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

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1 ... 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 = x + iy (x, y) (x, y) z = x + iy z = x + iy C (x, y)

2 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 sin θ) 3 () z z = z z, arg(z z ) = arg(z )+arg(z )() z z = z, arg( z z z ) = arg(z ) arg(z ) O Z K; X + Y + (Z ) = 4 K N (,, ) z = z(x, y) l l K P = (X, Y, Z) z P z (x, y) P (X, Y, Z) X = x x + y +, Y = x = y x + y +, Z = x + y x + y +, X Z, y = Y Z L L z P N N

3 ... 4 {z n } n= c ε >, n, n n, z n c ε z n c z n = x n + iy n, c = a + ib z n c {x n } n= {y n} n= a b 5 ()lim n (z n + w n ) = lim n z n + lim n w n ()lim n z n w n = lim n z n lim n w n (3)lim n ( z n w n ) = lim n z n lim n w n (lim n w n ).. x e x = n= x n n! x iθ e iθ (iθ) n = n! n= (iθ) m = ( ) m θ m, (iθ) m+ = i( ) m θ m+ e iθ ( ) m θ m ( ) m θ m+ = + i (m)! (m + )! n= 3 n=

4 cos θ sin θ e iθ = cos θ + i sin θ 7.. z = x + iy = r(cos θ + i sin θ) = re iθ.. 3 z n = α z = re iθ, α = r e iθ z n = α (re iθ ) n = r e iθ, r n e inθ = r e iθ, r n = r, nθ = θ + kπ, k =,,,... r = n r, θ = θ + kπ, k =,,,... n z k = n r e i θ +kπ n, k =,,,... z k = n r cos θ + kπ n + i sin θ + kπ, k =,,,... n 4

5 n z k = n r (cos θ + kπ n + i sin θ + kπ ), k =,,,.., (n ) n z 6 = z 3 = 8i.3.3. w = f(z) a ρ U ρ (a) U ρ (a) = {z; z a < ρ} a ρ ρ = D D a ρ U ρ (a) D D D D a, b a b D D D w = f(z) D z = z α lim z z f(z) = α ε >, δ >, z; < z z < δ, f(z) α < ε ε δ < z z < δ z f(z) α < ε ε >, δ >, z U δ (z ), z z, f(z) U ε (α) 5

6 ()lim z z (f(z) + g(z)) = lim z z f(z) + lim z z g(z) ()lim z z f(z)g(z) = lim z z f(z) lim z z g(z) f(z) (3)lim = lim z z f(z), lim z z g(z) lim z z g(z) z z g(z) w = f(z) z = z lim f(z) = f(z ) z z ε >, δ >, z; z z < δ, f(z) f(z ) < ε f(z), g(z) z = z f(z) ± g(z), f(z)g(z), f(z), g(z g(z) ), z = z.3. D z w = f(z) z f (z ) df dz (z ) z f(z) f(z) f(z ) lim z z z z = lim h f(z + h) f(z ) h D f(z) D z f (z) f(z) ()(f + g) = f + g ()(fg) = f g + fg (3)( f g ) = f g fg, g g 6

7 z = x + iy, w = f(z) = u(x, y) + iv(x, y) u(x, y) v(x, y) (x, y) 6 f(z) = u(x, y) + iv(x, y) z = x + iy (x, y ) u(x, y), y(x, y) u x = v y, u y = v x x, y h(x, y) (a, b) A, B h(x, y) = h(a, b) + A(x a) + B(y b) + ηr r = (x a) + (y b) η w = f(z) z = x + iy ρ U ρ (z ) f(z) z.w = z z = x + iy w = (x + iy) = x y + ixy u = x y, v = xy u x = x = v y, u y = y = v x (x, y) w = z.w = z z = x + iy w = x + y 7

