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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1 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 = + i i3 3! + 4 4! + i5 5! cos =! + 4 4!, sin = 3 3! + 5 5! e i = cos + i sin e i = cos i sin cos = ei + e i, sin = ei e i i e iθ n = e inθ cos θ + i sin θ n = cos nθ + i sin nθ e iθ = e iθ = + +. =.. f f = f.. Cauchy-Riemman f = Cauchy-Riemman u x = v y, u y = v x.3 w = log w = w.4 log = Log + i arg + nπ = re iθ, re i θ +nπ. f. f a b.3 c or a b Cauchy = f C

2 = A = B f = F B F A.3 Cauchy γ C α fα = f πi C α f n α = n! f πi α.5 Taylor C n , f = gh g, h 4. f α n f α n n OK. sin = 3! + 4 5! 6 7! + α < β α = β + α + α + α c n n Taylor f = c n α n n= c n = n! f n α.5 f = 3. c n α n + c n α n + + c α e / +c + c α +.7 = lim sup n cn or R = lim R n n.6 Laurent c n c n+ f = + c n α n + c n α n + + c α +c + c α + f = n= c n α n D D = { R < < R } c n = f.6 πi α n+ C C D c n f = α n + c n α n n= } {{ } n= } {{ } α c Res f, α = c α k C f C f = πi n Res f, α k k=

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

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

5 3. = + 3i, = 3 + i. arg / 4. + i. π 4 + = i = + 3 e π 4 i. = = arg + = π 4 = = 4 = e π 3 i, = e π 6 i. = 4e π 3 + π 6 i = 4e π i 3. = 4 + 3i + 3i 3 i = 3 + i 3 + i 3 i = = + i. 4. = e π 3 π 6 i = e π 3 6 i = + i + i = + 3i + i 3 i = + 3i + 3i + = + 3i 3. p.4. = 3 + i,,. n c Mandelbrot n+ = n + c = c = + 3i, arg, 5

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

7 6. e = n= n n! e + = e e j e e = k j! k! j= k= = j k.3 j! k! n= j+k=n + n = j+k=n n! j!k! j k + n = n! j+k=n j k j!k!.3,.4 e e = + n = e + n! n=.4 6. p.5. cos + sin = cos = ei + e i, sin = ei e i i. cos + = cos cos sin sin 7. i i = log i log = e nπi i = e nπi i = e n n 7. i i p.5 7

8 8. f =. f = u + iv u v x, y. f 3. f Cauchy-Riemman f f. f = x + y x + iy u = x + y x, v = y x =, y = = f =. u v = x, x y = u y = y, Cauchy-Reimman u x = v y, u y = v x v x = f = = c = 8. p.5. f =. f = 8

9 9.. I =, γ. I = γ 3. I 3 = =. = + it = it = + idt I = it + idt =. x = t = t = dt y = + it = it = idt I = tdt + itidt = + i + = + i 3. = = cos θ + i sin θ I 3 = = i π π = cos θ i sin θ = sin θ + i cos θ dθ cos θ i sin θ sin θ + i cos θ dθ dθ = πi 9. p.6. I =, γ. I = γ 3. I 3 = = Re 9

10 . n R >. I =,. I = γ γ, 3. I 3 = n n, 4. I 4 = =R =R, 5. I 5 = n n =R a., b., c., α log α log α α = α. x = t, y = t I = F i F = I = = t + it = + it + = + idt { + it + } + i = + i + i = I = = 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 = log log = arg = = i arg = π I = F i F = πi = πi 3. I 3 = F = 4. = Re iθ, = ire iθ dθ

11 I 4 = = i = e iθ R π e iθ π R ireiθ dθ dθ = πi I 5 = n = e inθ R n π e inθ = i R n R n π ireiθ dθ e n iθ dθ = lim R I 5 = I 5 n = R n I 5 R n = π R Rn n > R πr M R n n > πm R n R πm Rn C α n α n πi n = = C n. p.6 I =, I = γ γ, I 3 = =

12 . I = Cauchy = e, I = = + 3, I 3 = = Cauchy. Cauchy fα = f πi C α f = e, α = f = e πi = I =πif = πei = 3 = 3 3. I 3 = + = =, = =, = I 3 = =ε + =ε. Cauchy f =, α = 3 f = 3 πi = Cauchy =ε + =ε + = πi, + = πi I 3 = I = πif = πi + = +

