z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy

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みひなたけくま
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1 z fz fz x, y, u, v, r, θ r > z = x + iy, f = u + iv γ D fz fz D fz fz z, Rm z, z. z = x + iy = re iθ = r cos θ + i sin θ z = x iy = re iθ = r cos θ i sin θ x = z + z = Re z, y = z z = Im z i r = z = z z = x + y θ = arg z = arctan y x e iz = + iz z iz3 3! + z4 4! + iz5 5! cos z = z! + z4 4!, sin z = z z3 3! + z5 5! e iz = cos z + i sin z e iz = cos z i sin z cos z = eiz + e iz, sin z = eiz e iz i e iθ n = e inθ cos θ + i sin θ n = cos nθ + i sin nθ e iθ z = e iθ = z + z z + z. z z = z z.. fz fz = fz.. auchy-riemman f z = auchy-riemman u x = v y, u y = v x.3 w = log z w = z z w.4 log z = Log z + i arg z + nπ z = re iθ, re i θ +nπ. fz z. fz a b.3 c or a b auchy = fz

2 z = A z = B fz = F B F A.3 auchy γ α fα = fz πi z α f n α = n! fz πi z α.5 Taylor n , fz = gzhz gz, hz 4. f z α n fz z z α n n OK. sin z z = z 3! + z4 5! z6 7! + α < β α = β + α + α + α c n n Taylor fz = c n z α n n= c n = n! f n α.5 fz = 3. c n z α n + c n z α n + + c z α e /z +c + c z α +.7 = lim sup n cn or R = lim R n n.6 Laurent c n c n+ fz = + c n z α n + c n z α n + + c z α +c + c z α + fz = n= c n z α n z D D = {z R < z < R } c n = fz.6 πi z α n+ D c n fz = z α n + c n z α n n= } {{ } n= } {{ } z α c Res fz, α = c α k fz fz = πi n Res fz, α k k=

3 . Laurent /z α. n c = n! lim d n z α n {z αn fz} 3. fz = hz α g h gz hα c = hα g α.8 fz O R lim fz = R Γ R fx dx = fz R Γ R R R Γ R R R π Rcos θ, sin θdθ z = e iθ cos θ = z + /z, sin θ = z /z = ie iθ dθ i fx x α dx = πi.9 e πiα k fx Res x α, α k z = α Taylor + a z α + a z α + = a z α + α r z α = re iθ, = ire iθ dθ Jordan Im z > fz O Jordan lim fz e ipz = R Γ R fx e ipx dx = fze ipz R sin θ π θ p > R /z z = α z α πi z α = πi z = α n z α n = π e R sin θ dθ < π e R π θ dθ < π R 3

4 . P x, y =, P π 3 P z = + i e π 3 i = cos π 3 + i sin π 3 = + 3 i z 3i = + i + x, y = 3, = i. p.35 P x, y = 3, P π 4. cos 3θ + i sin 3θ = cos θ + i sin θ 3 cos sin cos θ + i sin θ 3 = cos 3 θ 3 cos θ sin θ + i 3 cos θ sin θ sin 3 θ cos 3θ = cos 3 θ 3 cos θ sin θ sin 3θ = 3 cos θ sin θ sin 3 θ. p.35 + i tan θ θ + i tan θ 3 3θ tan 4

5 3. π i i 4 arctan 5 + arctan i i 3. Machin 4 arctan 5 arctan 39 = π 4. z = x + iy arg z = arctan y x 39 + i arctan 39 n z n = x + iy n argz n = n arctan y x 5 + i4 4 arctan 5 arg z z = arg z arg z arg 5 + i i = 4 arctan 5 arctan i i i = 39 + i i39 i = 39 + i39 i i = 57 = + i arg 5 + i i = π arctan π 3. p.35 π 4 = arctan arctan 7 5

6 4. z = + 3i, z = 3 + i. argz + z. z z 3. z /z 4. z + i z. π 4 z + z = i = + 3 e π 4 i. z = z = arg z + z = π 4 z z = z z = 4 z = e π 3 i, z = e π 6 i.3 z z = 4e π 3 + π 6 i = 4e π i 3. z z = 4 + 3i + 3i 3 i = 3 + i 3 + i 3 i = = + i.3 4. z = e π 3 π 6 i = e π 3 6 i = z + i z + i z = + 3i + i 3 i = + 3i + 3i + = + 3i 4. p.35. z = 3 + i z, z, z z. n c Mandelbrot z n+ = zn + c z = c = + 3i z, arg z, z 6