8 u = x + y, v = u x = x, v y =, u y = y, v x = w = z.3.3 α, β, γ, δ αδ βγ w = αz + β γz + δ z w w = αz + β γz + δ = α βγ αδ γ + γ z + δ γ = α + β z + γ, α = α, β γ = βγ αδ, γ = δ, γ w = γ αz+β γz+δ z = z + γ, z = z, z 3 = β z, w = z 3 + α ()w = z + α, z α ()w = βz, z arg β z β (3)w =, z z z z O 8

9 () (3) w = z + z z r w r >.3.4 w=z n n w = z n z = re iθ, w = ρe iθ ρe iθ = (re iθ ) n = r n e inθ ρ = r n Θ = nθ z r z O, r w ρ = r n w O, r n Θ = nθ,z z < θ < π n w, < Θ < π z z arg z = θ w w arg w = nθ z < arg z < π n w w = z n w = n z 9

10 .4.4. n= c nx n, c n n= c nz n, c n 7 n= z n n= z n n= z n 8 n= c nz n lim c n+ = ρ n c n z < (= R ρ 9 n= c nz n ( c n z n ) (k) = n= n(n )...(n k + )c n z n k n=k

11 .4. w = e z x e x = x n n= x z n! e z z n = n! n= z e z z n n= n! d dz (ez ) = e z, e z +z = e z e z z = x + iy w = e z = e x+iy = e x e iy = e x (cos y + i sin y) w = e x, arg w = y z Rez = x y < y < π w w = e z e x z Imz = y, < y < π, x < x < w w = e z arg w = y z {(x, y); < y < π, < x < } w w = e z πi w = e z = e x (cos y + i sin y) = u + iv u = e x cos y, v = e x sin y u x = e x cos y = v y, u y = e x sin y = v x w = e z.4.3 w = log z w = z α α z = e w

12 z = re iθ, w = u + iv re iθ = e u+iv = e u e iv r = e u, e iθ = e iv u = log r, v = θ + nπ, n =, ±,... w = u + iv = log r + i(θ + nπ), n =, ±,... w = log z = log z + i(arg z + nπ)n =, ±,... log i = log i + i(arg i + nπ) = i( π + nπ) log( + i) = log + i + i(arg( + i) + nπ) = log + i( π + nπ), 4 log e = log e + i( + nπ) = log e + inπ = + inπ w = z α = e α log z w = z α, α; ) z α = e α log z = e α(log z +i(arg z+nπ)), n =, ±,..., z α i i = e i log i = e i(i( π +nπ)) = e ( π +nπ), n =, ±,...,

13 .4.4 w= k z α = k k z k (log z +nπi) = e k = k z (cos nπ k log z = e k e nπ i k nπ + i sin ), n =, ±,..., k k nπ z (cos k + i sin nπ ), n =,,..., (k ) k k w = z k k ( i) = ( i) = e log( i) = e {log i +( π +nπ)i} = e ( π +nπ)i = e π 4 i e nπi = ±( + i ).4.5 w = cos z, sin z cos x = xn n= ( )n, (n)! sin x = xn n= ( )n (n )! x = z ( ) n zn (n)!, ( ) n z n (n )! n= n= cos z, sin z cos z = sin z = ( ) n zn (n)!, n= ( ) n z n (n )! n= 3

14 d cos z = sin z dz d sin z = cos z dz cos( z) = cos z, sin( z) = sin z cos z = (eiz + e iz ), sin z = i (eiz e iz ) cos z sin z π cos z + sin z = cos z = cos z = (eiz + e iz ) = X = e iz X X + X = 4 X 4X + = 4

15 X = ± 3 e iz = ± 3, z = x + iy e y+ix = ± 3, e y (cos x + i sin x) = ± 3 sin x =, e y cos x = ± 3, x = nπ, e y = ± 3, y = log( ± 3), z = nπ i log( ± 3) + i 3 i log( + i) Log( ) ( + i 3) i i.5.5. D C z(t) = x(t) + iy(t), α t β, z(α) = a, z(β) = b 5