13 I 3 = = = πi πi = = + α πi R R ± > R I 3 < =R + < 4 R 4 8πR = R R R. I = p.7 = sinπ, I cosπ + sinπ = = 3

14 . Cauchy I = = e + 4, I sin π = = 3, Cauchy n! n + + f n α = n! f πi n+.5 α C Taylor Taylor Taylor α n+ f = fα + f α α + f 3! α α + + f n α α n + n! f α n+ = fα α n+ + f α α n + f α 3! α n + + f n α n! α + α α πi.5. Cauchy f = e, α =, n = 3 f 3 = 3! πi = f { } 4 I = πi 3 f 3 = πi 3e I = πif f = cos π 4π sin π π cos π 8. f = sin π, α = 8, n = f = f πi = 3 f = π 4 I = π 4 i. I p.7 I = + = + 3 4

15 3... Taylor = Taylor sin = Taylor 3. sin = Taylor Taylor Taylor Taylor f n α n!.. = = = { } n n= sin sin = = sin = sin = sin = 3 3! + 5 5! sin = 6 3! + 5! = n 4n+ n +! n= 3... p.7 = Taylor 3 sin = Taylor 5

16 4. f = n n + f = n n n + f 3 = +! + 3 3! + + n n! +. R R = lim n n n = R = Taylor Taylor = =. R R = R = lim n R = lim n n n = n n+ = lim n n + n = ln + Taylor Taylor = = = = = = = f = n + n = n= f = n= nn nn + dx xx = = f = n + = n= n dx + x 3. R = lim n n n! = R = R = lim n n! n+! = lim n n + = 4. p.8 f = f = + 3 n n + 6

17 5. Laurent. = 3,4 = = a e b cos c e. = Laurent a < < b < Laurent Taylor.6 α.a e = ! +.6 e = + + 3! + b e Taylor cos = + 4 4! cos = + 4! c.6 / e = ! 3 +.a b = = = n < n= = = = n= n 5. p.8. = 3,4 = = a sin b sin c sin. = Laurent a < < b < 7

18 6.. f = +,. f = 3, 3. f cos π 3 = sin π Laurent / α. n c = n! lim d n α n { αn f}. f = h α g h g hα c = hα g α. = f f = + + = + Res f, = lim + f =. = f c = lim f = lim 3 = = = f 3 c = lim d f = lim = lim 3 = = C 3 = C d 3. = n sin π = n cos π = n f cos π Res f 3, n = cos nπ sin π = =n π cos nπ = π = cos π = π +! [ sin π = π π 3! = + π π 3! f 3 = ] + + cos π sin π = π + = f π 6. p.8. a f a = 3 + 4, b f b = sin. a = n f n b = f f = cos π sin π 8

19 7. e I = + 4, I = = = sin π 3, I cos 3 = =4 sin, I I 4 I 3. = 4 e = e + e e + 3 +e + 3! e + 4 = e e e + + 6e + + = 6e e I = πires + 4, = πi 3e e Res + 4, = 3! lim d 3 3 e = 6e sin π = cos π 4π sin π π cos π sin π Res 8 3, = π 6 sin π I = πires 8 3, = π 4 i 3. = π,, π cos Res sin, = cos sin n =,, =nπ =. = 3 sin π Res 8 3, = lim d sin π 8 I 3 = πi + + = 6πi 7. p.9 I = = 9

20 8. I = dx + x 4 P x m Qx n Q n m + P x Qx dx = πi k Res P Q, α k α k P Q R R R Γ R C R C R R + 4 = dx + x C R R R.7 I O 4 I = + 4 C R = e 4 πi, e 3 4 πi α k 4 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 8. p.9 I = x + x 4 dx

21 9. Jordan I = sin x x dx f Im f R γ R p > lim e ip f = R γ R P m Q n n m + Qx P x eipx dx = πi k Res Qx P x eip, α k Qx cos pxdx = P x [ Re πi k Qx sin pxdx = P x [ Im πi k ] Qx Res P x eip, α k ] Qx Res P x eip, α k ei.8 lim R ε [ ε R e ix ] R x dx + e ix ε x dx = i sin x x dx C C C e i = e i ΓR = e i ε + + γε e i + R R ε e i e i.8.8 R Jordan e lim R γr i =.8 ε to e γε i = γ ε + e i γ ε?? = εe iθ ε γ ε = π ie iθ = πi eiθ.8 R, ε = i sin x dx πi = x sin x x dx = π