7 5. z 4 = 4 z = re iθ.4 z = + i, + i, i, i 4 = 4e in+π.4 z 4 = r 4 e 4iθ = 4e in+π r 4 = 4 r r = 4θ = n + π θ = nπ + π 4 5. p.36. z 3 =. z 4 = 4 6. e iθ θ z = e iθ e iθ θ θ = 6. p.36 a e iθ e iθ θ θ b e x e iy e x e iy e x e x x > 7

8 7. e z = n= z n n! e z+z = e z e z j e z e z = z k z j! k! j= k= = j k z z.5 j! k! n= j+k=n z + z n = j+k=n n! j!k! z j z k z + z n = n! j+k=n z j z k j!k!.5,.6 e z e z = z + z n = e z+z n! n=.6 7. p.36. cos z + sin z = cos z = eiz + e iz, sin z = eiz e iz i. cosz + z = cos z cos z sin z sin z 8. i i = log i log = e nπi i = e nπi i = e n n 8. i i p.37 8

9 9. fz = z z. fz = u + iv u v x, y. fz 3. fz auchy-riemman z f z z z z f z. fz = x + y x + iy x =, y = z = fz = z z z u = x + y x, v = y z. u v = x, x y = u y = y, auchy-reimman u x = v y, u y = v x v x = fz z = z z = 3. z = 9. p.37. fz = z z z. fz 9

10 .. I = z, γ. I = z γ 3. I 3 = z = z. z z = + it z = it = + idt I = it + idt =. x z = t z = t = dt y z = + it z = it = idt I = tdt + itidt = + i + = + i 3. z = main z = cos θ + i sin θ I 3 = = i π π z = cos θ i sin θ = sin θ + i cos θ dθ cos θ i sin θ sin θ + i cos θ dθ dθ = πi. p.37. I = z, γ. I = z γ 3. I 3 = z = Re z

11 . n R >. I = z,. I = γ γ z, 3. I 3 = z n n, 4. I 4 = z =R z =R z, 5. I 5 = z n n z =R a., b., c., z α logz α logz α z α z = α. x = t, y = t I = F i F = I = z = t + it = + it + = + idt { + it + } + i = + i + i = I = z = e iθ, π e iθ ie iθ dθ = i i = ie iθ dθ [ e iθ ] π =.. I = I = π ie iθ πi dθ = eiθ = + i t + it dt + i t it t + t dt F z = log z log z z = arg z = z = i arg z = π I = F i F = πi = πi 3. I 3 = F z = z 4. z = Re iθ, = ire iθ dθ

12 I 4 = = i z = e iθ R π e iθ π R ireiθ dθ dθ = πi I 5 = z n = e inθ R n π e inθ = i R n R n π ireiθ dθ e n iθ dθ = z lim R I 5 = I 5 z n = R n I 5 R n = π R Rn n > R πr M R n n > πm R n R πm Rn α n z α n πi n = = n. p.37 I = z, I = γ γ z, I 3 = z = z

13 . I = auchy z = e z z, I = z = z + z 3, I 3 = z = z auchy. auchy fα = fz πi z α fz = e, α = f = e z πi z = z I =πif = πei z = 3 z = 3 3. I 3 = z z + z = z =, z = z =, z = I 3 = z z + + z z + z =ε z+ =ε. auchy fz = z 3, α = f = z 3 πi z= z auchy z =ε z+ =ε z z + = πi, z z + = πi I 3 = I = πif = πi z z + = z z + 3

14 I 3 = z = = πi πi = z z = z + z α πi R R z ± > R I 3 < z =R z z + < 4 R 4 8πR = R R R. I = p.38 z = sinπz z, I cosπz + sinπz = z = zz 4

15 3. auchy I = z = e z z + 4, I z sin πz = z = z 3, auchy n! n + + f n α = n! fz πi n+.7 z α Taylor Taylor Taylor z α n+ fz = fα + f α z α + f 3! αz α + + f n α z α n + n! fz z α n+ = fα z α n+ + f α z α n + f α 3!z α n + + f n α n!z α + z α z α πi.7. auchy fz = e, α =, n = 3 f 3 = 3! πi z = fz {z } 4 I = πi 3 f 3 = πi 3e I = πif f z = cos πz 4πz sin πz π z cos πz 8. fz = z sin πz, α = 8, n = f = fz πi z = z 3 f = π 4 I = π 4 i 3. I p.38 I = z+ = z z + 3 5

