ {i} ) / $ c{ɛ}+{i} $ (z) = + z c(z) = ( + z) m cn = ) [z n ]( + z) m = ( m n m [m] = {,..., m} C m {i} i=

26 ( / /'+ ',* /, ' / c{i} (z) = z c {i} $ (z) = ( z) c(z) = ( z) m$ ) c n = [z n ]( z) m = ( m n ) ( ) n = ( m+n n " ' " n " " {,..., m} 0 & w(k) = k " " p{k} (z) = z k p{k} (z) = ( z k ) P (m) {} {}... {m} p (m) (z) = z z... z m ' " n " " " " P {k} k * p(z) = k ( z) k p 4 = [z 4 ] p(z) = [z 4 ] ( + z + z +...) ( + z + z ) ( + z 3 + z ) ( + z 4 + z ) ( + z 5 + z ) = [z 4 ] ( + z + z + 3z 3 + 5z ) = 5. " 4 = 3 + = + = + + = T {ɛ} + { } T T t(z) = + zt(z) zt(z) t(z) + = 0 t(z) = ± 4z ; z ( 4z)/ t(0) = 0 t(z) = z t n = [z n ]t(z) = n + ( ) n, n n (( " " " " )

27 ( / /'+ ',* /, ' / *%& " $ " " w(σ) = n $ σ # $ λ : [n] [n] # " σ = $ (σ, λσ ) τ = (τ, ( % λ τ ) -) w(σ) = n w(τ) = m σ τ ((σ, τ); λ) $ λ : [n + m] [n + m] # λσ λτ! $ "" $ θσ : [n] [n + m] θ τ : [m] [n + m] Im θ σ Im θ τ = i [n] $ j [m] λ(θ σ (i)) = θ σ (λ σ (i)), γ σ τ $ w(γ) = n + m λ(θ τ (j)) = θ τ (λ τ (j)). / σ τ " σ τ # σ τ = { 3 4 5, ,,, 4 5, 5 3 4, 3 4, ,, A B A B = α A, β B α β. #! %& A B %& â(z) ˆb(z) C A B C ĉ(z) = â(z) ˆb(z) 5 }

28 ( / /'+ ',* /, ' / $ # α = (α, " $ λα ) A β = (β, λ β ) B w(α) = h w(β) = k α λ α β λβ ( h+k ) " h (α, β) λ, (α, β) $ w(α, β) = h + k $ ( h+k ) h A B z w(σ) ĉ(z) = w(σ)! σ C = ( ) w(α) + w(β) z w(α)+w(β) w(α) (w(α) + w(β))! α A β B = z w(α) w(α)! zw(β) = â(z) ˆb(z). w(β)! α A β B," %& / *%& 0, ' / A + B, i 0 A i A B A A [ ] µa A[B] â(z) + ˆb(z), i 0 âi(z) â(z)ˆb(z) ( â(z)) eâ(z) z D â(z) â(ˆb(z)) / " " / C A ( % - ) ' A k = k {}}{ A A... A %&A k = (â(z))k A {ɛ} + A + A A +... = k 0 A k / " ( %& = ( â(z)) / C A [ ] ( % - ) ' A [k] = { %&A [k] = k! %& A k = k! (â(z))k {}}{ {α,..., α k } (α,..., α k ) A k }

29 ( / /'+ ',* /, ' / A [ ] {ɛ} + A + A [] + A [3] +... = k 0 A [k] %& A = [ ] k! (â(z))k = eâ(z) k 0 %& ' $ $ & S = {[n] : n } [n] = {,..., n} w([n]) = %& w({,..., n}) = n S ŝ(z) = n 0 s n zn n! = zn n! = ez. n # S [k] " # k %&S [k] k! (ez ) k = n 0 { } n z n k n!. %& * B = S " $ [ ] = %& B = exp(e z ) = z b n $ n %& n! n 0 / ' P C p n = n! $ P ĉ(z) ˆp(z) = n 0 p n z n n! = ( z). C [ ] $ ˆp(z) = e ĉ(z) " ĉ(z) = ln( z) = ln( z) = n [ ] z n c n = ĉ(z) = n![z n ]ĉ(z) = n! n! n n zn = (n )!,

30 ( / /'+ ',* /, ' / 0 ' C [k] $ k %&C = [k] k! (ĉ(z))k = ln( ) k ' k! z [ z n ] [ ] %& n C n!, [k] k ' D " # # $ C {()} + D ĉ(z) = z + ˆd(z) ˆd(z) = ln( %& ( z) z D [ ] D = e ˆd(z) = " e z ) [ ] z $ " $ " # ' " $ I I {(), ( )} [ ] %&I = exp ( z! + z! ) = exp (z + z ). T r T T T 3... T k ( ) $ T {r} T ˆt(z) = z ˆt(z) ˆt(z) = ( 4z) t n = [ z n n! ˆt ˆt + z = 0 ] ˆt(z) = n! n ( n ) n ( " ) ( ) # T {r} T ( ) [ ] ˆt(z) = zeˆt(z) ' ˆt(z)! ((% % +, " "$ ( (% %$ ) f(z) ϕ(u) (& ) $ " ϕ(0) 0 f(z) = zϕ(f(z)) # f ϕ " " [z n ]f(z) = n [un ]ϕ(u) n.