22 9. p.3 I = x sin x + x dx

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

24 3. P = 6 + i e π 4 i = cos π 4 + i sin π 4 = + i = 6 + i + = 3 + i + 3 i. = c = + 3i. x, y = 3, 3 + = + 3i + 3i = 3 3i + 3i = 3 3i = 3 3 i = { } cos 6 π + i sin 6 π = + i tan θ 3 = 3 tan θ + i3 tan θ tan 3 θ arg = 7 6 π tan 3θ = Im Re = 3 tan θ tan3 θ 3 tan θ = 3 Wikipedia a b c = 3 + i 3 i 3 i = = 3 i 3 + i 4 = 3 + i 3 i = 4 = e nπi = re iθ 3 = r 3 e 3θ = e nπi r = θ = nπ 3 =, e 3 πi, e 4 3 πi 4

25 =, ± 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. e x e x y = nπ n = re iθ 4 r =, θ = nπ + π 4 = ± ± i = ± i, ±i e cos + sin i + e i e i e i = + = i. cos cos sin sin = ei + e i e i + e i ei e i i e i e i i = ei+ + e i+ = cos e iθ e iθ θ = nπ n 7. i = e n+ πi i i = e n+ π n 8.. fx, y = x + y x iy x + iy x + y x + y = x i y 5

26 ux, y vx, y x + y ux, y = x x + y vx, y = y Cauchy-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 f =. = = = 9.. = + it I = = = t = + idt t + idt = + i 3. x = t y = + it = + it it = + t = idt I = t dt + = i + t idt 3. = = cos θ + i sin θ I 3 =. Re = cos θ = i = sin θ + i cos θ dθ π π cos θ sin θ + i cos θ dθ cos θdθ = πi = x I =. x dx = 3 F = 3 3 I = F F = 3 = t = dt 6

27 = e iθ, =ie iθ dθ I b = πif b = πi I = π ie iθ dθ = πi eiθ / = = =. I = I a + I b = 4πi 3. I 3 = π ie iθ dθ eiθ π = i e iθ dθ = = R I 3 = = π R = = R f = 8, α =, n = f f = πi + = I = πif f } 3 {.. f = sinπ, α = f = sinπ/ πi = I =πif = πi. =, = I a, I b = f a = cos π + sin π, α = I a = πif a = πi = f b = cos π + sin π, α = 3. f = 4 3 f = 7 I = 7 πi 3 = 3 = = = k k= k= = k+ k k= k 7

28 . = Taylor sin = 3 3! + 5 5! e, sin, cos Taylor sin = 4.. R =. R = = ! + 5 5! = R = lim n R = lim n = n R = lim n n n = n = lim + = n n n + = f = n + = π 6 5..a sin = 3 3! + 5 5! 3. sin = 3! + 4 5! sin = b 3. sin = 3! + 3 5! 3. c 3. / sin = 3! + 5! 3.a <, < b = + = = 3 3 = n n n n= n= <, < = + = + 6..a = = n n n= = 4 f a Res f a, 4 = lim 4 + 4f a = 4 5 = = f a Res f a, = lim d 3 f a 3 f a = + 4 =

29 d + 4 = = Res + 3, = lim + 3 = 7 Res f a, = 4 5 b = nπ Res sin, nπ = sin = =nπ cos nπ = n = 3 Res + 3, = lim d 8 = lim 4 3 = nπ f b n.a = n n sin π cos nπ n = n cos π sin π cos π / Res f, n = sin π = cos nπ /n π cos nπ =n = πn b = sin π 3 cos π = = cos π 3 sin π Res f, = lim = lim d 3 f d cos π sin π π cos π sin π = π lim sin 3 π 3 = π lim π3 3 + π = π 3 cos π = π +! sin π = π π 3! = + π π 3! + + cos π sin π = π π 3 + cos π sin π = π 3 π 3 + = 7 = I = πires + 3, = 7 πi 8. CR + 4 = R R x + x ΓR 4 dx R 3.3 I + 4 O I = CR + 4 = e 4 πi, e 3 4 πi α k Res + 4, α k = + 4 = =αk 4α k I = πi = πi 4e 4 πi + 4e 3 4 πi e 4 πi + e 3 4 πi = π π

30 9. Euler xe ix + x dx = x cos x + x dx + i x sin x + x dx Jordan xe ix I = Im + x dx xe ix [ ] e i dx = πi Res + x + ; i = πi ie i = i π e I 6 = π e. = e iθ cos θ = + I = a a + = = i = a a + + a, dθ = i + i = a, a < a < = > = a = a = I = πres a a + + a, a = π a 3

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 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

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x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s ... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z

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