16 4... Taylor z = Taylor z sin z z = Taylor z 3. sin z z = Taylor Taylor Taylor Taylor f n α n!.. z = z = + z z + z 3 = { z } n n= sin z z sin z = z z3 6 + z = + z + z + z sin z z = z + z z3 + sin z z = sin z = z z3 3! + z5 5! z z sin z =z z6 3! + z 5! = n z4n+ n +! n= 4... p.39 z = Taylor z z 3 sin z z = Taylor z 6

17 5. f z = + z + z + z zn n + f z =z z + z3 3 z4 4 + n zn n + f 3 z =z + z! + z3 3! + + zn n! +. R R = lim n n n = R = z Taylor Taylor z = z =. R R = R = lim n R = lim n n n = n n+ = lim n n + n = lnz + Taylor Taylor z = z = z = z = z = z = z = f = n + n = n= f = n= nn nn + dx xx z = z = f = n + = n= n dx + x 3. R = lim n n n! = R = R = lim n n! n+! = lim n n + = 5. p.39 f z = + z + z + z 3 + f z = z + z z 3 z n n + 7

18 6. Laurent. z = 3,4 z = z = a e z z b cos z z c e z. z = Laurent a zz < z < b zz < z Laurent Taylor.6 z α.a e z = + z + z + z3 3! +.8 e z z = + z + z 3! + b e z Taylor cos z = z + z4 4! cos z z = z + z 4! c.8 z /z e z = + z + z + 3!z 3 +.a b zz = z z = + z + z + z 3 + z = z n z < n= zz = z z = z + z + z + = n= z n 6. p.39. z = 3,4 z = z = a sin z z b sin z z c sin z z. z = Laurent a z z < z < b z z < z 8

19 7.. f z = z z +,. f z = z 3 z, 3. f cos πz 3z = sin πz Laurent /z α. n c = n! lim d n z α n {z αn fz}. fz = hz α g h gz hα c = hα g α. z = f f z = z + z + = z + Res f z, = lim z z + f z =. z = f c = lim f z = lim z z z 3 = z = z = f 3 c = lim z d f z = lim z = lim z z 3 = z = z 3 z = d z 3. z = n sin πz z = n cos πz z = n f z cos πz Res f 3 z, n = cos nπ sin πz = z=n π cos nπ = π z = cos πz = πz +! [ sin πz = πz πz 3! = + πz πz 3! f 3 z = ] + + cos πz sin πz = πz + z = f z π 7. p.4. a f a z = z z 3 z + 4, b f bz = sin z. a z = n fz n b z = fz fz = z cos πz sin πz 9

20 8. e z I = z + 4, I = z = z = z sin πz z 3, I cos z 3 = z =4 sin z, I I 5 I 3. z = 4 e z = e + e z + z + + e z + 3 +e + 3! e z z + 4 = e + z 4 + ez ez + + 6ez + + z = 6e e I = πires z + 4, = πi 3e e Res z + 4, = 3! lim z d 3 3 ez = 6e z sin πz = cos πz 4πz sin πz π z cos πz z sin πz Res 8 z 3, = π 6 z sin πz I = πires 8 z 3, = π 4 i 3. z = π,, π cos z Res sin z, = cos z sin z n =,, z=nπ =. z = 3 z sin πz Res 8 z 3, = lim z d z sin πz 8 I 3 = πi + + = 6πi 8. p.4 I = z z + 3 z+ =

21 9. I = dx + x 4 P x m Qx n Qz n m + P x Qx dx = πi k Res P z Qz, α k α k P z Qz R R R Γ R R R R + z 4 = dx + x z 4.9 R R R.9 I O z 4 I = + z 4 R z = e 4 πi, e 3 4 πi α k 4z 3 = αk 4α 3 k Γ R I = πi = πi 4e 3 4 πi + 4e 9 4 πi e 3 4 πi + e 4 πi = π 9. p.4 I = x + x 4 dx