31 ( / /'+ ',* /, ' / ( $ $ " * % %& ˆt(z) = zeˆt(z) ( " " [ z n t n = n! ] ˆt(z) = n![z n ]ˆt(z) = n! n [un ](e u ) n = (n )![u n ]e nu, e nu = (nu) $ k n tn n = (n )! k! (n )! = nn k 0 ( **)$ n ( ) n n

32 " z D log f(z) = n 0 f nz n % " & $ / fn " ( ) " " ( ) ' & zf (z) (z) = z D ln f(z) = f(z) ( ) ' & (z) ( ) (z) = ( ), # zf (z) = (z)f(z) n nf n z n = ( δ k z k)( f j z j), k j 0 k δ k z k nf n = n δ k f n k, n. k= *

33 ( / * *' / 0 '/ ' * ˆb(z) = e e z = n 0 b n n! zn (z) = z D lnˆb(z) = z D(e z ) = ze z z k+ = = k! k 0 k n n bn = n! (k )! b n = = k= z k (k )! b n k (n k)! n ( ) n b n k n k n ( ) n b k, n. k k= k= * $ " # 0 (( $ )$ & z f = n 0 f nz " ( n f z )$ " f 0 0, f(z) = z " ( z z = { f g! } : f, g z, g 0 (( " " $ (f, g) z " " (f, g) (f, g ) " fg = f g ) / z " & h(z) = h n z n, m. n=m / z h 0 " f = $ n m f nz " n fm $ & f(z) = z m $ f(z) f(z) = " n 0 f n+mz n z z f(z) g(z) = n # n 0 g nz f (z) = z m g(z) = n= m g n+m z n.

34 ( / * *' / 0 '/ ' * ( " # ( " % ) & * g 0 D f g = Dfg = fg g + f g = f g fg g, ( h(z) = h n z n Res(h(z)) = h ( n=m = 0 $ m 0 ) $ h(z) = h n $ z n hm $ & ( 0 # n=m ( ) Res(h (z)) = 0 ( ) Res(h (z)/h(z)) = m " $ (0 " ) f(z) = n f nz $ # n f0 = 0 f 0 g(u) = $ n g nu n g(f(z)) = z ( ( " ), g " ( ) g n = Res. nf n (z) " # z = g(f(z)) ( ) ( ) = D g k (f(z)) k k ( ) nf n (z), " ( ) Res nf n (z) nf n (z) = k = k = k k g k (f(z)) k f (z). k n g k (f(z)) k n f (z). k n g k Res ( (f(z)) k n f (z) ).

35 ( / * *' / 0 '/ ' * ( " (f(z)) k n f (z) = k n D( f(z) k n), k n f (z) f(z), k = n. ( " " Res ( (f(z)) k n f (z) ) = ( ) Res = n nf n (z) n g n = g n { 0, k n, k = n,

36 " f, g : [, b] $ P " [, b] = x0 < x <... < x n = b t 0,..., t n $ tk [x k, x k+ ' ] n S(P) = f(t k )(g(x k+ ) g(x k )). k=0 " A $ ɛ > 0 P ɛ (P P ɛ S(P) A < ɛ), " f ( ', % ) g " [, b] $ A = b f(t) dg(t). g(t) = t $ " ' %! $ $ $$%""&%$ / #

37 ( / 0 (, 0*', (/, t = ( &!) t ( &!) t = t ( &!) {t} = t t $ t () // b $ " f(t) dg(t) ' " $ " f g "!$ 0 k<n P 0 ( " b () ( ) (c f + c f ) dg = c ( ) f d(c g + c g ) = c () / b b g(x k+ ) g(x k ) < f dg + () b b f(t) dg(t) + dg(t) <!) $ f dg + c f dg f dg + c f dg f dg c f dg b c b f dg = c f dg. f dg$ # b b g(t) df(t) = / b g df f(t)g(t). () h : [, b] "$ " " & # b f(h(t))dg(h(t)) = h(b) h() (0) b [, b] $ f(t) dg(t). f dg g (t) " " b f(t) dg(t) = b f(t)g (t) dt.

38 ( / 0 (, 0*', (/, (), b # $ f " b b f(t) d t = f(t) dg( t ) = b f(k), k= f(k) g(k), g(k) = g(k + ) g(k). k<b " f " " b b f(t) d t = f(t) dg( t ) = b f(k), k=+ <k b " f # " f(k) g(k), g(k) = g(k) g(k ). b f( t ) dg(t) = " " " b f( t ) dg(t) = f(k) g(k) <k b f(k) g(k), k<b (*) b f(t) d t " $ %" & g(u) dh(u) = b f(t)g(t) dh(t). " f! b ' f(k) f(t) dt k<b " ', % () () " " f <k b f(k) = b f(t) d t = b f(t) dt b f(t)d{t}.