16 C w = f(z) C β α f(z(t))z (t)dt C f(z)dz O(, ) + i(= (, )) C C w = z + i x (= (, )) x + i(= (, )) C C w = z w = z C z(t) = t + it = ( + i)t, t, z (t) = ( + i) w = z zdz = ( + i)t( + i)dt C = ( + i) [ t ] = i. w = z C z dz = ( + i) t ( + i)dt = ( + i) 3 [ t3 3 ] = ( + i)3 3 6

17 C t z(t) = t, z (t) =, t z(t) = + i(t ), z (t) = i w = z C zdz = tdt + { + i(t )}idt = [ t ] + [( + i)t t ] = i w = z C z dz = t dt + = [ t3 3 ] + [ = 3 + ( + i)3 3 { + i(t )} idt { + i(t )}3 i] 3i 3 = ( + i)3 3 α r C C (z α) n C z(t) = α + re it, t < π 7

18 C z (t) = ire it π (z α) dz = n (re it ) n ireit dt = i π e i(n )t dt r n { πi, n = =, n.5. f(z) D D C f(z)dz = C.6 f(z) D D a D C f(a) = f(z) πi z a dz C 8

19 f(z) D f(z) D a D C f (n) (a) = n! f(z) dz, n =,...,... πi (z a) n+ C z C; z i = (z )(z i) z e C; z = z C; z = z 3 8 z sin z ze C; z + i = z C; z = z+i (z ) f(z) D D a D f(z) f(z) = f(a) + f (a)! (z a) + f (a)! (z a) f (k) (a) (z a) k +... k! a a =, log( + z) = ( ) n= n zn n!, z < ( + z) k k(k )...(k n + )z n =, z <, k n! n= 9

20 z = z n, z < n=, 4 D f(z), h(z) D z {z n } n= f(z j ) = h(z j ), j =,...,... D f(z) = h(z) 5 D f(z) D a f(z) f(z) 6 D {f j (z)} j= f(z) D f(z) D 7 8 n n n a z n + a z n +...a k z n k a n z + a n =, a j, j =,,..., n,

21 .7. a a 9 f(z) D = {z : R < z a < R } a f(z) D c n = πi C f(z) = n= n= c n (z a) n, f(z) dz, n =, ±, ±,.. (z a) n+ C z = a r R < r < R. f(z) z a < R c n = f (n) (a), n =,,... n! c n =, n =,, n z = a n < n= n= c n (z a) n (z a) p a f(z) p p a f(z)

22 3. f(z) = sin z z z = z = sin x sin x = ( ) n x n (n )! n= f(z) = xn n= ( )n (n )! z = ( ) n x n (n )! = x 3! +... n= f() = f(z) z = f(z) < z a < R a f(z) f(z) = (z )(z ) z =, 3 () z < f(z) f(z) = (z )(z ) = (z ) (z ) = z + z z <, z <

23 f(z) = ( z )n + n= n= = ( ( n ))zn = n= n= () < z < ( )zn n+ z n f(z) = z + z = z z z z <, z < (3) < z f(z) = ( z )n ( z z )n n= n= z n = ( n+ z )n+ n= n= f(z) = z z z z z <, z < 3

24 f(z) = z ( z )n z n= = z = n= n= n z n n z n+ ( z )n n= α (α = z 3 i) cos z (α = z + ) z+ (α = )(α = z 3 ( z) z 3 +z ) (α = (z ) (z 4) ) z (α = ) e z.7.3 z = a f(z) C a f(z) a Res[f; a] = f(z)dz πi C C { (z a) dz = n (z a) dz = πi, n = n, n C ρ (a) C ρ (a) a, ρ Res[f; a] = πi = πi = πi = c C f(z)dz n= C n= n= n= C c n (z a) n dz c n (z a) n dz 4

25 a a p c () a, f(z) = c (z a) + f(z)(z a) = c + n= n= n= n= c = lim z a f(z)(z a) () a p, c n (z a) n, c n (z a) n+ f(z) = c p (z a) p + c (p ) (z a) (p ) c (z a) + n= n= c n (z a) n, f(z)(z a) p = c p + c (p ) (z a) c (z a) p + n= n= c n (z a) n+p, c = (p )! lim d p z a dz {f(z)(z p a)p }.7.4 5

26 D w = f(z) D C n z = a j, j =,..., n, f(z)dz = πi C n Res[f, a j ] j= x 4 + dx f(z) = z 4 + f(z) C C = C + C, C ; Imz =, R < Rez < R, C ; z = R, Imz >. f(z) C z = + i = e i π 4 z = + i = e i 3π 4 z = z j, j =, f(z) z z j Res[f; z j ] = lim z zj z 4 + = lim z z j 4z 3 = { { 4 e i 3π 4, j = 4 e i 9π 4, j = = 4 i 4, j = 4 i 4, j = 6

27 z 4 + dz = πi(res[f; z ] + Res[f; z ]) C = πi( 4 i i 4 ) = π dz c z 4 + dz + C z 4 + c z 4 + dz C z = x R c z 4 + dz = R x 4 + dx R c z 4 + dz x 4 + dx. C z = Re iθ, < θ < π, dz = ire iθ dθ, z 4 + = (Re iθ ) 4 + R 4, π z 4 + dz = c π = πr R 4 (Re iθ ) 4 + ireiθ dθ R R 4 dθ 7 (as R )

28 π x 4 + dx = π dθ, (a > b > ) a + b cos θ z = e iθ, < θ < π, z O C C; z =,, cos θ = (eiθ + e iθ ) = (z + z ), dz = ie iθ = izdθ, < θ < π, a + b cos θ = a + b (z + z ) = z (bz + az + b), π dθ a + b cos θ = = C C z bz + az + b iz dz bz + az + b i dz = 4π (C f(z) = bz + az + b 8

29 f(z) = bz +az+b C bz + az + b = a ± a b b z = a + a b b z f(z) z z Res[f, z ] = lim z z bz + az + b = lim z z bz + a = a b π dθ a + b cos θ = π a b cos mx x + dx f(z) = eimz z + C C = C + C, C ; Imz =, R < Rez < R, C ; z = R, Imz >. C f(z) z = i i f(z) Res[ eimz z + eimz, i] = lim(z i) z i z + e imz = lim z i z 9 = e m i

30 e imz dz = z πe m + C C C C z = x C e imz z + dz = R R C e imx x + dx = R R z = Re iθ, θ π dz = ire iθ, cos mx + i sin mx dx x + e imz = e im(x+iy) = e imx e my = e my = e my, y, C e imz z + dz C eimz π π z + dz R R e iθ + dθ R R dθ = πr R (as R ) R cos mx + i sin mx dx = πe m x + cos mx x + dx = πe m, sin mx dx =, x + 3

31 sin x dx x C f(z) = eiz z C = C C C 3 C 4, C = {Imz =, ε Rez R}, C = { z = R, Imz > }, C 3 = {Imz =, R Rez ε}, C 4 = { z = ε, Imz > }, C f(z) e iz z dz = C C j, j =,.., 4, C, C 3 C e iz z dz + C3 e iz z dz = z = x, ε < x < R, R < x < ε = R ε R e ix = i ε R ε e ix ε x dx + x dx sin x x R R e iy ε dx i 3 e ix x dx y dy, (x = y) sin x dx(as ε and R ) x

32 C z = Re iθ = R(cos θ + i sin θ), < θ < π, dz = ire iθ dθ, e iz z eiz z = eir(cos θ+i sin θ) Re iθ = e R sin θ R C e iz z dz π π e R sin θ dθ = π e R sin θ dθ e Rθ π π dθ = R ( e R ) (as R ) θ π sin θ θ, ( θ π )...( ) ( ) F (θ) = sin θ θ F (θ) = θ cos θ sin θ θ G(θ) = θ cos θ sin θ G (θ) = θ sin θ, ( θ π ) G(θ) G(θ) G() = 3

33 F (θ) F ( π sin θ ) F (θ) F () = lim θ θ π F (θ), ( θ π ) = C 4 e C4 iz z dz = C 4 z dz + e iz dz C 4 z z = εe iθ, < θ < π, dz = iεe iθ dθ C 4 z dz = π iεe iθ dθ εe iθ = iπ e iz eiz = ( n= z n= z z n n! (iz) n n! ) z = e z = e ε dz e iz C 4 z e iz dz C 4 z π n= z n n! = + z! +... e ε εdθ = e ε επ (as ε ) 33

34 i sin x dx iπ = x sin x dx, sin x x dx = π cos x dx f(z) = e z C = C C C 3, C = {Imz =, < Rez < R}, C = { z = R, < arg z < π 4 }, C 3 = {z; arg z = π, < z < R}, 4 C f(z) C e z dz = C j, j =,, 3, C z = x, dz = dx C e z dz = R e x dx π (as R ) 34

35 C z = Re iθ, < θ < π 4, dz = ire iθ dθ, z = R e iθ = R (cos θ + i sin θ) = R cos θ + ir sin θ, e z = e R cos θ (cos(r sin θ) i sin(r sin θ)), e z = e R cos θ π e z 4 dz e R cos θ Rdθ = C (θ=φ) π π e R cos φ Rdφ = e R sin φ Rdφ (φ = π φ) ( 4 ( ) ) π C 3 z = ρe i π 4 e R π φ Rdφ = π R 4R [e π φ ] π = π ( e R) (as R ) 4R = ρ(cos π 4 + i sin π 4 ) = ρ( z = ρ e i π = iρ, e z = e iρ = cos ρ i sin ρ, 35 + i ), < ρ < R,

36 dz = ( + i )dρ, R e z dz = (cos ρ i sin ρ )( C 3 + i )dρ R R = (cos ρ + sin ρ )dρ i (cos ρ sin ρ )dρ (cos x + sin x )dx i (cos x sin x )dx(as R ) (cos x + sin x )dx = (cos x sin x )dx = π, cos x dx = π, sin x dx = π f(z) = ez z n+ O C π e cos θ cos(nθ sin nθ)dθ, π e cos θ sin(nθ sin nθ)dθ 36

37 f(z) = ez (n + ) z n+ Res[f, ] = n! lim ez z (zn+ )(n) zn+ = n! lim z ez = n!. z = e iθ = (cos θ + i sin θ), < θ < π, dz = ie iθ dθ e z dz = zn+ C i π = i = i π cos θ+i sin θ e e i(n+)θ π π ie iθ dθ e cos θ+i sin θ e inθ dθ e cos θ e i(sin θ nθ) dθ π i e cos θ e i(sin θ nθ) dθ = π n! i e cos θ (cos(sin θ nθ) + i sin(sin θ nθ))dθ = π n! i π π e cos θ cos(sin θ nθ)dθ = π n!, e cos θ sin(sin θ nθ))dθ = π +sin θ dθ π π dθ +sin θ x 6 + dx dx (x 4 +) cos x x + dx x sin x x + dx x k dx( < k < ) x+ dx (x +) n 37

38 . < x j <, j =,.., n f = f(x), x = (x,..., x n ) ( ) n... π f(x) ( ) n... π f(x)e ix ξ dx...dx n f(x)e ix ξ dx...dx n = F [f](ξ) x ξ = n j= x jξ j x = (x,.., x n ) ξ = (ξ,..., ξ n )... f(x) dx...dx n < L. ( ) n... π F [g](x) = ( ) n π F [f](ξ)e ix ξ dξ...dξ n = f(x)...() g(ξ)e ix ξ dξ...dξ n

39 F [F [f]] = f F [ F [g]] = g F F = F F = I F D λ (x) D λ (x) = λ cos ωxdω D λ (x) (i)d λ (x) = sin λx x (ii) lim λ b a D λ (x)dx = π, a < < b (iii) lim f(ξ)d λ (x ξ)dξ = {f(x + ) + f(x )} λ π (iv) π ( f(ξ) cos ω(x ξ)dξ)dω = {f(x + ) + f(x )} ( (i) (ii) sin x x dx = π 39

40 (ii) (iii) (iii) (iv) e iθ = cos θ + i sin θ (ii) ( D λ (x) λ δ(x) lim x D λ (x) = sin λx lim x = λ lim x x δ(x) = x = x lim λ D λ (x) δ(x) D λ (x) (ii) (iii) λ () b a δ(x)dx = π, a < < b, () π f(ξ)δ(x ξ)dξ = f(x) π D λ (x) F [f (α) ](ξ) = (iξ) α F [f](ξ)...() f (α) (x) = α x α x... α n x n f(x) = α δ(x) α x α x... α n x n f(x), (iξ) α = (iξ ) α (iξ ) α...(iξ n ) αn, α = α +...α n 4

41 π f (x)e ixξ dx = [f(x)e ixξ ] x= x= π π f(x)(e ixξ ) dx = (iξ) f(x)e ixξ dx π π [f(x)e ixξ ] x= x= lim f(x) = x ± ( ξ )α F [f](ξ) = F [( ix) α f](ξ)...(3) δ F [δ] = F [δ] =, F [] = F [] = δ...(4) F [f g] = πf [f]f [g] f g f g (f g)(x) = f(x y)g(y)dy (f g)(x) = 4 f(y)g(x y)dy

42 f(x) = e a x (a > ) π e a x e ixξ dx = π ( lim x ± e (±a+iξ)x = e (a+iξ)x dx + e ( a+iξ)x dx) = { π a + iξ [e (a+iξ)x ] x= x= + a + iξ [e ( a+iξ)x ] x= x=} = ( π a + iξ + a iξ ) = a π a + ξ lim x ± e ax cos ξx i sin cos ξx = lim x ± e ax = (a > ) π a a + ξ eixξ dξ = e a x cos xξ a + ξ dξ = π a e ax { sin xξ a + ξ dξ = {, x < a, f(x) =, x a f(x) = e a x, a > f(x) = x, x < a, a, x a { f(x) = x, x <,, x f(x) = 4 { x, x <,, x

43 .3.3. x t u = u(x, t) u(x, t) u t = u xx (t >, < x < )...() u(x, ) = φ(x)...() () x U(ξ, t) = π u(x, t)e ixξ dx u(x, t) t x U(ξ, t) du dt = ξ U...(3) ξ (3) U(ξ, t) = e ξt Φ(ξ)...(4) Φ(ξ) ξ () Φ(ξ) φ(x) e ξt = F [ 4t ] u(x, t) = e x 4t 4πt = 4πt 43 φ e (x y) 4t φ(y)dy t e x

44 .3. f(t)(t > ) F (s) = e st f(t)dt t < f(t) = s = σ + iς(σ, ς ) F (σ + iς) = e (σ+iς)t f(t)dt = e iςt e σt f(t)dt e σt f(t) = π π f(t) = π = σ+iς=s idς=ds F (σ + iς)e iςt dς F (σ + iς)e (σ+iς)t dς πi σ+i σ i F (s)e st ds (σ, ) R C Re(z) = σ, R < Im(z) < R C ( z σ = R, Re(z) < σ) F (s) a k (k =,,.., n) C F (s)e st ds (R ) n f(t) = Res[F (s)e st, a k ] s s +a s z = ±ia Res[ e st, ±ia] s +a Res[ s s + a est, ±ia] = lim s ±ia 44 = lim s ±ia = ±ia ±ia e±iat s(s ia) s + a s (s ± ia) est est

45 f(t) = (eiat + e iat ) = cos at a s a s s +a (s a) (s +a ) (s +a ) (s +a ) n a (s +a ) n 45

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