22 . Jordan I = x sin x + x dx fz Im z z f R γ R p > lim e ipz fz = R γ R P z m Qz n n m + Qx P x eipx dx = πi k Res Qx P x eipz, α k Qx cos pxdx = P x [ Re πi k Qx sin pxdx = P x [ Im πi k ] Qx Res P x eipz, α k ] Qx Res P x eipz, α k Euler xe ix + x dx = x cos x + x dx + i x sin x + x dx I = Im xe ix + x dx ze iz + z ΓR = ze iz + z + R R xe ix + x dx Jordan R R I z = ±i ze iz + z z = i xe ix [ ] ze iz dx = πi Res + x + z ; i = πi ie i = i π e I = π e. p.4 t + t fω = te iωt + t dt

23 . π dθ a + cos θ a > cos θ, sin θ π Rcos θ, sin θdθ z = e iθ = ie iθ dθ = izdθ dθ = iz Euler cos θ = z + z, sin θ = z z, i z = z + z z z R, i z = e iθ cos θ = z + z, = izdθ π dθ a + cos θ = i z = az + z + az + z + = z = a ± a a + a < a a > z = a + a π dθ a + cos θ = π a. p.4 π dθ a a cos θ + < a < 3

24 . sinc I = sin x x dx Jordan z = sin z eiz z z z = z = eiz R, ε z I. ε to γε e iz z = γ ε z + e iz γ ε z?? z = εe iθ ε e iz z = e iz z ΓR = e iz ε z + + γε e iz z + R R ε e iz z e iz. z. R Jordan lim R ε R e lim R γr iz z =. [ ε e ix ] R x dx + e ix x dx = i ε sin x x dx γ ε z = π ie iθ = πi eiθ. R, ε z = i sin x dx πi = x sin x x dx = π I = π 4

25 3. I = dx + x 3 fz Rez z fz fxdx = fzlog zdx πi z π < arg z < π π < Im Log z < π α = e πi 3, α =, α 3 = e 4 3 πi + z 3 hz/gz c = hz g z Res + z 3, e πi 3 = e πi 3 3 Res + z 3, = 3 Res + z 3, e 4 3 πi = e πi 3 3 Log z Log e πi 3 = 3 πi Log = Log e 4 πi 3 = 3 πi I = e πi 3 3 e πi πi 3 πi = 3π 9. p.4 I = x + x 3 dx 5

26 4.. fz = z z + A f A = 3 A B D B D f B, f, f D B D z = z = D B A. fz = z z + A f A = 3 A B D B D f B, f, f D B D z = z = D B A z α z α z α = re iθ z z α θ z = r e iθ, z + = r e iθ z z + = r r e i θ +θ f A = 3 A r =, r = 3, θ =, θ = B x θ r =, r = 3, θ = π, θ = π f = 3e = 3 B B r =, r =, θ = 3 4 π, θ = 4 π A f B = e πi = i B B A A θ 6

27 B f D = e πi = i D A D θ B θ π r =, r =, θ = 3 4 π, θ = 7 4 π A 3. p.4 fz = z z + A f A = / 3 A B D B D f B, f, f D B D z = z = D B A 7

28 5. m w t + EI 4 w x 4 = v, m, EI wx, t = ϕt expiωt 4 ϕ x 4 mω EI ϕ = µ 4 = mω EI 4 ϕ x 4 µ4 ϕ =.... x = x = L x = v = v = x = L v = v = µ ϕ = e λt λ 4 = µ 4 c + c 4 = c + c 3 = c sin µl c cos µl + c 3 sinh µl + c 4 cosh µl = µ c cos µl + c sin µl + c 3 cosh µl + c 4 sinh µl = λ = ±µ, ±µi sin µl cos µl sinh µl cosh µl cos µl sin µl cosh µl sinh µl c c c 3 c 4 = ϕ = Ae µt + Be µt + e iµt + De iµt = A + B cosh µt + A B sinh µt + + D cos µt + Di sin µt = a cosh µt + b sinh µt + c cos µt + d sin µt c, c, c 3, c 4 cos µl cosh µl + = 8

29 6. Fourier Fourier I = π e t e iωt dt fxe ix dx z fz = e z Jordan e z = e x +y y e z e t e iωt dt = e x+ ω i ω dt = e ω e x+ ω i dt } {{ } fz = e z fz = γ R ω < e R + ω 4 e R+iy dy R ω R fz = e x+ ω i dx γ T R R fz = e z R fz + fz R γ R + fz + fz = γ T γ L R π R R R lim e x dx = lim e x+ ω i dx R R R R I = e R ω lim π R = π e ω = e ω4 4 lim R R R R e x+ ω dx e x dx 9