39 ( / 0 (, 0*', (/,, " % () b b f(t) d{t} + {t} df(t) = " $, b, b f " " [, b] $ " / b f(t){t}. f(k) = <k b " # b b f(t) dt + f (t){t}dt, ( ) k b f(k) = b f(t) dt + b ( f (t) {t} ) f() + f(b) dt + k<b f(k) = b f(t)dt / b f(t) + b ( f (t) {t} ) } {{ } R f # " "$ " # R $ ( ) k<b f(k) = b f(t) dt + n m= Bm " $ Bn (x) n B n (x) = ( ) n B 0 x n + 0 dt. / b B b m f (m ) (t) + ( ) n+ B n ({t}) f (n) (t) dt m! n! }{{} R n ( ) n B x n ) ( ) n B n x 0. n # Rn " n " " n " ", " " ( ) & n! %

40 ( / 0 (, 0*', (/, 0 " " ln n! = k n ln k n n ( = ln t dt + {t} ) ln + ln n dt + t n ( = n ln n n + + ln n + {t} ) dt t = n ln n n + ln n! n ln n n + ln n + = e ( ) n n n! en ( n n e e). " " * " ( ) & ln(n )! ln n ln(n )! = k<n ln k n = ln t dt + = / n m= (t ln t t) / n B n m D (m ) ln t m! / n ln t + = n ln n n + ln n + O() / n t + B ({t}) n D () ln t dt {t} {t} + 6 t dt = ln n! = n ln n n + ln n + O() = n! = Θ ( ( n n n ) e). # &%$! $ $ '# " f, g : [, b] " ', % b f dg ', " " f = f + if $ g = g + ig $ ', b ( ) f dg = b f dg b f dg + i * ( ) " ', % () (*) b f dg + b f dg.

41 ( / 0 (, 0*', (/, D " $ f(z) D & f z0 D " f f(z) f(z 0 ) (z 0 ) = lim z z0 z z 0 '" $! z z0 f $ " " D z 0 $ D f A D A * f ) z0 $ z0 # B(z0 ; r) ' z = x + iy $ f(z) = u(x, y) + iv(x, y) $ f z = x + iy $ " & u(x, y) v(x, y) (x, y) " "$ " u x = v y, u y = v x. * " $ f z $ " " z f $ z & * e z z n " " & ( f(z) < M $ z ) " $ " / ) ( ) " " γ : [, b], b 0 γ " Γ = γ([, b]) 0 $ γ() = γ(b) $ $ $ " γ " " [, b) ( "!) " $ ) $ / " % " $ " ) $ & γ " " γ

42 ( / 0 (, 0*', (/, * D $ f D & γ : [, b] $ γ([, b]) D f ) γ f b f(γ(t)) dγ(t) = γ ( $ " γ " ) '# b γ f(γ(t))γ (t) dt. f(z) dz πθ z 0 γ z0 $ f(z) = z z 0 γ : [0, ] $ γ(θ) = z 0 + e πiθ # γ dz z z 0 = = = γ (θ) dθ γ(z) z 0 e πi πiθ eπiθ dθ πi dθ = πi. f(z) = (z z 0 ) n $ n > # " " " γ (z z 0 ) n dz = 0 = πi ( e πiθ ) n πi e πiθ dθ 0 e (n )πiθ dθ / πi = e (n )πiθ (n )πi 0 = ( ( ) e πi (n ) n }{{} ) = = 0. # $ " " z0 r ( r = )

43 ( / 0 (, 0*', (/, γ : [, b] γ : [b, c] $ γ (b) = γ (b) ' γ + γ : [, c] " (γ + γ )(t) = { γ (t), t [, b] γ (t), t [b, c] ', % $ f = f + f % " γ +γ γ γ, $ $ " γ : [, b] γ : [, b] ( γ)(t) = γ( + b t) f γ f = γ % γ γ f - γ : [, b] $ L(γ) = & $ f(z) M $ z γ([, b]) # γ f M L(γ)., ', % 0 γ, γ : [, b] " )) $ D b dγ f ( ) γ() = γ() $ γ(b) = γ(b) γ() = γ(b) $ γ() = γ(b) ( ) ( ) γ([, b]) D $ γ([, b]) D " " h : [0, ] [, b] D $ h(0, t) = γ(t) t [, b] h(, t) = γ(t) t [, b] $ h(s, ) = γ() h(s, b) = γ(b) s [0, ] h(s, ) = h(s, b) s ( ) [0, ]

44 ( / 0 (, 0*', (/, γ = h h3 4 γ() γ(b) h γ = h 0 - f D $ $ " " γ γ $ " D # γ f = γ - ( -) f D γ $ D # f = 0 γ (0 ) γ = γ +γ # γ γ $ f = f + f = f f = 0. γ γ γ γ γ f γ D γ γ, $ γ D $ D (/ #! ) D $ "$ ( D!) f(z) = z γ : [0, ] $ γ(θ) = e πiθ # γ z dz = 0 = πi = πi 4πi e πiθ (πi e πiθ) dθ e 4πiθ dθ 0 / e 4πiθ 0 = (e4πi ) = 0.

45 ( / 0 (, 0*', (/, -! ( - % "!) f D γ D " z0 # γ f(z 0 ) = πi γ f(z) z z 0 dz. $ γ D z 0 " " ( $ D $ γ " ' g(z) = f(z) f(z 0 ) z z 0, z z 0 f (z 0 ), z = z 0. # # g D z0 $ ",, 0 0 = = f(z 0 ) = πi γ f(z) f(z 0 ) g(z) = dz dz γ γ z z 0 γ z z 0 f(z) dz dz f(z 0 ) γ z z 0 γ z z 0 }{{} πi f(z) z z 0 dz. * % dz * ( *) ( γ z z 0 0 ' " $ γ # n(γ, z0 $ ) γ z0 - " πi γ f(z) z z 0 dz = n(γ, z 0 )f(z 0 ).

46 ( / 0 (, 0*', (/, - $ n 0 f $ D $ γ z0 ( 0 # f (n) (z 0 ) = n! πi γ & f z0 f(z 0 ) = n 0 f(z) dz, (z z 0 ) n+ f (n) (z 0 ) (z z 0 ) n. n!, ( 0 & f z0! " # D \ {z0 } lim f(z) = m & z z0 g(z) = (z z0 ) m f(z) z 0 $ z0 & ) $ " $ " f m z 0 * ( 0 g z0 # " " g(z) = n 0 b n (z z 0 ) n, f " " f(z) = (z z 0 ) m g(z) = n (z z 0 ) n, n m n b n+m " z0 m $ " m = b 0 0 ( ) $ $ f z0 ( f z 0 ' = Res(f; z 0 ) = Res z=z 0 f(z). # Res(f; 0) [z ]f(z) " " " Res f(z) = lim (z z 0 )f(z), z=z 0 z z0 m " Res z=z 0 f(z) = lim z z0 (m )! D(m ) ((z z 0 ) m f(z)).

47 ( / 0 (, 0*', (/, " # # γ D " $ f z0 # γ z 0 γ f(z) dz = n m n = πi γ (z z 0 ) n dz = πi Res z=z 0 f(z). f D $ D - - ( -!) γ " $ & f " z,..., z k $ γ & f # γ f(z) dz = πi k Res f(z). z=z j (( ) ' γ γ! " z,..., z k " " " % " γ..., γ k j= z z z 3 γ γ " " $ f

48 ( / 0 (, 0*', (/,, 0 = γ γ f = f + f γ γ f = f γ f γ k k = πi Res f(z). z=z j j= γ k f γ " # ' γ dz 4z ' & f(z) = " z = ± 4z 4z = / z + / + z = /4 z / + /4 z + / Res f(z) = 4 $ Res f(z) = z= 4 z= γ Res = 4 Res = 4, γ ( dz 4z = πi 4 + ) = 0. 4 ' dx + x & f(z) = + z = % (z i)(z + i) " γr $ r $

49 ( / 0 (, 0*', (/, γ R i ir R i R f " ±i $ z = i %, γ R γ R dz = πi Res + z z=i dz R + z = R, dx π + x + = πi lim + z z i dr e iϕ 0 + (R e iϕ ) }{{ } dx + x = π πr R z i + z = πi = π. i R dx + x.

50 " 0 # % " & ( ) f(z) = n 0 f nz n % # $ " ( 0 - " ( ) f n = πi γ f(z) dz, zn+ γ ( ) " " % / % ( ) " $ % " & f(z) = n 0 f nz n z R z = R f " z < R z,..., z k # P,..., P k $ f n = k j= z n j P j (n) + O ( R n). 0 Pj " zj mj " deg Pj = * " " " m j P j = z j Res z=z j f(z). 0

51 ( /, '0,* '* * *(' r ( ) ρ < r " % γ = γ + γ +γ 3 + γ 4 f " z,..., z k # " ' 00 γ f(z) k dz = πi zn+ γ γ 4 γ γ 3 ρ R z f(z) Res z=z j z n+ j= f(z) = dz + γ zn+ f(z) = πi f n + γ 3 z z z 3 f(z) dz + γ zn+ n+ dz, f(z) dz + γ 3 zn+ f(z) dz γ 4 zn+ " " $ γ γ4 " ( γ = γ 4 ) " f n = = j= ( ) { k }}{ f(z) Res z=z j z + f(z) dz n+ πi zn+ k j= z =R f(z) Res z=z j z + O ( R n), n+ ( ) π πr mx z =R f(z) R n+ = mx z =R f(z) R n. "$ " O (R n ) n " " " R z n j " $ $ f(z) Res z=z z n+ j P j (n) deg P j = m j ' sn h : [n] [k] $ k n $ ' S $ %& ŝ(z) = n 0 s n n! zn

52 ( /, '0,* '* * *(' * A = {} + {, } + {,, 3} +... $ $ %& â(z) = n zn = e z n!, S " # A "$ S $ A ŝ(z) = â(z) = e. z $ $ %& ŝ(z) " " z k = ln + k πi k ln + π ln ln π z z k Res ŝ(z) = lim z=z k z zk e = lim z z z k e = z e z k =. ('" " ( # ) ( " %& ŝ(z) " " ( " ) % * " " ln < R0 < ln + 4π $ " ŝ n = s n n! = z n 0 ( {}}{ Res ŝ(z) ) + O ( ) R n 0 = z 0 z=z 0 (ln ) (n+) + O ( ) R0 n ; " " ln + 4π < R < ln + 6π $ " ŝ n = (ln ) (n+) + " ln + k π < R k < ŝ n = (ln ) (n+) + ( (ln + πi) (n+) + (ln πi) (n+)) + O ( ) R n ; k j= ln + (k + ) π ( (ln + j πi) (n+) + (ln j πi) (n+)) + O ( ) R n k. * " sn = n! ŝ n ln ( ln ) n n! 0.7 (.44)n n!

53 ( /, '0,* '* * *(' f(z) ( ) z0 $ α & g(z) = (z0 z) α f(z) z 0 0 α z0 ) ' F # " " " f n n # " n 3 " %& ˆf(z) = f n z n n! n 0 # " " # ' C $ # " %& ĉ(z) = c n z n n! n 0, " n 3 $ %& = ln = ( " ) z n zn n ĉ(z) = ( ) ln z z z ˆf(z) = eĉ(z) = e z z 4 z $ & ˆf(z)! % z = ( ( z) ˆf(z) ) 3 6 f(z) " & z0 0 f(z) = f(z0 z) $ " z0 =

54 ( /, '0,* '* * *(' # $ z = f B(0; + η) η > 0 # & g(z) = ( z) α f(z) B(0; + η) z = g(z) = k 0 g k( z) k B(; η) \ {} " f(z) = ( z) α g(z) = k 0 g k ( z) k α, f " " f n = [z n ]f(z) [z n ] g k ( z) k α k 0 = [z n ] ( ) k α g k ( z) j j k 0 j 0 = ( ) ( ) k α n g k n k 0 = ( ) n k + α g k? n k 0 ( ) & f(z) = & $ n 0 f nz n α \ {0,,,... } g(z) = ( z) α f(z) $ # B(0; + η) η > 0 g z = g(z) = g k # ( z) k m 0 k 0 { m } f n = [z n ] g k ( z) k α = m k=0 k=0 ( ) n k + α g k n " &! h(z) = f(z) m g k ( z) k α = k=0 k m+ + O (n m +α ) + O (n m +α ) g k ( z) k α, z <. # h(z) = ( z) m+ α h(z) $ & h(z) B(0; + η) ( &$ % %& - %!$ )$ [z n ]h(z) = O ( n (m+ α) ) = O (n m +α )

55 ( /, '0,* '* * *(' { m } " fn = [z n ] g k ( z) k α + [z n ] h(z) k=0, " ( & ˆf(z) = e z z 4 ( z) " m = / & ĝ(z) = ( z) ˆf(z) = e z z 4 z = ĝ(z) = n 0 ĝ (n) () ( ) n ( z) n n! = e e 3 4( z) + e 3 4 ( ( m = ) ˆf n = f n n! = e 3 4 ( ) n + e 3 4 n ( ) n 3 + e 3 4 n 4 4 ( z) +... ( ) n 5 + O ( n 7/), n ", % " " # f n n! { e nπ 8n + } 8n ( " "" ( + $, ' '" ) (, %#) & f(z) = n 0 f nz n B(0; r) $ r > 0 $ z = r & " % z,..., z k " " α,..., α k \ {0,,,... } g,..., g k B(0; r) & $ z,..., z k g j (z) = ( z z j ) αj f(z). = mx{r # (α j) j =,..., k} f n = n k j= g j (z j )n α j Γ(α j )z n j + o ( r n n ).