30 7. 3

31 8. I = + x 3 dx.. R + z 3 z = e 3 πi [ ] I c = πires + z 3 ; e 3 πi = πi 3z z=e πi 3. Γ = 3 πie 3 πi z = te 3 i t = e 3 i dt 3. R + z 3 + I I = I R R I e 3 πi I 3 πie 3 πi I e 3 πi I = 3 πie 3 πi I = = 3 πie 3 πi e 3 πi 3 πie 3 πi 3e 6 πi = π 3 3 I = = e 3 πi I + t 3 e 3 i dt 3

32 9. fz = z cos πz sin πz. γ T z = x + ir fz R e R + e R e R e R. fz n γ T z z = n x + i + > n z < n cos πz Euler sin πz cos πz = e iπz + e iπz sin πz e iπz e iπz e iπz + e iπz e iπz + e iπz = e πy + e πy e iπz e iπz + e iπz e iπz e iπz + e πy e iπz e iπz + e iπz = e iπz = e πy y e iπz e iπz e πy e πy > cos πz = eπy + e πy sin πz e πy e πy 3

33 3. xy ω x y t =. z = x + iy, z = x + iy z, ω, t z. F z = F x + if y, F z = F x + if y F z, ω, t F z 3., m z = F z z = z e iωt F z = F ze iωt ż = ż e iωt + iωz e iωt z = z e iωt + iωż e iωt ω z e iωt m z = F z e iωt m z = F z i mωż + mω z mẍ = F x mωẏ + mω x mÿ = F y mωẋ + mω y 33

34 3. W = u iv X, Y X iy = iρ W fz W = df W x U a X Y fz = U z + a i Γ z π ln z z df W = [ = U a z i Γ ] πz =U i UΓ πz U a z Γ 4π z + Γ iua πz 3 + U a 4 Γ z 4 X =, X iy = iρuγ Y = ρuγ 34

35 3. P z = 6 + i e π 4 i = cos π 4 + i sin π 4 = + i arg + i 7 + i = π 4 π 4 = arctan arctan 7 z = 6 + i + = 3 + i + 3 i 4.. a b z = 3 + i x, y = 3, 3 + c 3 i 3 i z = = 3 i 3 + i 4 z z = 3 + i 3 i = 4. z = + i tan θ 3 = 3 tan θ + i3 tan θ tan 3 θ tan 3θ = Im z Re z = 3 tan θ tan3 θ 3 tan θ. z = c z = + 3i 3. Machin arg + i 7 + i + i 7 + i = arctan arctan 7 = 3 + 4i 7 + i = 3 + 4i7 i 5 = + i z = + 3i + 3i = 3 3i + 3i = 3 3i z = 3 3 i = { } cos 6 π + i sin 6 π argz = 7 6 π z = 3 35

36 Wikipedia 5.. z = re iθ = e nπi z 3 = r 3 e 3θ = e nπi r = θ = nπ e iθ e iθ θ = nπ n z =, e 3 πi, e 4 3 πi z =, ± i 3 e iθ e iθ θ = i e it dt θ θ θ e iθ e iθ θ θ e it dt = θ θ θ < θ e iθ e iθ θ θ. z = re iθ. e x e x y = nπ n 5 r =, θ = nπ + π 4 z = ± ± i z = ± i, ±i e cos z + sin iz + e iz e iz e iz z = + = i 36

37 . cos z cos z sin z sin z = eiz + e iz e iz + e iz = eiz+z + e iz+z = cosz + z eiz e iz i e iz e iz i... z z = + it 8. i = e n+ πi i i = e n+ π n 9.. fx, y = x + y x iy x + iy x + y x + y = x i y ux, y vx, y x + y ux, y = x x + y vx, y = y auchy-riemann u x = 3 x + y v y = x x 3 y + u x = v y x + y = u y = v x = xy u y = v x fz z = z z z = I = z = z z = t = + idt t + idt = + i 3. x z = t z = t = dt y z = + it z = + it it = + t = idt I = t dt + = i + t idt 3. z = z = cos θ + i sin θ I 3 =. Re z = cos θ = i = sin θ + i cos θ dθ π π cos θ sin θ + i cos θ dθ cos θdθ = πi z = x I = x dx = 3 37