56 ( /, '0,* '* * *(' & f(z) = n 0 f nz n $ " ( ) fn " - % " (( 0 " ) f n = πi γ f(z) dz, zn+ γ % " γ = " ρ ( " )$ f n 0 n z = ρ f(z) n 0 mx f(z) = f(ρ) z =ρ " ( ( ) f n π f n z n = n 0 f n z n = f(ρ) mx z =ρ ) f(z) πρ z n+ = π f(ρ) f(ρ) πρ = ρn+ ρ. n / " ( ) " ρ > 0 $ " " $ " / f(ρ) $ lim = ρ 0, ρ n ρ " Df(ρ)/ρ n Df(ρ) ρ n ρf (ρ) f(ρ) = n ( ). = f (ρ) ρ n nf(ρ)ρ n = ρ n (ρf (ρ) nf(ρ)) = 0 * $ " n # ( ) " ( ) " ρ > 0 " " " f(z) = e z = n # n! zn ρf (ρ) f(ρ) = ρeρ e ρ = ρ,

57 ( /, '0,* '* * *(' " n % ρ = n " " f n = n! f(ρ) ρ n ( = en n ) n n n!. n e * $ % " ( ) & " " " " " "" ( &$ % %& - %!$ *)! ( 0) f(z) = n 0 f nz n! ( ) & ' & (ρ) = ρf (ρ)/f(ρ) b(ρ) = ρ (ρ) n # (ρ) = n " ρn # f n f(ρ n) ρ n n πb(ρn ). " ( &$ * *)$ " " ( ) & e P(z)$ P(z) 0 [z n $ " ]e P(z) > 0 n ( ) f g " $ # fg e f " ( ) f P $ " $ f +P $ f P P(f) " f(z) = e z " ( " f n = n! en n, n πn n! ( n ) n πn. e

58 ( /, '0,* '* * *('!"# $% %$ $$ () " " % ) ' " " " " % I = e h(z) dz, πi γ γ " % " $ " h(z) " "! & $ mx h(z) = h(r), R > 0. z =R ( h(z) = h n $ ) z n hn 0 n 0 n 0 * - " % " & f(z) " % [z n ] ( ) h(z) h n (z) = ln f(z) (n + ) ln z. ' % R > 0 " $ ( ) h (R) = 0, h (R) > 0. * " ( ) ( ) (" " ) f (R) f(r) (n + ) R = 0 R f (R) = n +. f(r) ( )$ R γ = γr & e h(z) z = R ## $ θ " " ( ) γ e h(z) dz e h(z) dz γ[θ] ( ) e h(z) e h(r)+ h (R)(z R) $ z γ[θ] h (R)=0 }{{}

59 ( /, '0,* '* * *(' θ γ[θ] γ R, % R # γ[θ] z = R + it $ $ " ( ) γ[θ] e h(z) dz R+i θ π R R i θ π R e h(z) dz R+i R i e h(z) dz. θ R R + it ( ) ( ) " " $ % I " I = πi γ πi π γ[θ] = eh(r) π e h(z) dz e h(r)+ h (R)(z R) dz e h(r) e h (R)(it) dt = eh(r) π e t h (R) dt π h (R) = e h(r) πh (R)., " % " & f(z) [z n ] " ( ) [z n ]f(z) e h(rn) πh (R n ) = f(r n) R n+ n πh (R n ),

60 ( /, '0,* '* * *(' 0 h(r) = ln f(r) (n + ) lnr h (R n ) = 0 R nf (R n ) f(r n ) = n +. " ( ) " " "$ ( ) ( ) " % I[f] & f(t)! " " & 0 bs (t) I[f](s) = f, b s " & f / I [ & ˆf] f & b s (t) = e st $ L[f](s) = 0 f(t) e st dt & bω (t) = e iωt $ F[f](ω) = f(t) e iωt dt & bp (t) = t p $ M[f](p) = / % " 0 f(t) t p dt I[αf + βg] = αi[f] + βi[g]. % " " (- L ± [f](s) = f(t) e st dt,

61 ( /, '0,* '* * *(', " ' F[f](ω) = L ± [f](iω) M[f] & g(t) = f(e (- " t ) L ± [g](p) = = 0 g(t)e pt dt = f(x)x p ( dx x ) = f(e t )e pt dt f(x)x p dx = M[f](p), 0 " x = e t dx = e t dt = xdt "&$ & $ * % " & &! * f(t) = e λt$ λ $ L[e λt ](s) = 0 = λ s e λt e st dt = / R (λ) < (s) L " $ R f(t) = n i= ie it$ λ 0 e (λ s)t dt e (λ s)t = (0 ) =, λ s s λ 0 L[f](s) = i= & f! " " & L[f] "$ n i s λ i, " & L[f] " " " f " " '! " M[t λ ](p) = p+λ % $ % ˆf(p) = f(t)t p dt 0 $ " $ f(t) t p = o (t ) t 0 f(t) t p = o (t ) t $ { f(t) = o (t p ), t 0 f(t) = o (t p ), t. " & f { f(t) = o (t α ), t 0 f(t) = o ( t β), t

62 ( /, '0,* '* * *(' * α, β $ α < β ' M[f](p) #! α < R (p) < β { f(t) = δ(t) t λ$ δ(t) = #, 0 x 0, M[f](p) = = 0 δ(t) t λ t p dt t λ+p dt 0 = p+λ, λ < R (p) <. " $ M t 0 f(t) = i t λ i, λ < λ <... i= t ( ) f(t) = O t β, β < λ $ M[f](p) i= i p + λ i, λ < R (p) < β. * & f % # $ f(t) = k 0 f kt k M[f](p) = k 0 " p = 0,,,... f(t) = e t # M[e t ](p) = 0 f k p + k e t t p dt = Γ(p) (= (p )!! ). M[f](p) p = λi i p = λi

63 ( /, '0,* '* * *(' / % # e t = ( ) $ k k 0 t k e t = o (t α ) k! α > 0 t 0 e t = o ( $ & t β) β > 0 t Γ! Γ(p) = M[e t ](p) = k 0 = ( ) k k! p + k, k 0 " R (p) > 0 $ "" &% & ( ) k k! M[t k ](p) ' " " " (- ( ) " " ˆf = M[f] c & ˆf # ( ) f(t) = πi c+i ˆf(p)t p dp c i (c) % ( ) " ˆf % ˆf(p)t p dp. " " " " ( ) M ( ) M[f(t)] = ˆf(p), M[f(t)] = p ˆf(p). ( ) " " M[f(t)] = ˆf(p) $ f(k) = k (c) ˆf(p) k p dp = k ( ) M[f(t)] = ˆf(p) $ k (c) [ ] M λ k f( k t) = ˆf(p) k k ( ) λ k f( k t) = ˆf(p) λ k p k dp (c) k ˆf(p)ζ(p) dp; λ k p k,

64 ( /, '0,* '* * *(' 0, S = ( ), " " n n n & $ f(t) = cos πt t, ' ( % ' ) ˆf(p) =, " " S = (c) = (c) = π = π = π cos pπ 0 cosπt t t p dt = = cos ( p π ) 0 cosπt t p 3 dt Γ(p ) π p ( < R (p) < 3). Γ(p ) π p ζ(p) dp, < c = R (p) < 3 π Γ(p ) p ζ( p) dp Γ(p) (c) p ζ( p) dp (p )(p ) p=0,, Res p ( = π ( ( ) ( 0 ( )( ) Res p=0 ) ζ( p) + ζ(0) + ( ) ( ) ζ( ) ) ( )) ( + 4 = π + 3) = π ( # $ ζ & (' ) ζ(p) ζ( p) = πp p Γ(p) cos πp, ζ(p) cos pπ = ζ( p)πp p Γ(p), $ & ζ( p) p = 0 $

65 " ' (h) t n n $ h ( $ $ # ) 0 " # T (h+) { } ( T (h)) 0 T (0) { } " % " & t (h+) (z) = z ( t (h) (z) ) t " " # (0) (z) = z { z t h+ = t h t 0 = z z { t h+ t h+ t h t 0 = z = z / " th = h /b h h+ b h+ h+ b h+ h h+ h+ b h = z h h+ = z, " bh = h+ h+ h = z h+ ( " th = z( h+ / h+ ) # # { h+ z h = h+ h+ h+ + z h = 0 0 = 0, = " % " &! ( ) q q + z = 0 q = ± 4z. 0

66 ( / *,*(( /, 0 ' q + q ' " c c h = c q h + c q h { 0 = c + c = 0 = c ( + 4z) + c ( 4z) = c c 4z = c = 4z, c = 4z h = ( ( + 4z 4z ) h ( ) ) h 4z t h (z) = z h+(z) h+ (z) = z qh+ q h+ = z q h+ q h+ q = z ρh+ q ρ h+, ρ = q = 4z q + 4z. th (z) ( q ρ h+ (z) = ρ(z) = e πi h+ k, k = 0,,..., h +. q ) h+ ( q q ) h+ % ρ(z0 ) = z 0 $ = 4 " $ lim t h (z) = z z 0 h + h +. ' # ρ(z k ) = ω k, k =,,..., h +, ωk " = e πi h+ k (h + ) # ( ρ(z) = 4z + 4z = ω 4z = ω ( + ω z = ( ) ) ω 4 + ω = = = ω ( + ω) = ω + ω + ω = + ω + ω ω ' + ω + (ω) +. ( ω # )

67 ( / *,*(( /, 0 ' zk " $ z k =, ϕk = π + cosϕ k % k, k =,,..., h +. h + z = z h+ =, ϕ = π + cosϕ h +. / & ρ h+ (z) z $ z # & ρ h+ (z) = ρ h+ (z ) + (z z ) Dρ h+ (z ) + O((z z ) ) = + (z z ) Dρ h+ (z ) + O((z z ) ). (z z ) ρ h+ (z) = Dρ h+ (z ) + O(z z ) z = z $ $ Dρ h+ (z ) 0 z & ) ( " th (z) z = z h+! $ % " % $ # " ( (h) t n " " Res t h (z) = z=z = z " ) q (z ) ( ρ h+ (z ) ) Res ρ h+ (z ) z q (z ) ( ρ h+ (z ) ) ( ). Dρ h+ (z ) ( " " " z=z Dρ h+ (z) = (h + ) ρ h+ (z) ρ (z) = (h + ) ρ h+ 4 (z) 4z ( + 4z) / " Res t h (z) = z=z = = = 4(h + ) 4z q (z) ρh+ (z). z q (z ) ( ρ h+ (z ) ) ( 4z 4(h + ) q (z ) 4(h + ) z q (z ) ( ) 4z ρ h+ (z ) 4(h + ) z q (z ) 4z (ρ(z ) ), ) ρ h+ (z )

68 ( / *,*(( /, 0 ( t (h) n t (h) z =, + cosϕ ϕ = π h +, c = " n c ( ) n + c h+ ( ) n z z h+ = c ( z ) n, 4(h + ) ( + 4z )( 4z ) = 4z (h + ) = (h + ) cosϕ. + cosϕ / " t (h) n h + cosϕ + cosϕ ( + cosϕ ) n. ( ) 4z + 4z 0 $ " * ( $, ' * " 0 # " n ( ) l ' t " " " (!) t0 (!) t / " * " " " s[t] = s[t 0 ] + s[t ] + ( ) = U[t ]s[t 0 ] + U[t 0 ]s[t ] +,

69 ( / *,*(( /, 0 U[t] ( ) "! ' sn = (s[t] n ) % " & ŝ(z) = s n zn n! n 0 t n " $ # $ t0 k t n k ( ) n p n,k = k n. " n $ #! P (!) = u, v, w " " " %&$ û, ˆv, ŵ ( ) u[t] = v[t] + w[t] û(z) = ˆv(z) + ŵ(z) ( ) u[t] = v[t0 ] w[t ] û(z) = ˆv(z/) ŵ(z/) ( ) " $ ( ) n ( ) n u n = k v n k w n k k=0 n ( ) ( v k = n! k! wn k k (n k)! k=0 n v ( k z ) k k! w ( n k z (n k)! = un n! zn = k=0 n k ) n k /" ( ) " %& # ( û(z) = e z) ) ( ) ŝ(z) = e z/ ŝ ( z ) + e z }{{ z }. " ŝ(z) = k 0 s n = k 0 k (e z s 0 =s =0 ( + z k )e ( k )z ) ( k ( ) n n ( ) ) n. k k k

70 ( / *,*(( /, 00 / " ( ) n e n (" ) & σ(x) = k 0 k ( e x k ( + x k )) ( $ sn σ(n) s n = σ(n) + o( ) n) σ ' " ( $ ( )) "" $ ' σ(x) = k λ kf( k x) M[σ](p) = M[f](p) ( ) p k = k k 0 }{{} (p+)k & f(x) = e ' x ( + x) M[f](p) = 0 ' $ < R (p + )Γ(p) p+, ( e x ( + x) ) x p dx = (p + )Γ(p). (p) < σ " "" = M[σ] & σ(x) x " " p = 0 (& Γ(p) ) $ pk ( = + (πi/ ln ) k k ) ( p+ ( Res p=0 σ (p) = lim p + ) Res p 0 Γ(p) = = p+ p=0 Res p= σ (p) = (p + )Γ(p) lim (p + ) p p+ p + = lim p = lim p = lim p = ln Res σ (p) =... = p=p ln k e ( (p + ) ( ) (p+)ln! p + e (p+)ln ln e (p+)ln (( ) p + } {{ } )

71 ( / *,*(( /, 0 ' $ & σ(x) σ(x) n k= kx λ k$ x $ ' σ (p) " λ,..., λ k,..., k ( ) * " $ n s n σ(n) n ln ( + Q(log n)), Q(t) & $ $ Q(t) t n p k = n n k (πi)/ln = n e πi log n k = n ( + Q k (log n)).

72 ( ( n 0), n ( ),..., n n ( X) $ * ) $,,,... $,, 4, 8,... $ # " $ $ % $ $ & $ + $ $ + "$ + $ $ - $ - % "$ - $ - ' $ $, %# $ "$ " " & $ $ * "$ -- "$ $ $ 0$ & $ & % $ $ 0 % " & $ % $ 0* $ & $ $ % $ % $ 0 " "$ 0$ $ " $ 0 # $ $ $ * $ $ & $ 0 $ 0 $ % $ * $ $ $ $ 0$ $ $ $ & $ & $ $ " % $ $ $ " $ " $ (- $ 0 ( $

73 /*', ( $ $ 0 $ ( " $ ' $ 0 ' "$ $ & $ $ $ "$ " $ $ $ $ $ * $ $ % $ $ & $ $ $ $ $ $ % $ * % $ $ $ 0 #$ & $ $ $ ', % $ ', $ $ $ $ 0, % "$, % $ $ 0, % $ $ $ $ $ $ $ $ $ " % " & $ $ $ 0 $ $ $ $ $ 0