38 . F z = z3 3 I = F F = 3. z =, z = I a, I b z = f a z = cos πz + sin πz, α = z I a = πif a = πi z = f b z = cos πz + sin πz, α = z I b = πif b = πi z = e iθ, =ie iθ dθ I = π ie iθ dθ = πi eiθ /z z = z = z = 3. I = I a + I b = 4πi 3. I 3 = π ie iθ dθ eiθ π = i e iθ dθ = z = R I 3 = = π R z = z z = z R fz = 8z, α =, n = fz f = πi z+ = I = πif fz } 3 { z.. fz = sinπz, α = f = sinπz/ πi z = z I =πif = πi f z = 4z 3 f = 7 I = 7 πi 38

39 4. z z 3 = z 3 z = z z = z z = z k z k= k= = k+ z k k=. z = Taylor sin z = z z3 3! + z5 5! k e z, sin z, cos z Taylor z z = + z + z 4 + sin z z z = z3 3! + z5 5! + z + z = z z3 + z5 + z = f = n + = π 6 6..a sin z = z z3 3! + z5 5! 3. sin z z sin z z = z 3! + z4 5! = z 3 + z b 3. sin z z = z z 3! + z3 5! 3. c 3. z /z sin z = z 3!z + 5!z z 3 R = R = lim n = n.a z <, z <. R = R = lim n R = lim n n n = n = lim + = n n n + z z = z + z = z z z = z z z 3 z z 3 = z n z n n n= n= 39

40 b z <, z < z z = z + z = z z + z z = z z z 3 + z + z + z 3 + = n z n n= b z = z sin πz 3 cos π = z = cos πz z sin πz 3 Res f, = lim z = lim z d z 3 f d z cos πz sin πz πz cos πz sin πz = π lim z sin 3 πz 3 = π lim π3 z 3 + z π 3 z 3 + = π 3 7..a z = 4 f a Res f a z, 4 = lim z 4 z + 4f az = 4 5 z = z = f a b Res f a z, = lim d z 3 z f a z z 3 f a z = z z + 4 = 4 z + 4 d z z + 4 = 8 z Res f a z, = 4 5 z = nπ Res sin z, nπ = sin z = z=nπ cos nπ = n z = nπ f b n.a z = n n sin πz cos nπ n z = n cos πz z sin πz Res f, n = = cos πz /z sin πz cos nπ /n π cos nπ z=n = πn cos πz = πz +! sin πz = πz πz 3! = + πz πz 3! + + cos πz sin πz = πz πz 3 + cos πz z sin πz = πz 3 π 3z + π 3 8. z = Res z z + 3, = lim z z + 3 = 7 z = 3 Res z z + 3, = lim z d 8z = lim z 4z 3 = 7 z = I = πires z z + 3, = 7 πi 9. R z + z 4 = R R x + x ΓR 4 dx + z + z

41 R 3.3 I z + z 4 O z xe ix + x dx [ ze iz = πi Res + z ; e π 4 ] ze iz + Res + z, e 3π 4 I = R z + z 4 z = e 4 πi, e 3 4 πi α k z Res + z 4, α k = z + z 4 = z=αk 4α k = πi??? I = π e I = πi = πi 4e 4 πi + 4e 3 4 πi e 4 πi + e 3 4 πi = π. z = e iθ cos θ = z + z I = a a z + z z = = i z = az a + z + a, dθ = iz + iz. x = t I = xe iωx + x dx e iωt ze iz + z ΓR = ze iz R + z + R xe ix + x dx Jordan R R I z = e +n 4 πi n =,,, 3 zeiz + z z = e πi 4, z = e 3πi 4 z = a, a < a < z = > z = a z = a z = I = πres az a + z + a, a = π a. 3. z = r e iθ, z z + = z + = r e iθ r r e i θ +θ f A = 3 A r =, r = 3, θ =, θ = 4

42 B r =, r =, θ = 3 4 π, θ = 4 π B f B = e πi = i A B θ B A A B x θ r =, r = 3, θ = π, θ = π D θ B θ π r =, r =, θ = 3 4 π, θ = 7 4 π f = 3e = 3 f D = e πi = i B D A A 4

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =