( V V dv = ˆx + x y ŷ + V ) z ẑ (dxˆx + dyŷ + dzẑ) (gradient) ( V V V = ˆx + x y ŷ + V ) z ẑ (infinitesimal displacement) dl = (dxˆx + dyŷ + dzẑ) θ dv

Transcription

1 ,2 () Needham WIKI WEB... C = A B A, B θ C C = (A B) (A B) C 2 = A 2 + B 2 2AB cos θ..2 f(x) df dx = lim f(x) x x df = (Oridinary Derivatives) ( ) df dx dx 3 V (x, y, z) ( ) ( V V dv = dx + x y ) dy + ( ) V dz z ˆx, ŷ, ẑ 2

2 ( V V dv = ˆx + x y ŷ + V ) z ẑ (dxˆx + dyŷ + dzẑ) (gradient) ( V V V = ˆx + x y ŷ + V ) z ẑ (infinitesimal displacement) dl = (dxˆx + dyŷ + dzẑ) θ dv = V dl = V dl cos θ V V V V 2..3 Operator (Operator) Graduent ˆx x + ŷ y + ẑ z r r r = x 2 + y 2 + z 2 r = ˆx r x + ŷ r y + ẑ r z ( = 2 r = (x x ) 2 + (y y ) 2 + (z z ) 2 2x x2 + y 2 + z 2 ˆx + 2y x2 + y 2 + z 2 ŷ + ) 2z x2 + y 2 + z ẑ 2 = xˆx + yŷ + zẑ x2 + y 2 + z 2 = r r ˆr r 2

3 ( r 2) = ( ˆx x + ŷ y + ẑ ) ((x x ) 2 + (y y ) 2 + (z z ) 2) z = 2 {(x x )ˆx + (y y )ŷ + (z z )ẑ} = 2r 2 (r n ) = ( ˆx x + ŷ y + ẑ ) ((x x ) 2 + (y y ) 2 + (z z ) 2) n/2 z = n 2 {2(x x )ˆx + 2(y y )ŷ + 2(z z )ẑ} ( (x x ) 2 + (y y ) 2 + (z z ) 2) n 2 2 = n ((x x )ˆx + (y y )ŷ + (z z )ẑ) ( (x x ) 2 + (y y ) 2 + (z z ) 2) n 2 ((x x ) 2 + (y y ) 2 + (z z ) 2 ) /2 = n ˆr r n ( ) r ( = ˆx x + ŷ y + ẑ ) ((x x ) 2 + (y y ) 2 + (z z ) 2) /2 z ( ) = 2(x x )ˆx + 2(y y )ŷ + 2(z z )ẑ 2 ((x x ) 2 + (y y ) 2 + (z z ) 2 ) 3/2 = ˆr r 2.2 [57] (Gauss s theorem) V S n x, t u(x, t) q(x,t).: [57] 3

4 ρ c cρ u t dv = V S q nds (.) AdV = A nds (.2) V V ( cρ u ) t + q dv = S cρ u t + q = (.3) q = k u (k > ) ().3 cρ u t = u u t = κ u, κ = k cρ, = κ() κ = iħ.2: 4

5 Maxwell divb = (.4) 2,4 B + rote = t (.5) divd = ρ (.6) roth D t = j (.7) D = ϵe, B µh J = σe E = µ H t H = σe + ϵ E t H, E E ( E) = ( E) 2 E E = E σ = 2 E = µ t ( H) ( = µ t ϵ E t + σe) = µσ E t + µϵ 2 E t 2 2 E t 2 = c2 2 E H H = 2 H = µσ H t = µσ H t σ = 2 H t 2 H, E ϵ t ( ) µ H t + µϵ 2 H t 2 = c2 2 H 5

6 .3 d P(t) = F(x, t) dt F = ax mẍ = ax ẍ + ω 2 x = m y = u(x, t) ( v 2 g 2 ) t 2 2 x 2 u(x, t) = (.8) v g v p A ω = a m v p = Aω mẍ = ax + mg (.9) x = x x = mg a mẍ = a(x + x ).9 λ k v g = ω k v g v p ω = v g k.8 k a m = ω = v p A = v gk (.) v 2 g = E = a/k 2 (.) a mk 2 = E m 6

7 ( m E 2 t 2 2 x 2 mv 2 g = E ) u(x, t) = mc 2 = E x 2 ϕ(x) = 2 mẍ max = ax max.8 ( v 2 g 2 mẍ = κẋ t 2 + κ t 2 x 2 ) u(x, t) = (.2) ( ) κ t 2 x 2 u(x, t) = (.3) κ.8 9 t iħ E m ( iħ ) t 2 x 2 u(x, t) = 7

8 .3. y = f(x, y) ω ω ω (exact) ω P (x, y)dx + Q(x, y)dy = (.4) 2 x, y xy 2 (x, y), (x + x, y + y) x y U = U(x + x, y + y) U(x, y) (partial diffrential) U x = lim U(x + x, y) U(x, y) x x U y = lim U(x, y + y) U(x, y) x y x, y U = U(x + x, y + y) U(x, y) = U U x + x y y du = U U dx + x y dy du U (total diffrential).4 P (x, y) = U U, Q(x, y) = x y ω = du = (.5) (exact diffrential equiation).5 U(x, y) = Const P y = 2 U x y = Q x P y = Q x 8

9 P dx + Qdy = du P, Q 2 C, C 2.3: 2 C : U(x, y) = C 2 : U(x, y) = x y P (u, y )du + P (x, v)dv + U(x, y ) x y y x P (x, v)dv + P (u, y)du + U(x, y ) y x C (P dx + Qdy) = C (P dx + Qdy) C (P dx + Qdy) = C 2.4 ω = (P dx + Qdy) = (.6) C C D i a y D C a x D I.4: D=C C ω = lim ω = a i D i 9

10 C P P OP ω = y x 2 + y 2 dx x x 2 + y 2 dy U = tan ( x y ) { tan x } = + x 2 U x = y + (x/y) 2 = y x 2 + y 2 U y = x y 2 + (x/y) 2 = x x 2 + y 2 ω ω = ( ) x U = tan = Const (.7) y y = Ax x = r cos θ y = r sin θ.7 ( ) cos θ U = tan sin θ = tan ( sin [ θ + π/2] cos [ θ + π/2] ( ) sin [θ π/2] = tan cos [θ π/2] ) = 2 π θ C ω = du = dθ ω = dθ = C

11 θ y y P x x.5: C C dθ = dθ = 2nπ n θ xy ω xy y x.6: x n ξ 2π θ

12 .4 D (t, x) = (R R n ) G(t, x) C (D, R) ẋ = f(t, x) (first integral) G c = G(ξ R n ) S S (level set) (τ R, ξ R n ) t S = G (c); G (c) := {x R n G(x) = c} d G(t, x(t, τ, ξ) = (.8) dt d dt = t + ẋ x.8 G d G(x(t)) = G(x(t)), ẋ(t) = dt ẋ(t) G S G S (regular) x G(x) = c x = x (x 2, x n ) (.9) S n S n C.8 G n t + G f(t, x) = (.2) x k k= G, f t ( G G = t, G ) x x n G(x(t)), f(x) = G f XG, X = n k= f k (x) x k (.2) G (differntial oprator) ẋ = e k 2

13 x k n { x, x n } R n (tangent space) T x R n f(x).7: t = ξ S γ : t x(t), x() = ξ ξ x(t, ξ) ẋ(, ξ) = f(ξ) T ξ S x S f(x) T x S ẋ = f(x).9 n ẋ k = f k (x (x 2,, x n ), x 2, x 3, x n ) k = 2, n C 2 U(x) x = x(x,, x n ) n R n ẋ k m k ẍ k = U x k (k =, n) (.22) m k ẍ k ẋ k U x k ẋ k = ( n ) d dt 2 m kẋ 2 k U = k= 3

14 H H = n k= 2 m kẋ 2 k + U(x).2 (x, p) ẋ k = H p k ṗ k = H x k.22 H ẍ = ω 2 x (.23) x(t) = C cos ωt + C 2 sin ωt.23 ẋ ẋẍ = ω 2 ẋx ( d 2ẋ2 + ) dt 2 ω2 x 2 = ẋ ω 2 = k/m 2 ω E ẋ(t) 2ẋ2 + 2 ω2 x 2 = E ẋ(t) = ± 2E ω 2 x 2 (t) z(t) x(t) = 2Eω 2 z(t) (.24) ż(t) = ± ω 2E ω2 x 2 (t) = ±ω z 2 2E 4

15 ωdt = z z 2 dz t z z z t ωdt = z z z z 2 dz = [ sin (z) ] z z ωt = sin z sin z ωt sin z(t) = sin(ωt + α) α = sin z.24 x(t) = 2Eω 2 sin(ωt + α) R 3 P m,x = (x, x 2, x 3 ) G mẍ = GM mx x 3 k = GM ẍ = k x 3 x x x (t), x 2 (t), x 3 (t) R x x x = Rx x x 2 -sin cos r 2 r x x 2 r x x cos sin x x Rx = x ẍ = k x 3 x (.25).8: 5

16 r(x, x 2 ) r = rˆr 2 ṙ = ṙˆr + r ˆr = ṙˆr + r ϕ ˆϕ (.26) ˆr = d dt (cos ϕ, sin ϕ)t = ϕ( sin ϕ, cos ϕ) t = ϕ ˆϕ.25 ẋ.26 ( ( ẍ ẍ 2 x,2 () = ξ,2 ẋ,2 () = η,2 x 3 () = ẋ 3 () = (ẋ2 2 + ẋ 2 2) = Const x 2 + x 2 2 ẋ ẋ 2.25 ( ) = ) = ṙ x = r cos ϕ x 2 = r sin ϕ ( cos ϕ sin ϕ ( r r ϕ 2) ( cos ϕ sin ϕ ẍ ẍ 2 ) ) ) = kr2 ( + r ϕ ( sin ϕ cos ϕ ) ( + 2ṙ ϕ + r ϕ ) ( sin ϕ cos ϕ cos ϕ sin ϕ 2 ) ) 2 r r ϕ 2 = k r 2 2ṙ ϕ + r ϕ = d 2 dt r 2 ϕ = Const ( ) r 2 ϕ = 2 (x, x 2 ) x ẋ 2 = r 2 ϕ sin 2 ϕ x 2 ẋ = r 2 ϕ cos 2 ϕ r 2 ϕ = x2 ẋ x ẋ 2 6

17 Au xx + 2Bu xy + Cu yy + Du x + Eu y + F u + E = (.27) A G x, y P (x, y) 2 ξ(x, y) = Const, η(x, y) = Const (x, y) (ξ, η) (x, y) (ξ, η) ξ x ξ y η x η y (ξ x, ξ y ) (η x, η y ) ξ(x, y) = Const, η(x, y) = Const.9:.27 au ξξ + 2bu ξη + cu ηη + O(u ξ, u η, u) = a b c = Aξx 2 + 2Bξ x ξ y + Cξy 2 = Aξ x η x + B(ξ x η y + ξ y η x ) + Cξ y η y = Aηx 2 + 2Bη x η y + Cηy 2 b 2 ac = (B 2 AC)J 2 (.28) a = Aξx 2 + 2Bξ x ξ y + Cξy 2 = ξ(x, y) = Const ξ x dx + ξ y dy = 7

18 ξ x, ξ y dy/dx 2 Ady 2 + 2Bdxdy + Cdx 2 = dy dx = B ± B2 AC A.27 (.29). B 2 AC > 2. B 2 AC = 3. B 2 AC <.28 A, B, C B 2 AC > A a = c =, b u ξη + O(u ξ, u η, u) = (.3) A =, C u ξη + O(u ξ, u η, u) = A =, C = ξ = X + Y, η = X Y.3 u ξη = (u XX u Y Y )/4 u XX u Y Y + O(u X, u Y, u) = 2 B 2 AC = u tt c 2 u xx = Ady Bdx = ξ(x, y) = Const () η(x, y) = Const a =.28 b = u ηη + O(u x, u y, u) = 8

19 u t κu xx = 3 B 2 AC < 2 Ady (B + i AC B 2 )dx = 2 Ady (B i AC B 2 )dx = f(x, y) = Ay (B + i AC B 2 )x = Const f (x, y) = Ay (B i AC B 2 )x = Const Af 2 x + 2Bf x f y + Cf 2 y = Af 2 x + 2Bf xf y + Cf 2 y = ξ = 2 {f(x, y) + f (x, y)} a = c, b = η = 2i {f(x, y) f (x, y)} u ξξ + u ηη + O(u x, u y, u) = 2 2 Tristan Needham Visual Complex Analysis[] 9

20 2. z = re iθ M(z) = az + b cz + d C (2.) z z + d (2.2) c z z (2.3) ad bc z c 2 z (2.4) z z + d (2.5) c 4 4 S 2 = T 2 (X 2 + Y 2 + Z 2 ) L 4 L 2.. [] [ M(z) = az + b [M] = cz + d a c b d ] (2.6) R 2 [M] = [ ] = z (2.7) 2 z = c c 2 (2.8) z [c, c 2 ] z k [kc, kc 2 ] = k[c, c 2 ] [c, c 2 ] C 2 [ c c 2 ] [ a = c ] [ b d c c 2 ] [ ac + bc 2 = cc + dc 2 ] 2

21 w = c c = ac + bc 2 (2.9) 2 cc + dc 2 w = ac /c 2 + b cc /c 2 + d = az + b cz + d [ c ] [M] λ [ a c b d ] [ c c 2 ] = λ [ c c 2 ] 2.8 M(z) = λc λc 2 = z z = c /c 2 [c, c 2 ] [M] [M] det{[m] λ[ε]} = [ε] = [ ] 2.3 a, d λ 2 (a + d)λ + = λ = 2 {(a + d) ± (a + d) λ λ 2 =, λ + λ 2 = a + d N λ λ 2 λ n = detn, λ + λ λ n = tr[n] 2

22 2..2 C 2 p, q 2 p q = p q + p 2 q 2 = p(, i), q(, i)c 2 p, q = p q + p 2 q 2 (2.) p q + p 2 q 2 = p / p 2 = q /q 2 p(p /p 2 ), q(q /q 2 ) q = p p, q C 2 2 C 2 [R] 2. [R] p, [R] q = p, q p {[R] [R]}p = p q [R] [R] = [ε] [R] = [R] [ a [R] = b b ā ] (2.) R(z) = az + b bz + ā a, b (2.2) 22

23 2..3 [8] M(z)M n M n M P SL(2, C) L F ix(m) {z S 2 Mz = z} M id () M(z). (Loxodromic):F ix(m) 2 z, z 2 lim M n (z) = z, n lim n M n (z) = z 2 2. (elliptic):f ix(m) 2 z, z 2 S 2 /{z, z 2 } L t L t M(L t ) = L t 3. (parabolic):f ix(m) z S 2 z L t M(L t ) = L t lim M n (z) = lim M n (z) = z n n M(z) ξ +, ξ C C p ξ +, ξ p K ξ +, ξ K ξ + ξ F (z) = z ξ + z ξ z = F (z), w = F (w) z w = M(z) F F z w = M(z) M w = F (w) = F (M(z)) = F (M[F (z )]) M = F M F M, m M (z ) = mz (2.3) m = ρe iα (2.4) α ρ m = e iα M 23

24 2.2 O A, B A B AB O O A B 2.: O AB A B C (A B) C B C A (B C) 2 (A B) C = A (B C) 2.2: (A B) C = A (B C) 2.2. R K T z q q z ρ z T (z) q z q ρ R 2 /ρ T (z) 24

25 2.3: [] z q T (z) q = ρρ = R 2 (2.5) (T (z) q)(z q) = R 2 T (z) = q =, R = R2 z q + q = q z + (R2 q 2 ) z q (2.6) T (z) = z (2.7) O A,B A B O B A K A B O R 2.4: ( A B = R 2 OA OB ) AB (2.8) O A,B A B OAB OB A 25

26 2.6 T L K L L K K L K K L A A B B BAq π/2 A B q π/2 qa L q K B A B A L q 2.5: C K q K C K T K K R L L 2 R L a R L R L = id 3 a T K 26

27 2.6: [] q q K ACB A B C q 2.7: K K A,B,A,B q,c,c q A A z /z =, = (2.9) z /z 27

28 2.8: q C L C C q [] C K 2 a, b a T K q K K 2.9: [] q C z z C b K z z z K 2 z 2.: [] 28

29 K pz p z = K q = ir K z p T k (z) T R T (z) z < R i z/r T K (z) = z + i z2 R z3 = z (i z/r) R C S R P P P P S R = C R 2.: p, q p p 29

30 q z P h P 2.2: p(ϕ, θ) θ = ON p p = re iθ = cot ( ) ϕ e iθ (2.2) 2 q p (ϕ, θ) q (ϕ + π, θ) 2.2 cot(θ + π/2) = tan(θ) 2.7 ( ) ϕ + π q = cot e iθ (2.2) 2 = tan(ϕ)e iθ = p p (x, y, z ) = T (p) (2.22) x 2 + y 2 + z 2 = (2.23) p C hn z NO p = h p h (2.24) p h = z (2.25) h = x + iy (2.26) p = x + iy = x + iy z (2.27) p 2 = x 2 + y 2 ( z ) 2 = z 2 ( z ) = + z z 3

31 x + iy = z = p 2 p 2 + 2p 2x + i2y = + p 2 + x 2 + y 2 (2.28) (2.29) S S 2 p p T, T 2 p S, S 2 T T 2 2.3: [] p K z K 2 2.4: [] a p K 2 T, T 2 K p = T K (p) θ 3

32 b b 2 2.5: a b [] z T T : re iθ r e iθ (2.3) 2.6:. z = re iθ z z r eiθ = z 2. z z K C 2. K K K 32

33 /3 2.7: /3 2.8: /3 2.9: 2 3 C C K C C 33

34 K C T(C) C C C C K T(C) 2.2: C K 3 M(z) = az + b kaz + kb = cz + d kcz + kd 3 ad bc = ac ad = bc M(z) = acaz + acb accz + acd = a2 cz + a 2 d ac 2 z + bc 2 = a2 (cz + d) c 2 (az + b) = a2 c 2 M(z) M(z) 2 = a2 c 2 a/c z z M( d c ) = ad bc = (2.3) M (z) = dz b cz + a ξ z = M(z) (2.32) 34

35 ξ ± = (a b) ± (a + d) 2 4 2c (2.33) z q,r,s 3 C q r s C r q s q s r 2.2: 3 p,q,r,s 3 p,q,r,s [w, q, r, s ] = [z, q, r, s] = (z q)(r s) (z s)(r q) (2.34) 3 q,r,s C C 3,, 2.3 O L L M L (2.35) P 2 L,L2 p, ˆp 35

36 2.22: p p L,L2 θ/2 θ R θ p (2.36) 2 p Rp θ 2 R θ p = M L2 M L ab a b P ˆP a b L ˆP = M L P P = R ϕ a Rθ p (2.37) L Rf L R P ϕ R ϕ P p, q R ϕ P = R ϕ q (2.38) Rf L2 Rf L = R ϕ P : p θ p θ/2 36

37 T s S 3 S q 2 S q 3.29 p q ρ T s (p) q p R 2 ρ q Π q 2.23: [2] 2 q 2.4 z f dx df = f dx (2.39) or π 2π z 37

38 2.24: f (z) z f (z) f(z) f (z) = f (z) exp(i arg[f (z)]) df(z) = f (z)dz dz f (z) argf (z) f f = f (z) (2.4) = argf (z) (2.4) 2.25: 38

39 f : z z (2.42) 3 f(z) p f(z) f(z) 2.26: 2.4. z z = p f(z) f(p) z m z = (m ) f (p) = p f = z m m f (z ) = w = f(z) w = f(z ) z z w w m z w z = x + iy w = a + ib w z = (ax by) + i(bx + ay) R 2 (x, y) A 39

40 A = ( a b b a ) (2.43) Jacobi J ( ) x u y u J = x v y v (2.44) (CR ) x u = y v (2.45) x v = y u (2.46) 2 K ϵ x ϵ f 2.27: x f = f x y x f = ϵ x f f y f = ϵ y f CR f = u + iv (2.47) i x f = i x u x v y f = y u + i y v i x f = y f π/2 4

41 z r dr r θ dθ 2.28: e iθ dr (2.48) f e iθ irdθ = izdθ (2.49) dr r f (2.5) dθ θ f (2.5) dθ θ f = idr r f (2.52) dr = rdθ (2.53) 2.47 CR θ f = ir r f (2.54) θ v = r r u (2.55) θ u = r r v e iθ dr f = dr r f izdθ f = dθ θ f f = e iθ r f 4

42 f = i z θf dθ dr u, v 2.29: v r u θ v(θ), u(r) 2.55 v r =, u θ = (2.56) θ v(θ) = r r u(r) (2.57) C θ v(θ) = C r r u(r) = C v(θ) = Cθ + C u(r) = C log r + C C u + iv = C(log r + iθ) + C 42

43 z = e iθ f(z) = C log z + C f = log z = log r + i(θ + 2nπ) f = e iθ r f = e iθ r = z f = i z θf = z /r θ log dθ K2 f(e2) e K f(e) 2.3: dθ Log θ ± 2mπ e z x,y e x θ e z = e x e iθ (2.58) 43

44 δ 2.3: e z 2.6 p κ K f κ K p κ K p ξ f ξ ζ ϵ q A C K B B C A 2.32: p, p κ = ϵ/ ξ κ = ϵ/ ξ (2.59) 2.4 ξ = f (p) ξ p p ϵ σ ϵ = ϵ + σ 44

45 σ p f(p) f (p) p f (p) f (p) z ξ f (p) χ χ = f (p)ξ (2.6) K p f f (p) g 2.33: f (p) σ 2.6 σ = Im [χ/f (p)] σ = Im [f (p)ξ/f (p)] (2.6) v θ u r θ v(θ) = r r u(r) (2.62) A c θ v(θ) = A (2.63) r r u(r) = A (2.64) v = Aθ + c u = Alog r + c u + iv = A(log r + iθ) + c z = re iθ 45

46 u + iv = Alog z + c 2.34: log z,, z K p κ = Im [f (p)ξ/f (p)] + ϵ f (p) ξ ˆξ = ξ ξ κ f κ = ( [ ] ) f Im f (p)ˆξ/f (p) + κ (p) [ ] f (p)ˆξ = Im f (p) f (p) + κ f (p) (2.65) 2 2 R r = /k r = R/κ K [ ] f (p)ˆξ k(ˆξ) = Im f (p) f (p) K = i f f f κ(ˆξ) = K ˆξ + κ f (p) K K = K + i K2 (2.66) K, K2 e, e2 f K2 f(e2) e K f(e) 2.35: K f

47 f K f 2.36: K ξ 2.6 χ, K ξ K (2.67) χ f K ī f if (2.68) f π/2 f Q f Q Q K Q Q ik Q ˆξ Q R Q K R = K ˆξ (2.69) S K ˆξ θ S = K ˆξsinθ (2.7) S = ˆξ K (2.7) p ξ ζ 2 σ 47

48 2.7 f(r) f(r) r m m r = f(r) r = r r z r = pe it + qe it (2.72) z = a cosθ + ib sinθ a > b a = p + q b = p q ± a 2 b 2 = ±2 pq a b a b r 2 z z : z = e it +.3e it z 2 48

49 2.8 Kasner,Arnol d F r z z m (A + 3)(Ã + 3) = 4 m = A A = m = 2 2 A = 2, m = /2 Kasner,Arnol d F r A 3 < A < (2.73) 2.9 P n Π Π p κ Π n κ max κ min k(p) k(p) = κ min κ max (2.74) T π E E(T ) (T ) π p da E( ) = k(p)da (2.75) E, da k E(T ) = k(p)da k E(T ) = k(p)da = ka(t ) T T 49

50 For 2.38: : π Thomas Harriot T α, β, γ T A(T ) 5

51 A(T ) + A(T α ) + A(T β ) + A(T γ ) = 2πR 2 3 α, β, γ A(T ) + A(T α ) = 2πR 2 α 2π = 2αR2 A(T ) + A(T β ) = 2πR 2 β 2π = 2βR2 A(T ) + A(T β ) = 2πR 2 β 2π = 2βR2 3A(T ) + A(T α ) + A(T β ) + A(T γ ) = 2(α + β + γ)r 2 A(T ) = (α + β + γ π)r 2 (8 ) E k = /R 2 E(T ) = ka(t ) (2.76) 2. S ẑ C Z ds ˆds dz z z, ϕ ˆds = Λ(z, ϕ)ds ϕ ˆds = Λ(z)ds (2.77) S Λ C N C K 2 ds ^ ds z z^ N 2 K C 2.4: 5

52 2.8 3 ˆds = 2 (Nz) 2 ds (Nz) 2 = + z 2 ˆds = 2 ds (2.78) + z2 f(z) 2.4 Λ( z)d s = Λ(z)ds (2.79) z = f(z) d z = f dz (2.8) 2.79 f (z) = Λ(z) Λ(f(z)) (2.8) k = S Σ 2.78 f (z) = + f (z) 2 + z 2 (2.82) Σ C â, ˆb Σ â θ 2 Σ 2 â, ˆb ˆL, ˆL2 R θ â = Rf ˆL2 Rf ˆL Ra θ = Rf L2 Rf L Rf T Ra θ 2.4 m m = e iθ Σ Râ θ C Rθ a Rθ a a, /ā a m = e iθ ( ) [ ] R θ e iθ/2 a 2 + e iθ/2 2iasin(θ/2) a = m 2iāsin(θ/2) e iθ/2 a 2 + e iθ/2 52

53 m = ( e iθ e iθ ) 2.2 â z a z + B B z + Ā l 2 + m 2 + n 2 = v 2.28 v = li + mj + nk a 2 = + n n [ R θ v ] = ( cos(θ/2) + insin(θ/2) ( m + il)sin(θ/2) (m + il)sin(θ/2) cos(θ/2) insin(θ/2) ) (2.83) , I, J, K I 2 = J 2 = K 2 = (2.84) IJ = K = JI, JK = I = KJ, KI = J = IK (2.85) 4 V 4 V = v I + v 2 J + v 3 K V = v + V v = V 4 4 W 2.85 V = v + V W = w + W VW = vw V W + vw + wv + V W (2.86) 53

54 V, W 4 VW = V W V W I, J, K 2.83 θ = π ( [Rv π ] = in m + il m + il in ) (2.87) = [ ] [, I = i i ] [, J = i i ] [, K = i i ] (2.88) R θ V = cos( θ 2 ) + V sin(θ 2 ) V = li + mj + nk i π/2j π/2 R π/2 j R π/2 i = 2 ( + J) 2 ( + I) = ( + J + I K) 2 θ = 2π/3 v = 3 (i + j k) 3 P R θ V P 3 P = R θ V PR θ V 2.4: 54

55 w w w w ˆp w w R θˆp = Rϕ â Rθˆp R ϕ â 4 R θ P = Rϕ V Rθ PR ϕ V θ = π /R Y R X = Re σ/r (2.89) 2.42: tratrix dσ dx = R x Y R R (2.9) 55

56 r o P r S B C A R Q P r o S r 2.43: P r Y O OP r P O S r r 2 k k = rr AC R = P Q r AC QP = R r P A = QB = R QP = BC BCA APO ABC OAP r R = AC BC = AC QP = R r k = rr = R 2 x 2π σ (x, σ) (x, y) z (x, σ) σ X 2.9 dx dσ = X R σ = X = R log X = σ/r + C R = X = Re σ/r X = e σ 56

57 σ= x = σ= y(σ) = p(x, σ), q(x + dx, σ) dx X Xdx 2 dx /X 2.77 ˆds = Xds (2.9) ϵ ϵ/x y σ = X = σ = y = dσ = Xdy dy dσ = X y = e σ (2.92) ˆds = Xds = dx2 + dy 2 Ŝ y dŝ = dxdy y 2 (2.93) r x= x rdx y x= dx x 2.44: σ = σ = y t 2.92 y = e σ(t) σ(t) = t 57

58 y = σ = t Beltrami dx2 + dy ˆds = 2 = ds/y (2.94) y y = () ˆds x + iy y = 2.3 ϕ ϕ = kq r z (2.95) z = re iθ f(z) = kq z

59 e 2.45: iθ r 2θ 2π π/2 3 2n z = x + iy (x, y) z h f(z + h) f(z) lim = f (z) (3.) h h h = h e iϕ (3.2) h ϕ ε δ δ = z z (3.3) z f (z) f(z ) f(z) z z < ε (3.4) 59

60 f (z) f (z) ε 3.: ε ε dω = ε (3.5) 3. ϵ D z ω = f(z) = u(x, y) + iv(x, y) z h = x + i y f (z) ω f(z + h) f(z) = u + i v f (z) = a + ib u + i v = f (z) z + O(h) u = u(x + x, y + y) u(x, y) = a x b y + O( x 2 + y 2 ) 6

61 v = v(x + x, y + y) u(x, y) = b x + a y + O( x 2 + y 2 ) (3.6) (x, y) u x = v y = a u y = v x = b (Cauchy-Riemann s relation) h = x + i y u x v y = u y + v x = (3.7) f(z + h) f(z) = u + i v = (a + ib)( x + i y) + O( k ) f(z + h) f(z) lim = a + ib (3.8) h h f(z) D 3.3 Green z S AdS = A dl (3.9) X(x, y), Y (x, y) C D C (Xdx + Y dy) = Green D u(x, y), v(x, y) D l [ X y + Y ] dxdy (3.) x C f(z) = F (z, z) = u(x, y) + iv(x, y) F (z, z)dz = Green C (udx vdy) + i (vdx + udy) C 6

62 C F (z, z)dz = D = i = i ( u y v x ) dxdy + i (u x + iv x ) dxdy D D ( x F + i ) y F dxdy D D ( v y + u x ) dxdy (u y + iv y ) dxdy C 2 z = x i y, 2 z = x + i y F (z, z)dz = F (z, z)(idy + dx) (3.) C = i F (z, z)dxdy F (z, z)dxdy D x D y { = i D x F (z, z) + i } F (z, z) dxdy y = 2i F (z, z)dxdy z D 3.3. D S = dxdy F (z, z)dz = z 2iS D S = dxdy = zdz (3.2) D 2i D x 2 + y 2 = C z = e iθ z = e iθ dz = ie iθ dθ S = zdz = 2i 2i C C ω = u(x, y) + iv(x, y) idθ = π 62

63 S 3. S = dudv = ωdω D 2i C = ω dω 2i C dz dz ( = ω dω ) dxdy D z dz d ω dω = d z dz dxdy = ω (z) 2 dxdy D z z G = x G + iy G = S 3.2 D D zdxdy (3.3) z G = = z zdxdy 2iS D (3.4) C z zdz C zdz (3.5) a x 2 + y 2 = a 2 3.2: z = ae iθ z = ae iθ dz = iae iθ dθ 63

64 π z zdz = ia 3 e iθ dθ = 2a 3 C z = z = x z zdz = a C2 a x 2 dx = 2 3 a3 S C+C2 z zdz = 4 3 a3 S = 2i = 2i = 2i zdz C π { i } a 2 dθ + { iπa 2 } = πa2 2 z G = z zdxdy 2iS D = 4ai 3π i y 3.4, 3.4. C D f(z) D D C f(z)dz = (3.6) P r r(p) C ˆD ˆD = D r(p ) (3.7) 3. C f(z) z = (3.8) f(z) f(z)dz = 2i = (3.9) D z 64

65 C, C 2 D D C A B C2 D C A B C2 C2 C3 D C 3.3: f(z)dz = f(z)dz (3.2) C C 2 f(z)dz f(z)dz (3.2) C C 2 γ n D n + n C n + C C D 2 D : C2 C C 65

66 C D C C f(z)dz = n C j (3.22) j= n j= C j f(z)dz 3.6 f(z)dz = 3.2 f(z)dz = f(z)dz C C C D z z C D F (z) = C f(ξ)dξ = z f(z)dz = F (z) z f(ξ)dξ z f(z) = (z a) n a r C z a = re iθ f(z) = r n e inθ n C (z a) n dz = n = 2π dz = ire iθ dθ (3.23) ir n+ e i(n+)θ dθ 2π = ir n+ [cos ((n + )θ) + i sin ((n + )θ)] dθ [ ] 2π sin ((n + )θ) = ir n+ sin ((n + )θ) + i n + n + = 66

67 C (z a) n dz = = i z a dz C 2π = 2πi C α β γ C γdz = lim n k= n γ (z k z k ) n z re iθ = lim n γ(z k z k + z k z k + z ) = lim γ(z n z ) = γ(β α) dθ z = re iθ (3.24) α = r e iθ (3.25) β = r 2 e iθ2 (3.26) dz = z z dr + r θ dθ = ireiθ dθ + e iθ dr (3.27) r, θ C θ2 r2 γdz = iγ re iθ dθ + γ e iθ dr θ r = γr [ e iθ2 e iθ] + γe iθ (r 2 r ) C γdz = γr [ e iθ2 e iθ] + γe iθ (r 2 r ) = γr r=r [ e iθ 2 e iθ] + γe iθ θ=θ2 (r 2 r ) = γ(β α) C 2 γdz = γr [ e iθ2 e iθ] + γe iθ (r 2 r ) = γr r=r2 [ e iθ 2 e iθ] + γe iθ θ=θ (r 2 r ) = γ(β α) 67

68 r, θ 3.5: 2 γ ξ k C zdz = lim n k= = lim k= n ξ k (z k z k ) n z k + z k (z k z k ) 2 = { z 2 2 k zk 2 + zk 2 zk z 2 } = 2 ( β 2 α 2) C θ2 r2 zdz = i r 2 e 2iθ dθ + γ re 2iθ dr θ r = r2 2 [ r=r e 2iθ 2 e 2iθ] + e 2iθ θ=θ = ( β 2 α 2) 2 ( r r ) D [z, z] F (z) z F (z) = f(ξ)dξ = f(ξ)dξ C z 68

69 D df (z) dz = f(z) (3.28) D 2 z, z + z ( F (z + z) F (z) f(z) = z+ z ) z f(ξ)dξ f(z) z z 2 D f(z) = z F (z + z) F (z) z z z+ z z f(z) = z = z = ϵ f(z)dξ ( z+ z z z+ z z z z z ) {f(ξ) f(z)} dξ f(ξ)dξ f(z) 3.28 F (z) f(z) = /z 2 π < arg z < +π 3.6: π < arg z < +π F (z) = C ξ dξ = z C+C2 ξ dξ = ξ dξ z = re iθ C r = ξ = e iϕ C2 θ z = ρe iθ dξ = iξdϕ dz = e iθ dρ 69

70 F (z) = θ r dρ idϕ + ρ = iθ + log r = i arg z + log z = log z F (z) z = f(z) D D z C z f(z) = 2πi c z f(ζ) dζ (3.29) ζ z C z z γ z : ζ z = r 3.7: 2 ζ f(ζ)/(ζ z) D ζ = z 2 γ z C z f(ζ) ζ z dζ = γ z f(ζ) dζ = f(z) ζ z γ z ζ z dζ + γ z ζ = z + re iθ f(ζ) f(z) dζ ζ z dζ = ire iθ dθ = (ζ z)idθ 7

71 γ z ζ z dζ = (ζ z)idθ γ z ζ z = idθ γ z = 2πi f(z) r ζ z = r f(ζ) f(z) < z 2πi C z f(ζ) dζ f(z) ζ z = 3.5 2πi 2πi < ε 2π γ z γ z 2π f(ζ) f(z) ζ z f(ζ) f(z) ζ z dθ = ε dζ dζ = 2πi γ z f(ζ) f(z) r rdθ ε 3.29 f (n) (x) = dn f(z) dz n = n! 2πi C z f(z) D D C C f(ζ) n+ dζ (3.3) (ζ z) f(z)dz = (3.3) f(z) D (Morera s law) f(z) z z M f(z) M f (n) (z ) n!m R n (3.32) 3.3 C z : z z = R f (n) (z ) n! 2π n! 2π C z M R n+ f(ζ) dζ (ζ z) n+ dζ = n! C z 2π M R n+ 2π Rdθ = n!m R n (Liouville s law) n = f (z ) M/R R f (z) = (3.33) 7

72 f(z) f(z) D D z {z n } f(z n ) = D f(z) (3.34) f(z) f(z) D D f(z) D B D = D B f(z) B M ax[ f(z) ] on D (3.35) D a f(z) M a D B d z a = re iθ f(a) = 2πi = 2π z a =r 2π M = f(a) 2π f(z) z a dz f(a + re iθ )dθ ( < r < d) (3.36) 2π f(a + re iθ ) dθ M (3.37) z a = r f(a+re iθ ) = M r : < r < d z a < d f(z) = M f(z) f(z) D B D {f n (z)} D f(z) D k {f n (k) (z)} D f (k) lim f n (k) (z) = f (k) (z) (k =, 2, ) n D k= g k(z) g(z) = lim n k= n g k (z) D g (k) (z) = D n= g (k) n (z) (k =, 2, ) 72

73 3.5.2 f(z) r < z a < R (Laurent) f(z) = n= a n (z a) n, r < z a < R (3.38) a n = f(ζ) dζ, n =, ±,, r < ρ < R (3.39) 2πi ζ a =ρ (ζ a) n+ z r a C R 3.8: r < z < R < R r, R γ, Γ f(z) = 2πi c f(ζ) ζ z dζ = 2πi Γ f(ζ) ζ z dζ 2πi γ f(ζ) ζ z dζ Γ ζ z a ζ a ζ z = = = ζ a z a ζ a ( z a ζ a ζ a n= n= (z a) n (ζ a) n+ ) n z a ζ a < γ ζ z a ζ a 73

74 ζ z = z a ζ a z a ( ζ a = z a z a = n= n= (ζ a) n (z a) n+ ) n ζ a z a < f(z) = (z a) n 2πi n= Γ f(ζ) (ζ a) n+ dζ + n= r < ζ a < R γ, Γ f(ζ) (ζ a) n+ (z a) n 2πi γ f(ζ) dζ (ζ a) n+ ζ a = ρ r < ρ < R a ( < z a < R) f(z) a a f(z) (isolated singularity) f(z) = a n (z a) n a n + ( < z a < R) (z a) n n= n= z a < R 2 2 f(z) (singular part) () z = k (zero point) n= b n z n = b k z k + b k+ z (k+) + = g k(z) z k (k, b k = g( ) ) (3.4) ˆ a n = a (removable singularity) 74

75 f(z) = a n (z a) n ( < z a < R) n= f(a) = a f(z) a g(z) = sin z z z = g() = ˆ a (pole) k a k,a n = (n = k +, k + 2, ) f(z) = a n (z a) n + n= k (pole) k n= a n (z a) n = g k(z) (z a) k, g k(a) = a k z a < R g k (z) a n k (z a) n n= ˆ (essental singularity) a f(a) a {z n } {f(z n )} f(z) = exp(/z n ) = n= n!z n z = a = {z n = /(α + 2nπi)} exp(/z n ) = exp(α + 2nπi) = exp(α) = Const α Const D (meromorphic) ( < z a < R) a f(z) = a + a (z a) + a 2 (z a) 2 + z a f(z) a z a = r a n = 2πi r f(ζ) dζ ( < r < R, n =, 2, ) (ζ a) n+ f(z) M r a n = 2πi ζ a =r ζ a = re iθ, dζ = ire iθ dθ f(ζ) (ζ a) n+ dζ 2π r 2π f(a + re iθ )r n dθ Mr n 75

76 ( < z a < R) f(z) a f(z) < z a < R z a < R z = a a {z n } f(z n ) λ(n ) λ f(z) λ = R > < z a < R f(z) (Picard theory) ( < z a < R) w = f(z) a a f(z) ( 2 ) z = a = w = f(z) = e /z w w = re iθ e /z = re iθ log r+iθ = e z n = log r + i(θ + 2nπ) (n =, ±, ±2, ) f(z n ) = w n z n a = f(z) w w (r ) w (r ) z n < z < w = w = f(z) 2 w = f(z) = z + e/z z = a = w = 76

77 3.9: f(z) = e /z, f(z) = z + e/z, f(z) = z e/z, < z a < R f(z) Res(a) a (residue) n = f(z) = Res(a) = f(z)dz (3.4) 2πi z a =r n= a n (z a) n, r < z a < R a n = f(ζ) dζ (3.42) 2πi ζ a =ρ (ζ a) n+ a = f(ζ)dζ 2πi ζ a =ρ n = 3.42 R 77

78 3.: 3.4 Res( ) = f(z)dz (3.43) 2πi z =R Res( ) = b (3.44) n= b n/z n b /z Res(a) = a (3.45) Res(a) Res( ) < z a < f(z) = c n (z a) n (3.46) n= Res(a) = 2πi = 2πi c = c z a =r 2π f(z)dz idθ Res( ) = f(z)dz 2πi z a =R n = n n < z a < dz = ire iθ dθ R n = Res( ) = 2πi c 2π idθ = c i 78

79 3.6.4 p N f (z) 2πi C f(z) dz = d arg f(z) 2πi C = d log f(z) 2πi C = N P (3.47) z f(z) k a g(z) 2 a a f (z) f(z) f(z) = (z a) k g(z) f (z) = k(z a) k g(z) + (z a) k g (z) f (z) f(z) = k z a + g (z) g(z) 2πi C k f (z) f(z) dz = N b f(z) k b h(z) 2 b b f (z) f(z) f(z) = h(z)/(z b) k f (z) = k(z a) k h(z) + (z a) k h (z) f (z) f(z) = k z b + h (z) h(z) 2πi C 2πi C log f(z) C k f (z) dz = P f(z) f (z) dz = N P (3.48) f(z) f(z) = f(z) exp [i arg f(z)] 2 f (z) f(z) dz = d log f(z) = d log f(z) + i arg f(z) 79

80 f (z) 2πi C f(z) dz = d log f(z) 2πi C 2π d arg f(z) ω ω = f(z) ω = Γ N = d arg ω = dω 2π Γ 2πi Γ ω (3.49) C D D C f(z), g(z) C f(z) > g(z) C f(z), f(z) + g(z) (Rouxhe ) f(z) + g(z) f(z) g(z) > f(z) f(z) + g(z) f(z), f(z) + g(z) D f(z) + g(z) f(z) C d arg(f(z) + g(z)) 2π 2π C C d arg f(z) = ( d arg + g(z) ) 2π C f(z) C g(z)/f(z) < w = + g(z)/f(z) C w < (3.5) 3.: w = + g(z)/f(z) z C w 3.5 8

81 z n (n =, 2, : z n < z n+ ) (3.5) z k (k =,, 2 ) p k (z) p (z) C n R n : z n < R n < z n+ (3.52) C n z z n f(ζ) (z/ζ)n ζ z n z j = f(ζ) ζ j+ ζ = z k (ζ z k ) h k (z) f(z) = n k= 2 C n j= (p k (z) h k (z)) + zn f(ζ) 2πi C n ζ n dζ (3.53) (ζ z) R(z) = f(ζ) 2πi C n ζ n (ζ z) dζ z, z, z z n n + 2 ζ = z f(z) ζ = z k z n f(ζ) ζ n dζ = f(ζ) n (ζ z) ζ z z j ζ j+ j= (ζ z k ) = f(ζ) m+ n (ζ z k )(z z k ) m+ m= = f(ζ) m= p k (ζ) = (ζ z k ) m n (z z k ) m+ ζ = z k p k (z) + h k (z) i= z j ζ j+ j= z j ζ j+ j= a i (ζ z k ) i (3.54) 3.53 n R(z) = f(z) + ( p k (z) + h k (z)) (3.55) k= 8

82 : Im z f(z) θ π f(re iθ ) (R ) m > K R : z = Re iθ ( θ π) lim e imz f(z)dz = (m > ) R K R 3.35 f(z) M(R) e imz f(z)dz K R π e mr sin θ M(R)Rdθ = 2RM(R) 2RM(R) π/2 M π/2 e mr sin θ dθ e 2mRθ/π π dθ = 2RM(R) 2mR ( e mr ) < π m M(R) α a ρ K α ρ ( α 2π) 82

83 a+iy a 3.3: 2πiRes(i) α K αires(a) f(z) a α lim α ρ = α (3.56) ρ lim f(z)dz = αires(a) (3.57) ρ K Res(a) = a (3.58) f(z) = a + g(z) (3.59) z a g(z) z a ρ < ρ < ρ K K 3.57 K g(z) M (3.6) g(z)dz 2πρM (3.6) lim g(z)dz = (3.62) ρ K a + ρe iθ, a + ρe iθ2 a θ2 z a dz = a dθ = a i(θ 2 θ ) = iαa (ρ ) (3.63) θ 83

84 3.7.2 f(x) J = ε f(x)dx + ε f(x)dx (3.64) J (improper Riemann integral) J = f(x)dx ε lim f(x)dx + X X lim X 2 X2 ε f(x)dx (3.65) 2 (principal value) R p.v.j = p.v. f(x)dx lim f(x)dx (3.66) R R f(x) = x J R p.v.j = p.v. f(x)dx = lim f(x)dx R R p.v. xdx = lim R a R R xdx = lim R ( R 2 2 R2 2 ) = ( b a ϵ ) b p.v.j = p.v. f(x)dx = lim f(x)dx + f(x)dx ϵ a a+ϵ f(x) = /x 2 [, ), (, 2] f(x) 2 p.v.j = p.v. = lim ϵ ( ϵ dx x = lim ϵ ) = log 2 ( log ϵ + log 2 ϵ dx 2 x + ϵ f(z) ) dx x f(z) P, Q m > imz P (z) f(z) = e Q(z) 3.63 (3.67) ( ρ ) p.v. f(x)dx = lim f(x)dx + f(x)dx ρ ρ ( ) = 2πi Res(z) + Res() 2 z> z= (3.68) 84

85 m < ( ρ ) p.v. f(x)dx = lim f(x)dx + f(x)dx ρ ρ ( ) = 2πi Res(z) + Res() 2 z< z= P (x), Q(x) R(x) = P (x)/q(x) I = R(x)dx Q(x) q p + 2 R(z) I = R(x)dx = lim R(z)dz = 2πi ResR(x) (3.69) R C R y> K R [ R, R] z 2 R(z) M R(z) π z 2 R(z) 3.69 R(z)dz πm/r (R ) K R I = iy Kr R(x)dx = 2πi y> Res R(x) -R R x 3.4: 85

86 I = I = 2 dx x 2 (a > ) + a2 dx x 2 (a > ) + a2 R(x) = z 2 + a 2 = (z + ia)(z ia) Res(ia) = /2ia I = 2 2πi 2ia = π 2a t = ax I = f(z) = cos t t 2 dt (a > ) + a2 I = cos ax 2a x 2 + dx eiax z 2 + = e iax (z + i)(z i) e iax 2a x 2 + dx = 2a 2πie a = πe a 2i 2a I = cos ax πe a 2a x 2 dx = + 2a f(z) = e imx x 2 dx (3.7) + eimz z 2 + = e imz (z + i)(z i) 86

87 Res(+i) = e m 2i e imx x 2 dx = 2πie m = πe m + 2i (Step function) e imz θ(m) = lim ε 2πi x iε dx = (m > ) (m < ) f(z) = eimz z iε θ(m) = lim ε 2πi 2πiRes(iε) = e = m > m < (3.7) (3.72) ε sin x x dx 3.68 f(z) = eiz z lim f(z)dz = (3.73) R C e ix p.v. x dx = 2πi 2 Res() = πi sin x x dx = sin x 2 x dx = π 2 87

88 3.7.4 G(z) 2 z = (Mellin) I = x α G(x)dx ( < α < ) (3.74) z α < arg z α < 2πα f(z) = z α G(z) D < ρ < < R, < ϵ < π/2 D : ρ r R, ε θ 2π ε α D iy iy D -R R x D -R R x 3.5: D R ρ f(re iε )e iz dr + 2π ε ε f(re iθ )ire iθ dθ + ρ R ε f(re i(2π ε) )e i(2π ε) dr + f(ρe iθ )iρe iθ dθ 2π ε = 2πi x D Res(z) (3.75) z R ρ r α G(r)dr + 2π ε ε R α+ G(Re iθ )ie i(α+)θ dθ + ρ R ε r α G(re 2πi )e 2πi(α+) dr + ρ α+ G(ρe iθ )ie i(α+)θ dθ 2π ε = 2πi x D Res(z) (3.76) z α G(z) M (z ), zg(z) N(z ) ε 2π R α+ G(Re iθ )ie i(α+)θ dθ 2πM R α R, ρ 24 2π ρ α+ G(ρe iθ )ie i(α+)θ dθ < 2πNρα (3.77) 88

89 ( e 2πiα ) r α G(r)dr = 2πi x D Res(z) (3.78) ( e 2πiα ) x z α arg z α 2πα f(z) = z α G(z) D z = e iπ = x ρ dx ( < ρ < ) (3.79) x + f(z) = zρ z + (3.8) 3.78 Res(e iπ ) = e iπ(ρ ) = e iπρ ( ) = e iπρ (3.8) ( e 2πiρ x ρ ) dx x + = 2πi Res(z) = 2πie iπρ x ρ [ ] 2πie iπρ 2π sin ρπ 2π sin ρπ dx = Real x + e 2πiρ = = cos 2ρπ 2 sin 2 ρπ = π sin ρπ I = b a dx (b x)(x a) (a < b) (3.82) C dz J = (a < b) (3.83) (z a)(z b) C f(z) = (z a)(z b) (3.84) f(z) = z + a + b 2z 2 + (3.85) 89

90 J = C dz (z a)(z b) = 2πiRes( ) (3.86) = 2πi (3.87) Cu Cd 3.6: x z a = x a z b = e iπ/2 b x x z a = e i3π/2 x a z b = b x 3.87 J = a = 2i b b e π/2 dx b + (b x)(x a) a dx (b x)(x a) a e i3π/2 dx (b x)(x a) I = b a dx (b x)(x a) = π (3.88) 9

91 I x = e x2 dx (3.89) I x I y = = 4 = 2π e x2 dx dθ = 2π 2 = π e x2 dx e r2 rdr e y2 dy e y2 dy e x2 dx = π (3.9) I = π log(sin θ)dθ z = x + iy F (z) = e 2iz = 2ie iz sin z = e 2y (cos 2x + i sin 2x) F (z) y < x = nπ(n =, ±π, ±2π ) f(z) = log F (z) = log( 2ie iz sin z) π f(z + π) = f(z) P (z; a) c + c (z a) + c 2 (z a) 2 + ( z a < R) (3.9) 9

92 (function element) a K : z a < R b P (z; a) z = b P (z; a) = n= P (n) (b; a) (z b) n ( z a < R) (3.92) n! = c + c (b a)(z b) + c 2 2 (b a)2 (z b) 2 + (3.93) K R R b K l R l = R b a R l > K : z b < R b K P (z; b) = n= P (n) (b; a) (z b) n ( z b < R ) n! P (z; b) P (z; a) K K c K K K R > l P (z; b) P (z; a) P (z; a) z K F (z) P (z; b) z K 92

93 3.7: F (z) K K P (z; a), P (z, b), P (z, c) K f(z) K K F (z) K F (z) = f(z) F (z) f(z) (analytic continuation) P D D F (z) (analytic function) 2 P (z; a) K 2 a, b P (z; a i ), P (z, b j ) m, n P (z; a), P (z; a ), P (z; a m ); P (z; a), P (z; b ), P (z; b n ) P (z; a m ), P (z; b n ) K m, L n Ω F (z) 93

94 3.8: 2 Ω P (z; a m ) = P (z, b n ) K m, L n n n n F (z) n F (z) n n 2 w = z 2 z = re iϕ w = re iϕ/2, w = re i(ϕ±2π)/2 = w θ ±2nπ ±π 2 Π, Π D F (z) z = re iϕ D (3.94) n x x = F (z) = z = re iϕ/2 ( π < ϕ 3π; mod 4π) F (n) (z) := ( )n ( ) 3 (2n 3) 2 n z 2n 2 (n ) n = k n z 2n 2 k n = ( )n 2 n (2j 3) (3.95) x = + (x )!2 2!2 2 (x ) !2 3 (x ) !2 4 (x ) !2 5 (x )5 + z = R = j=2 94

95 P (z; ) = + (z ) 2 2!2 2 (z ) !2 3 (z )3 kn (z )n n! n ( ) j (z ) j j = j!2 j (2i 3) = j= i=2 n (z ) j F (n) (z) j! z < n j= P (z; ) = w = z Π () Π () 2, γ Π γ Π z 2π 3.9: z Π γ Π = e 2iπ = e 2iπ F (n) ( ) P (z; ) = + = e 2iπ + n= F (n) ( ) (z ) n n! n= F (n) (e 2iπ ) (z e 2iπ ) n n! = (z ) + 2 2!2 2 (z )2 3 3!2 3 (z )3 + kn (z )n n! n ( ) j (z ) j j = j!2 j (2i 3) j= = P (z; ) i=2 95

96 z = R = w = P (z; ) = w = P (z; ) P (z; ) a z = P (z; ) F (z) 2 D (natural boundary) f(z) = z 2n = z + z 2 + z 4 + z 8 n= z 2n = z + z 2 + z 4 + z 2n + n= m= f(z) = z + z 2 + z 4 + z 2n + f(z 2n ) z 2n+m f(z) z = = z 2n = e 2mπi z n,m = exp[ 2mπi 2 n ], m =, 2, 2n ; n =, 2, z = e x = + x! + x2 2! + e z = + z! + z2 2! + x z e z 2 f (x) = f 2 (x) f (z) = f 2 (z) 2 D D 2 Γ 96

97 3.2: Γ f (z) D D + Γ z D F (z) F (z) = f (z) F (z) D Γ f (z) D D + Γ z D 2 + Γ, z D + Γ f 2 (z) = f ( z) z D 2 F (z) = f 2 (z) z D df (z) dz f 2 (z + z) f 2 (z) f (z + z) f ( z) = lim = lim z z z z ( ) f (z + z) f ( z) = lim = f z z ( z) F (z) D 2 Γ D + D 2 Γ γ = γ 2 C + C 2 x 2 F (z)dz = F (z)dz + F (z)dz C +C 2 C +γ C 2+γ 2 γ D δ 97

98 3.2: D iδ C +γ F (z)dz = = ( b+iδ + b a b a+iδ b+iδ a ) + + F (z)dz a+iδ γ [f (x) f (x + iδ)] dx + i δ [f (b + iy) f (a + iy)] dy δ 2 C +γ F (z)dz = C 2+γ 2 F (z)dz = C +C 2 F (z)dz = 3.3 F (z) D + Γ + D 2 Γ f (z) D D + Γ z D 2 + Γ, z D + Γ f 2 (z) = f ( z) f Γ D z > P (z; ) = z n = + z + z 2 + z < F (z) n= 98

99 F (z) = /(z ) F (z) P (z; ) z < t n e t dt = n! P (z; ) = G(z) = n= z n n! e z e t dt = t n e t dt = G(z) e (z )t dt Re[z] < D P (z; ) G(z) Γ 2 z = w /2 = w w ρe iϕ ( π < ϕ π) z = ρe iϕ/2, z = ρe i(ϕ/2+π) = z 2 w ( π < ϕ π) Π Π 3.22: z = w 2 +π 2 w, w 2 4π w mod 4π w = z 2 z α n α ϕ < α + 2nπ; (mod 2nπ) w = f(z) = 2 ( z + ) z (3.96) 99

100 2wz z 2 = (z w) 2 = w 2 ± z = f (w) = w + w 2 w ± 2 2 [, +] w = f(z) = f(/z) 2.7 z = re iθ, w = u + iv 3.96 u = 2 ( r + ) cos θ r v = 2 z r = r w z θ = θ w ( r ) sin θ r u 2 (r + /r ) 2 + v 2 (r + /r ) 2 = 4 u 2 cos 2 + v2 θ sin 2 = θ z r = w (u, v) = (cos θ, ) u :

101 3.9 Γ(z) Re[z] = a > Γ(z) = t z e t dt (Re[z] > ) (3.97) n z n! Γ(z) = lim n Π n k= (z + k) (3.98) t z e t dt t z e t dt < (3.99) D : Re[z] > Γ(z) = t z e t log tdt (3.) D : Re[z] > Γ(z) D Γ(z) = = z [ z tz e t ] + z t z e t dt (3.) t z e t dt (3.2) Γ(z) = Γ(z + ) = zγ(z) (3.3) Γ(z + n) (3.4) z(z + )(z + 2) (z + n ) Γ(z) z =,, : z=-5 5 z = Γ(n + ) = n! (3.5)

102 L Γ(z) = t z e t dt (3.6) L z Γ(z) Γ(z) z t = re iϕ t z = r z e iϕ(z ) = e (z )(log r+iϕ) ϕ 2π (3.7) ϕ = F (t) = e t t z = e reiϕ e (z )(log r+iϕ) (3.8) F (r) = e r r z (3.9) b r c a 3.25: L ρ ( a I(z) = + + abc Re[z] > a lim ρ F (t)dt = 3 ϕ = 2π 3.8 c ) F (t)dt (3.) e r r z dr = Γ(z) (3.) F (r) = e reiϕ e (z )(log r+iϕ) = e 2πiz e r r (z ) log r (3.2) 2

103 lim ρ c F (t)dt = e 2πiz e r r z dr = e 2πiz Γ(z) (3.3) 2 ρ 3.8 dt = iρe iϕ dϕ abc 2π F (t)dt = iρe (z ) log ρ e ρeiϕ e iϕ dϕ 2π = iρ z e ρeiϕ e iϕ dϕ Re[z] > ρ F (z) = ( e 2πiz ) Γ(z) = (i sin 2πz cos 2πz ) Γ(z) = ( 2i sin πz cos πz 2 sin 2 πz ) Γ(z) = 2ie iπz sin πz Γ(z) Γ(z) = e iπz t z e t dt (3.4) 2i sin πz L t = B x + x (3.5) B(α, β) = = = t a ( t) β dt x α ( + x) α dx β ( + x) x α ( + x) α+β dx Re[α] >, Re[β] > 4 Pi π 4. σ µ G(x) = 2πσ 2 e (x µ)2 /2σ 2 3

104 4.: x + µ =, σ = N(x) = 2π e x2 /2 [,x] Ef(x) = x N(t)dt 4.2: (error function) erf(x) = 2 x e t2 dt π Ef(x) = 2 erf(x/ 2) erfc(x) = erf(x) 2 = e t2 dt = e x2 erfc(x) π x erf( z) = erf(z) 4

105 erf(z ) = erf(z) erf(x) = π Γ( 2, x2 ) = 2 π n= ( ) n x 2n+ n!(2n + ) 5 5. π < x < π a n = π π π f(x) cos nxdx b n = π π π f(x) sin nxdx f(x) = a 2 + (a n cos nx + b n sin nx) n= c n = π f(x)e inx dx 2π π f(x) = c n e inx n=,±, 5.2 Gibbs π 4 ( π < x < ) g(x) = (5.) π 4 ( < x < π) (g(+) + g( )) /2 = +,- a n = a = π π 4 b n = 2 π π 4 π π dx + π π 4 π dx = sin nxdx = ( ) n 2 n 5

106 g n (x) = + ( ) n sin nx 2 n n= ( ) sin x + + = 2 2 = n= sin 3x 3 sin (2n ) x 2n n = 3, n = 5.: g n (x) = sin(2n )x n= 2n n= n=3 n N Gibbs x g(x) g n (x) < ϵ ϵ n > N z = e iθ log( + z) = z z2 2 + z3 3 log( + e iθ ) = e iθ e2iθ 2 + e3iθ 3 e inθ = cos (nθ) + i sin (nθ) Im log( + e iθ ) = sin θ sin 2θ 2 + sin 3θ 3 Im { log( + e iθ ) log( e iθ ) } sin 3θ sin 5θ = sin θ = g (θ) (5.2) log re iθ = log z + i arg z g (θ) g (θ) = arg { } + eiθ log 2 e iθ 6

107 arg log( + e iθ ) = α arg log( e iθ ) = β ImZ ImZ e i e i o ReZ o ReZ -e i -e i 5.2: e iθ, e iθ π α β = π 2 ( < θ < π) α β = π 2 ( π < θ < ) 5. θ = π, 5.2 π 4 ( < θ < π) g (θ) = π 4 ( π < θ < ) z m=,,2, m=, 2, = 2 ( ) m 2m + = ( ) m 2m + = e iπz + 7 m=,±,±2 ( ) m 2m +

108 z = n(n = ±, ±3, ) z n Res = lim z n e iπz + = lim z n d dz d (z n) dz (eiπz + ) = iπ e inπ = i π e iπm m,, 2, e i π 2 m, i,, i 3.42 f(z) = Res m (z a) m + Res m (z a) m + + Res 2 (z a) 2 + Res z a + f(z) f(z) m + n z = a Res f(z)dz = Res 2πi C z = a C z = ±, ±3,, ±2n n C n (5.3) 5.3: z = ±, ±3,, ±2n i/z e iπz + ei π 2 z 5.3 i/π 2πi 2 2 N = 2 lim = 2 lim N N n= N N N n= N C n 2 C n 2 i z i z e i π 2 z e iπz + dz e i π 2 z e iπz + dz 2 C + 8

109 ImZ : C + 4 i z e iπz + dz = i 2πi[ ei ] = π e i π 2 z 6 6. u tt c 2 u xx = dx 2 c 2 dt 2 = 2 dx cdt =, dx + cdt = ξ = x ct = Const. η = x + ct = Const. 9

110 7 [57] Green cohomology 2 x x 2 n (7.) f x = ict 2 f = (7.2) 2 y = c2 t2 x 2 y (7.3) u v ω = f(z) u =, v = (7.4) 3.7 Dirichlet D U = (7.5) D U(x, y) C U/ n (Neumann) S V S 7. ( = x i, y j, ) z k = 2 x y z 2 2, 3 V S V V C 2 2 u u xx + u yy + u xx = (7.6) 2 u u xx + u yy + u xx = ϕ(x, y, z) (7.7)

111 3 r = x 2 + y 2 + z u x 2 = ( ) u x x = ( ) r u x x r ( ) r r = = x ( r x ( u 2 x r r + r x 2 ) 2 2 ( u 2 ) r 2 + r u x 2 r ) u r 2 u y 2 = ( ) 2 r 2 u y r 2 + ( 2 ) r u y 2 r 2 u z 2 = ( ) 2 r 2 u z r 2 + ( 2 ) r u z 2 r r x = x r, r y = y r, r z = z r 2 r x 2 = z2 + y 2 r 3, ( ) 2 r + x 2 r y 2 = z2 + x 2 r 3, ( ) 2 r + y 2 r z 2 = x2 + y 2 r 3 ( ) 2 r = z 2 r x r y r z 2 = 2 r 2 u = 2 u r 2 = 2 u r 2 + u r = r 2 ( ( ) 2 ( ) 2 ( ) ) 2 r r r u ( 2 ) r x y z r x r y r z 2 ( ) 2 r ( r 2 du ) = (7.8) dr d dr c r r 2 du dr = c du dr = c + c r 2 (7.9)

112 c = /4π u(r) = 4πr 3 r PQ (7.) 7.6 r = (x x ) 2 + (y y ) 2 + (z z ) 2 x y z ( ) = r r r 2 x = x x r 3 ( ) = r r r 2 y = y y r 3 ( ) = r r r 2 z = z z r 3 r = x x ( ) = r r r 3 (7.) q u(r) = q 4πϵ r E(r) = u(r) = q 4πϵ r r 3 GM 7. r = 2 r = x 2 + y 2 ( ) 2 r + x ( ) 2 r = y 2 r x r y 2 = r 2 2 u = 2 u r 2 = 2 u r 2 + u = r d dr ( ( ) 2 r + x ( ) = r r ( r du ) = dr ( ) ) 2 r + u ( 2 ) r y r x r y 2 2

113 u r = r u(r) = c + c log r (7.2) u(r) = 2π log r PQ P Q, Q 2, Q n q, q 2, q n 3 r j = x x j u(r) = 4πϵ ρ, σ u(r) = 4πϵ V u(r) = 4πϵ V (surface distribution) n j= q j r j (7.3) ρ(x ) x x dv σ(x ) x x ds ql = l, q 7.: Q Q2 l r = r 2 + ( ) l 2 r 2 + rl cos θ ( l cos θ ) 2r 3

114 r 2 = r 2 + ( ) l 2 r 2 rl cos θ ( + l cos θ ) 2r u(r) = ( q lim + q ) ql cos θ = 4πϵ l r r 2 4πϵ r 2 p = ql r = x x i ( ) x = r u(x) = p r 4πϵ r 3 (7.4) ( x x r 3, x y r 3, x z ) ( ) r 3 = x r 7.4 n n x i u(x) = p r 4πϵ r 3 = p x 4πϵ ( ) = p r 4πϵ n ( ) r n P u(x)r = x x θ 7.2: u(x) = u(x ) 4πϵ S n ( ) ds(x ) = u(x ) cos θ r 4πϵ S r 2 ds(x ) P S 2 (double distribution) 4

115 7.2 f(x) = x < (7.5) s = a + i b a > (7.6) F (s) = + f(x) exp( sx)dx (7.7) x < (7.8) 7.7 F (s) = + f(x) exp( ax) exp( ibx)dx (7.9) f(x) exp( ax) (7.2) f(x) exp( ax) = 2π + exp( ax) s F (s) exp( ibx)db (7.2) a f(x) = 2πi db ds = i F (s) exp(sx)ds (7.22) < b < (7.23) Dirac a > f(x) = δ(x) F (s) = δ(x) exp( sx)dx = 5

116 2. a > f(x) = F (s) = exp( sx)dx = s 3. a > f(x) = x F (s) = x exp( sx)dx = s 2 4. a > 5. a > f(x) = exp(x) F (s) = exp(x sx)dx = s 6. a > f(x) = cos(x) F (s) = cos(x) exp( sx)dx = s + s 2 f(x) = sin(x) F (s) = sin(x) exp( sx)dx = + s F (s) = f(x) = 2πi = δ(x) exp(sx)ds 6

117 2. F (s) = s exp(sx) f(x) = 2πis ds = 3. (2 ) F (s) = s 2 exp(sx) f(x) = 2πis 2 ds = x 4. F (s) = s exp(sx) f(x) = 2πi(s ) ds = exp(x) 5. s F (s) = + s 2 exp(sx)s f(x) = 2πi( + s 2 ) ds = cos(x) 6. F (s) = + s 2 exp(sx) f(x) = 2πi( + s 2 ) ds = exp(x) f(x) F (s) f (x) f (x) exp( sx)dx = [f(x) exp( sx)] + s f(x) exp( sx)dx = f() + sf [s] F [f (x)] = sf [s] f() 2 7

118 F [f (x)] = s{sf [s] f()} f () = sf [f (x)] f () p τ < τ < t χ p(t) = χ(t τ)e(τ)dτ (7.24) t P (s) = exp( st) χ(t τ)e(τ)dτdt (7.25) z = t τ (7.26) χ(s) = E(s) = exp( sz)χ(z)dz (z ) exp( sτ)e(τ)dτ (τ ) s s ω P (s) = χ(s)e(s) (7.27) Im [X(ω)] = Re [X(ω)] = Re [X(s)] π(s ω) ds Im [X(s)] ds (7.28) π(s ω) t = δ(t) h o (t) h e (t) h(t) = t < h(t) = h(t) + h( t) 2 + h(t) h( t) 2 = h e (t) + h o (t) (7.29) h o (t) = sign h e (t) (7.3) h e (t) = sign h o (t) (7.3) 8

119 H(ω) = = + + dt h e (t)e iωt + + dt h e (t) cos(ωt) + i + dt h (t)e iωt dt h (t) sin(ωt) + + F(sign(t)) = dt h (t)sign(t)e iωt = 2π dt h e (t)sign(t)e iωt = 2π P ˆ sign(t) (7.32) + + dω ih I (ω ) ˆ sign(ω ω ) dω H R (ω ) ˆ sign(ω ω ) H R (ω) = = + + = 2π + dt h e (t)e iωt dt h (t)sign(t)e iωt = π P + dω ih I (ω ) ˆ sign(ω ω ) dω H I(ω ) ω ω H I (ω) = + = 2π + dt h e (t)sign(t)e iωt = πi P + dω H R (ω ) ˆ sign(ω ω ) dω H R(ω ) ω ω (7.33) 7.28 ω R r lim R,r { ω r R } H(z) R z ω dz + H(z) ω+r z ω dz = P dz H(z) = iπh(ω) (7.34) z ω 9

120 7.3: V S u(x) 2 u = (in V ) u s = f (on S) (Dirichlet) S 2 u = (in V ) u n S = g (on S) u(x) 2 S V S Dirichlet (maximum principle) u(x) V V S u M m V x m u(x) M S 2 V S V u V V S M m O V R V r(x) ϵ 2

121 7.4: v(x) = u(x) M ϵ(r 2 x 2 ) V R 2 x 2 M S v(x) S v(x) V u V x x v x = v y =, 2 v x 2, 2 v y 2 2 v x 2 u =, 2 r 2 = 4 2 v 4ϵ > v(x) V u(x) M + ϵ(r 2 x 2 ) ϵ u(x) M u(x) = u (x) u 2 (x) u V V S u = m = M = 2

122 V u = u = u 2 x ˆ 3 u(x) ˆ 2 u(x) Const 7.3. f, g S u(x) u n C x = r cos θ, y = r sin θ 2 u r 2 + u r r + 2 u r 2 θ 2 = u r θ Dirichlet D f(x) Re[f(x)] C C P (x, y) ds dn n x + in y = i(s x + is y ) n x = s y n y = s x iy C = s x s x + s y y n = n x x + n y y = s y x + s x y (7.35) D n P s x 7.5: C 22

123 D f(x) = U(x, y) + iv (x, y) (7.36) Im[f(x)] U(x, y) U x U y V = y = V x 7.35 U s U n V = n = V s D C U/ n U V V/ s C P C P(x,y) C V (x, y) = P P V ds (7.37) s V C D V C U V C Im[f(x)] D Re[f(x)] z = x + iy D U(x, y) D U D B K 3.6 Green 3. B ( B (Xdx + Y dy) = (7.38) X = U y Y = U x U ) U dx + y x dy = K Udxdy = (7.39) D Q Q V (x, y) = Q(x,y) Q ( ) U U dy x y dx (7.4) 23

124 (x, y) x V (x + dx, y) = V (x, y) + x+dx x ( U ) dx (7.4) y V x V (x + dx, y) V (x, y) = lim dx dx = lim dx dx = U y x+dx x ( U y ) dx y V y = U x (7.42) D Re[f(x)] = U(x, y) f(z) = U(x, y) + i Q(x,y) Q ( ) U U dy x y dx (7.43) z D z = x + iy w = u + iv w = f(z) (7.44) dw = f (z)dz dw = f (z) dz arg[dw] = arg[f (z)] + arg[dz] 2 z dz w f (z) f (z) dw (conformal mapping) 7.44 u = u(x, y) v = v(x, y) u v 24

125 D 3.7 f (z) = f(z) x = i f(z) y f (z) = u x (x, y) + iv x (x, y) = v y (x, y) iu x (x, y) F (z, z) f (z) = 2 z = x i y, 2 z = x = z + z, x + i y ( y = i z ) z ( z + ) ( F (z, z) = z z ) F (z, z) z F (z, z) z z = x iy = f(z) z = (7.45) F (z, z) = f(z) = F (x + iy) f(z) = u(x, y) + iv(x, y) u x 2 = 2 v x y = 2 v y x = u 2 y 2 2 v x 2 = 2 u x y = 2 u y x = v 2 y 2 u =, v = u v (conjyugate harmonic function) C : z(t) = x(t) + iy(t) z = z(t ) x θ Γ Γ : w(t) = u(x(t), y(t)) + iv(x(t), y(t)) Γ u ϕ w = f(z) α ϕ = θ + α tan θ tan(θ + α) = tan α + tan θ = tan ϕ (7.46) tan θ tan α 25

126 tan ϕ v/ u t dv dt = dv dx dx dt + dv dy dy dt = dx ( dv dt dx + dv ) dy dt dy dt dx du dt = du dx dx dt + du dy dy dt = dx ( du dt dx + du dy dy dt ) dt dx z = z(t ) 7.46 tan ϕ = dv/dt du/dt t=t = v x + v y (ẏ/ẋ) u x + u y (ẏ/ẋ) t=t = v x + v y tan θ u x + u y tan θ u x : u y : v x : v y = : tan α : tan α : u x = v y u y = v x (Riemann) z D 2 D ω D w = f(z) z w z D C D w w = f(z) D = D C C w = f(z) = z z w z = z = 26

127 7.6: w = f n (z) = z n n = 2 z = x + iy w = u + iv x 2 y 2 = u, 2xy = v w u = Const, v = Const z y = ±x, y =, x = x = a, y = b w v 2 = 4a 2 (a 2 u), v 2 = 4b 2 (b 2 + u) dw/dz = z = n = 2 z 2 w x + iy, x iy u + iv 2 7.7: u = ±, v = ± z x = ±, y = ± n z = fn (w) = n w n w = e z z = x + iy 2π z n = x + iy + 2inπ(n =, ±, ±2, ) 27

128 w n = e x+iy+2inπ = e x+iy w = dw dz = ez = w = z = log w w ρe iϕ e z = w z = x + iy e x+iy = ρe iϕ 2π z z N = x + y N = log ρ + i(ϕ + 2Nπ) (N =, ±, ±2, ) w z 7.8: w = e z z 2π w R V Q(x, y, z) P (x, y 2, z 3 ) O ( R 2 (x, y, z ) = r ( R 2 (x, y, z) = r 2 r R2 R2 x, y, 2 2 x, R2 r 2 ) r 2 z, r = x 2 + y 2 + z 2 ) y, R2 r 2 z, r = x 2 + y2 + z2 28

129 P(x,y,z) r r o Q(x,y,z) R 7.9: V P Q 2.8 rr = R 2 O u(x) u (x, y, z ) = R ( ) R 2 u r r 2 x, R2 r 2 y, R2 r 2 z = R u (x, y, z) (7.47) r O (Kelvin transformation) (R = ) u(x) = a r r = x x u (x ) = r u(/r ) = r a /r = a u(x) = a u (x ) = r u(x /r 2 ) = a r u(x) = ax u (x ) = r u(x /r 2 ) = ax r 3 29

130 7.4 u(x) = ax r 3 u (x ) = ax u(x) S V S C 7.7,7.73 ( u) 2 dv = V... u(x) = ( 4π S r / n V S u n u n u u ds (7.48) n ( )) ds r 2 C R = r2 R 2 ( ) R 2 u (x, y ) = u r 2 x, R2 r 2 y = u (x, y) (7.49) 2 u (x, y ) = 2 u (x, y) = r4 R 4 2 u(x, y) u(x) C u (x ) O u(x) r u (x ) O O u (x ) u u x, u y = O( r 2 ) u(x) C S ( u) 2 ds = S C C u n dl = u(x) = (( log ) u 2π C r n u ( log )) dl + c n r u u dl (7.5) n c = lim r u a ϕ n = 3

131 U ϕ Ux ϕ r r=a = ϕ = Ur cos θ (r ) ψ ϕ = Ux + ψ = Ur cos θ + ψ ψ r r=a = U cos θ ψ (r ) r > a 2 ψ = 2 ψ r ( ψ r r + r 2 sin θ ψ ) = sin θ θ θ ψ(r, θ) = f(r) cos θ d 2 f dr df r dr 2 r 2 f = df dr r=a = U f(r) = r n n =, 2 n = 2 f(r) = Ua3 2r 2 ψ(r, θ) = U 2 2 a 3 r 2 cos θ ϕ = U(r + a3 2r 2 ) cos θ x U a 7.6 z = re iθ u(z) v(z) z < R f(z) = u(z) + iv(z) (7.5) f(z) = f(ξ) dξ ( z < ρ < R) (7.52) 2πi ξ =ρ ξ z 3

132 z = re iθ ξ = ρ ẑ = (ρ 2 /r)e iθ f(ξ)/(ξ ẑ) ξ ξ ρ f(ξ) dξ = (7.53) 2πi ξ =ρ ξ ẑ ξ = ρe iϕ f(re iθ ) = u(re iθ ) = 2π = 2π = 2π = 2π = 2π 2πi = 2π 2π 2π 2π 2π 2π ξ =ρ 2π dξ = iρe iϕ dϕ (7.54) ( f(ξ) f(ρe iϕ ) ) dξ ξ re iθ ± ξ (ρ 2 /r)e iθ ( r ± ρ ei(θ ϕ) ρ r ei(θ ϕ) [ ] u(ρe iϕ )Re r ρ ei(θ ϕ) ρ dϕ r ( ei(θ ϕ) ) u(ρe iϕ ) r ρ cos(ϕ θ) ρ dϕ r cos(ϕ θ) ( ) ρ 2 +r 2 u(ρe iϕ rρ cos(ϕ θ) ) ( ) dϕ ρ 2 +r 2 rρ cos(ϕ θ) + cos 2 (ϕ θ) ( u(ρe iϕ ρ 2 r 2 ) ρr/ cos(ϕ θ) + (ρ 2 + r 2 ) ρr cos(ϕ θ) ( ) u(ρe iϕ ) dϕ (r < ρ < R) ρ 2 r 2 ρ 2 2ρr cos(ϕ θ) + r 2 u(z) z R ρ R u(re iθ ) = 2π 2π ( u(re iϕ ) R 2 r 2 R 2 2Rr cos(ϕ θ) + r 2 ) dϕ (7.55) ) dϕ ) dϕ (7.56) z R 7.55 f(re iθ ) = 2π 2π v(re iθ ) = v() π ( f(ρe iϕ ) i 2π ( u(ρe iϕ ) 2ρr sin(ϕ θ) ρ 2 2ρr cos(ϕ θ) + r 2 ρr sin(ϕ θ) ρ 2 2ρr cos(ϕ θ) + r 2 ) dϕ ) dϕ v() 3.36 ρ R v(re iθ ) = v() π 2π ( u(ρe iϕ ) Rr sin(ϕ θ) R 2 2Rr cos(ϕ θ) + r 2 ) dϕ (7.57) 32

133 ζ = Re iϕ f(z) = u(z) + iv(z) f(z) = iv() + u(ζ) ζ + z dζ 2πi ζ z ζ ζ =R (7.58) 7.56 z = re iθ, ζ = Re iϕ R 2 r 2 R 2 2Rr cos(ϕ θ) + r 2 = ζ 2 z 2 ζ z 2 ζ 2 z 2 ζ z 2 = Re ζ + z ζ z = + 2Re z ζ z = + Re z/ζ z/ζ (7.59) R 2 r 2 R 2 2Rr cos(ϕ θ) + r 2 = + 2 = + 2 = + 2 Re n= ( ) n z ζ ( r ) n cos(n(ϕ θ)) R n= ( r ) n (cos(nϕ) cos(nθ) + sin(nϕ) sin(nθ)) R 7.56 n= u(re iθ ) = a 2 + ( r ) n (an cos(nθ) + b n sin(nθ)) (7.6) R n= a n = π 2π u(re iϕ ) cos(nϕ)dϕ, b n = π 2π u(re iϕ ) sin(nϕ)dϕ (7.6) 7.56 Q(R, α) P (r, θ) PQ ρ ρ 2 = R 2 + r 2 2Rr cos(α θ) r < R ρ > 33

134 7.: P Q u(r, θ) C 2π R 2 r 2 ρ 2 dα = z < 2π k= R 2 r 2 R 2 + r 2 dα = 2π (7.62) 2Rr cos(α θ) z k = z z z = r R ei(α θ) [ ] ( r ) k ( r ) k Re (cos k(α θ) + i sin k(α θ)) = cos k(α θ) R R k= k= ρ 2 = R 2 + r 2 2Rr cos(α θ) [ ] z Re z = r cos(α θ) R r cos(α θ) = R 2 + r 2 ρ 2 2 R 2 Rr cos(α θ) = R 2 + r 2 ρ 2 2 R 2 Rr cos(α θ) = ( R 2 r 2 ) 2 R 2 + r 2 2Rr cos(α θ) R 2 r 2 R 2 + r 2 2Rr cos(α θ) = 2 ( r ) k cos k(α θ) + R k= 34

135 2π (r, θ) = 2π 2π ( f(α) R 2 r 2 R 2 2Rr cos(α θ) + r 2 ) dα (7.63) u(r, θ) = f(θ) (r R) P (r, θ) P M(R, θ) 7.62 u(p ) f(β) = R2 r 2 2π 2π f(α) f(β) ρ 2 dα M δ C C C 2 C Q(R, α) f f(α) f(β) < ϵ 7.: C C C 2 C R2 r 2 2π 2π f(α) f(β) ρ 2 dα < ϵ R2 r 2 2π dα 2π ρ 2 = ϵ P M δ/2 C 2 Q2 C f < K() R2 r 2 2π 2π ρ = P Q > δ 2 f(α) f(β) ρ 2 dα < 4K(R2 r 2 ) 2π dα 2π ρ 2 = 8K(R2 r 2 ) δ 2 35

136 P M r R C 2 ϵ P M u(p ) f(m) f(θ) u(r, θ) 2 f(θ ) f(θ 2 ) < ϵ ( θ 2π) u (r, θ) u 2 (r, θ) < ϵ R2 r 2 f(θ) 2π 2π dα ρ 2 = ϵ 7.7 P P P δ u(p ) = uds 2πδ 3 u(p ) P u u S S u S P 2 3 r = x x Q(x ) 2 c 3 u(r) = a log r u(r) = a cos θ r x x u(x) log r, r 7.8 u(x ) V C 2 V C V P (x) Q(x ) r = x x AdV = A nds V S A = ψ ϕ 36

137 n (ψ ϕ) = ψ ϕ + ψ 2 ϕ ψ(n ϕ) = ψ ψ n Green (Green s first identity) (ψ 2 ϕ + ψ ϕ)dv = ψ, ϕ V V (ψ 2 ϕ ϕ 2 ψ)dv = Green 2 (Green s second identity) S S ψ ϕ ds (7.64) n ( ψ ϕ ) n ϕ ψ ds (7.65) n 2 C A = u, 2 u = u dl =, n ϕ = ψ = u V ( 2 u)dv = S 3 2 V S u u n ds, ( 2 u)dv =, S S u n = u ds = (7.66) n ( 2 u)ds = S C u u dl (7.67) n ( 2 u)ds = (7.68) 7.2: C 3 u(x) = a r r = x 37

138 2 u = Ω n V u ds = a rdω = 4πa (7.69) S n Ω r 2 3 u(x) Const ( x ) (7.7) u(x) ( x ) V u = u u 2 2 u = u = u 2 u (x) u(x) = u (x) + u (x) 2 u = 7.65 ϕ, ψ u, v 2 u = τ(x) V C v(x) r = x x 2 v = 4πτ(x) v(x) = V τ(x ) dv (x ) r 7.7 ϕ(x) C 2 σ(x) C u (x) = 4π V v(x) = S ϕ(x ) dv (x ) r σ(x ) log r ds(x ) 2 v(x) = 2πσ(x) 38

139 7.9 u(x ) V C 2 V C V P (x) Q(x ) r = x x ϕ = u, ψ = r Q P ψ P δ V δ V V V δ S δ V ( ) 2 = r 7.3: n V V δ V 7.65 (ψ 2 ϕ ϕ 2 ψ)dv = V ( r 2 u)dv = V S S ( ψ ϕ ) n ϕ ψ ds n ( u r n u ( n r )) ds + S δ ( u r n u n ( )) ds (7.7) r δ V r dv = 2πδ2 V ( r 2 u)dv S δ n r r = δ, n ( ) = r r ( ) = r δ 2 S δ S δ P, P 39

140 S δ ( u r n u n S ( )) ds = u r δ S δ n ds δ 2 uds S δ = 4πδ u n (P ) 4πu(P ) u(p ) u(x) 7.7 δ u(x) = ( u 4π r n u ( )) ds n r 4π V ( r 2 u)dv 2 C S u(x) = {( log ) u 2π C r n u ( log )} dl ( log ) 2 uds n r 2π S r u V, S 2 u = u(x) = ( 4π S r u(x) = 2π C u n u n ( )) ds (7.72) r {( log ) u r n u ( log )} dl (7.73) n r 8 Green Green 8. ϕ = ρ (8.) ϵ ρ ϵ ϕ S = ϕ Green r r G(r, r ) δ(r r ) (8.2) G(r, r ) r on S = 4

141 Green 3 G(r, r ) = 4π r r Green Green 8.2 S (ϕ G G ϕ)ds = (ϕ G G ϕ)dv V = ( ϕδ(r r ) + G ρ )dv V ϵ = ϕ(r) + G(r, r )ρ(r )dv ϵ /r 2 ϕ(r) = G(r, r )ρ(r )dv ϵ V ρ(r ) = 4πϵ r r dv V Green V (8.3) 8.2 L L a = b a = L b L G G (Greenian) a = G b l x l l x = δ(x x ) l a = l G b = l G l l b dl = G(l, l )ϕ(l, b)dl G(l, l ) (Green function) Green x C 2 u(x), v(x) Lu(x) = f(x), a < x < b (8.4) 4

142 L L = p(x) d2 dx 2 + q(x) d + r(x) (8.5) dx B u = γ, B 2 u = γ 2 (8.6) 8.5 b a vludx = = = b a ( ) vpu + vqu + vru dx [ p(vu v u) + (q p )uv [ p(vu v u) + (q p )uv ] b a ] b a + + b a b a ( ) u pv + 2p v qv + p v q v + rv dx ul vdx (8.7) L L (formal adjoint differential operator) ( ) L = p d2 dx 2 + 2p d q dx + q + r p [ ] b p(vu v u) + (q p )uv L = L (formally self adojint) 8.6 γ = a B u =, B 2u = (adjoint boundary conditions) 8.3 Green L a < x, ξ < b x ξ LG(x, ξ) = δ(x ξ) B G = B 2 G = L G (x, ξ) = δ(x ξ) B G = B2G = G (x, ξ) 8.4 u(x) Lu = f(x) G (x, ξ) 8.2 v = G b G (x, ξ)f(x)dx = b ul G dx = b a a a u(x)δ(x ξ)dx = u(ξ) 42

143 u(x) = b a G (ξ, x)f(ξ)dξ (L, B, B 2 ) G (ξ, x) G (ξ, η) = G(η, ξ) G(ξ, η) = G(η, ξ) ξ L D Lu(x) = f(x) Bu(x) = 8.2 D x D x D vludv (x) ul vdv (x) = [boudary term] (8.8) D V (ψ 2 ϕ ϕ 2 ψ)dv = LG(x, ξ) = δ(x ξ) BG = L G (x, ξ) = δ(x ξ) B G = u(x) u(x) = D S ( ψ ϕ ) n ϕ ψ ds n G (ξ, x)f(ξ)dv (ξ) G G (x, ξ) E (x, ξ) G (x, ξ) = E (x, ξ) + g (x, ξ) E (x, ξ) g (x, ξ) E (x, ξ) L x = ξ δ L L = L 3 2 (8.9) E(x, x ) = E (x, x ) = 4πr (r = x x ) 3dimention (8.) 43

144 E(x, x ) = E (x, x ) = 2π log r (r = x x ) 2dimention g (x, ξ), g(x, ξ) u x D Bu = u = B v = v = (L, B) 2 S (v 2 u u 2 v)ds = S ( v u n u v ) dl n u = G(x, ξ) v = G (x, η) G (ξ, η) = G(η, ξ) 8.3. Lu(x, y) = 2 u(x, y) = f(x, y) (x, y) D u(x, ) = ϕ(x) (x > ) u n (, y) = ψ(y) (y > ) u ( x 2 + y 2 ) / n D 8.: (ξ, η) 44

145 LG(x, ξ) = 2 G(x) = δ(x ξ) (x D, ξ D) G G(x, ; ξ, η) =, n (, y, ξ, η) = (x >, y > ) G(x, y; ξ, η) ( x 2 + y 2 ) 2 u(x, y) = f(x, y) S (v 2 u u 2 v)ds = S ( v u n u v ) dl n v = G 2 G(x) = δ(x ξ) u n (, y) = ψ(y) ( G(x, ξ) 2 u u 2 G(x, ξ) ) dv = G(x, ξ)f(x, y)dv u(ξ) D = = D D ( ψ ϕ n ϕ ψ n ϕ G(x, ; ξ, η) dx + x ) dl (8.) G(x, ξ) (ξ, η) 2 G(, y, ξ, η)ψ(y)dy (8.2) G(ξ, η) = 2π log r (8.3) u(x, y) = + ϕ(ξ) G(x, y; ξ, ) dξ η G(x, y; ξ, η)f(ξ, η)dξdη G(x, y,, η)ψ(η)dη G(x, y; ξ, η) = G(ξ, η; x, y) 8.3 G(x, y; ξ, η) = ( ) log 4π [(x + ξ) 2 + (y + η) 2 ] + log [(x ξ) 2 + (y + η) 2 ] ( ) log 4π [(x ξ) 2 + (y η) 2 ] + log [(x + ξ) 2 + (y η) 2 ] = [ (x ξ) 2 4π log + (y η) 2] [ (x + ξ) 2 + (y η) 2] [(x ξ) 2 + (y + η) 2 ] [(x + ξ) 2 + (y + η) 2 ] 45

146 8.3.2 Green D a D Lu(x) = 2 u = f(x) (x D) (8.4) u(x) = ϕ(x) (x D) LG(x, ξ) = 2 G(x) = δ(x ξ) (x D) G(x, ξ) = (x D) (8.5) 8.2 G(x, ξ) ψ = G D ( G(x, ξ) 2 u u 2 G(x, ξ) ) dv = G(x, ξ)f(x)dv u(ξ) D ( = ψ ϕ ) S n ϕ ψ ds n G(x, ξ) = ϕ(x, ξ) ds x / n D u(x) = Green D D G(ξ, x) G(ξ, x)f(ξ)dv ϕ(ξ, x) ds (8.6) D ξ a G(x, ξ) 8.5 ξ o P(x) r Q r Q P(x) o r Q r Q 8.2: OQ OQ = a 2 ξ Q x P Q OQ OQ = a 2 Q Q x G(x, ξ) P Q = r, P Q = r G(x, ξ) = 4πr + q 4πr 46

147 q P P OQ Q OP r = a ρ r q = a ρ 8.5 P OQ = γ, OP = ρ G(x, ξ) = 4πr + a/ρ 4πr r = P Q = ρ 2 + ρ 2 2 ρρ cos γ r = P Q = ρ ρ2 ρ 2 + a 4 2a 2 ρρ cos γ (8.7) G(x, ξ) = G(ξ, x) 8.4 f(x) = ξ ρ < a G n = G Surface ρ ρ=a a 2 ρ 2 = 4πa (a 2 + ρ 2 2aρ cos γ) 3/2 ξ x ξ = ρ sin θ cos α, η = ρ sin θ sin α, ζ = ρ cos θ x = a sin θ cos ᾱ, y = a sin θ sin ᾱ, ζ = a cos θ x ξ 2 = a 2 + ρ 2 2aρ cos γ = a 2 + ρ 2 2aρ ( cos θ cos θ + sin θ sin θ cos(ᾱ α) ) cos γ = cos θ cos θ + sin θ sin θ cos(ᾱ α) u(ρ, θ, α) = a(a2 ρ 2 ) 4π 2π π ϕ( θ, ᾱ) sin θd θdᾱ a 2 + ρ 2 2aρ ( cos θ cos θ + sin θ sin θ cos(ᾱ α) ) 3/ k L = t k x 2 47

148 Lu(x, t) = f(x, t) ( < x <, t > ) u(x, ) = ϕ(x) (8.8) L L = t k x 2 vlu ul v = (u t ku xx ) + u(v t + kv xx ) = (uv) t k(vu x uv x ) x (x, t) D = (a, b) (t, t 2 ) t2 b t a (vlu ul v) dxdt = b k a t2 [uv] t=t2 dx t b [uv] t=t dx a t2 [vu x uv x ] x=b dt + k t [vu x uv x ] x=a dt (8.9) E (x, t; ξ, τ) t > τ L E (x, t; ξ, τ) = Et kexx = δ(x ξ, t τ) ( < x <, t > ) (8.2) E = (t > τ) F [δ(x ξ, t τ)] = δ(x ξ)δ(t τ)e iκx dx = e iκx δ(t τ) te ˆ (k, t) kκ 2 E ˆ (k, t) = e iκξ δ(t τ) Eˆ (k, t) = (t > τ) t H(x) H(x) = ˆ E (k, t) = x δ(ξ)dξ ae t (t < τ) (t > τ) = ah(τ t)e kκ2 t a a = e kκ2 τ ikξ ˆ E (k, t) = H(τ t)e kκ2 (τ t) e κξ 48

149 E (x, t; ξ, τ) = = H(τ t)e kκ2 (τ t) e κξ edκ ( ) H(τ t) (x ξ) 2 exp 4πk(τ t) 4k(τ t) 8.9 t2 b t a (vlu ul v) dxdt = a =, b = b a [uv] t=t dx 8.2 t ul vdxdt = = t vludxdt + [uv] t=t dx ul E (x, t; ξ, τ)dxdt = uδ(x ξ, t τ)dxdt = u(x, t) 8.8 E u(x, ) = ϕ(x), Lu = f, t =, t 2 = t > τ 8.6 u(x, t) = = t t ( f(x, t) (x ξ) 2 exp 4πk(τ t) 4k(τ t) ( f(ξ, τ) (x ξ) 2 exp 4πk(τ t) 4k(τ t) ) dxdt + ) dξdτ + ϕ(x) 4πkt exp ϕ(ξ) 4πkt exp ( (x ξ) 2 4kt ( (x ξ) 2 4kt ) dx ) dξ L = L = 2 t 2 c2 2 LE(x, t; ξ, τ) = 2 t 2 E c2 2 E = δ(x ξ, t τ) (8.2) τ E = (t < τ) t > τ δ(t τ) = d 2 Ê dt 2 + c2 k 2 Ê = e ik ξ δ(t τ) (k = k ) 49

150 Ê(k, t) = (t < τ) A sin(ckt) + B cos(ckt) (t > τ) t = τ Ê(k, t) = A sin(ckτ) + B cos(ckτ) = cka cos(ckτ) ckb sin(ckτ) = e ik ξ e ik ξ A = cos(ckτ) ck e ik ξ B = sin(ckτ) ck H(t τ) Ê = ck e ik ξ sin [ck(t τ)] H(t τ) k(k, k 2, k 3 ) ck = κ, κ = κ E = e ik (x ξ) (2π) 3 sin (ck(t τ)) dk dk 2 dk 3 H(t τ) ck E = H(t τ) (2πc) 3 e iκ/c (x ξ) κ sin (κ(t τ)) dκ dκ 2 dκ 3 E = = = = H(t τ) (2πc) 3 H(t τ) (2πc) 3 π H(t τ) 4π 2 c 2 x ξ H(t τ) 4π 2 c 2 x ξ iκ/c x ξ cos θ e sin (κ(t τ)) sin θdθdκ κ iκ/c x ξ cos θ e sin (κ(t τ)) d (cos θ) dκ κ ( ) x ξ 2 sin κ sin (κ(t τ)) dκ c { ( ) ( cos κ t τ cos κ t τ + x ξ c )} x ξ dκ c δ π cos κxdκ = δ(x) r = x ξ ( ( E(x, t; ξ, τ) = δ t τ = H(t τ) 4πc 2 x ξ (t 4πc 2 r δ τ r c ) ( x ξ δ t τ + c ) )) x ξ c (8.22) 5

151 τ ξ c x t τ = x ξ /c t L τ L E (x, t; ξ, τ) = 2 t 2 E c 2 2 E = δ(x ξ, t τ) (8.23) E = (t > τ) E t t, τ τ E (x, t; ξ, τ) = (t 4πc 2 r δ τ + r ) c 2 z [ ( )] 2 2 t 2 c2 x y 2 E(x, y, t; ξ, η, τ) = δ(x ξ)δ(y η)δ(t τ) z E z ζ, E(x, y, t; ξ; η, τ) = δ ( ) t τ r c 4πc 2 dζ r z ζ = s, r 2 = ρ 2 + s 2, ρ = (x ξ) 2 + (y η) 2 α = r c = ρ 2 + s 2 /c ds r = ds cα = cdα cdα = s c2 α 2 ρ 2 E(x, y, t; ξ; η, τ) = = = 2πc 2 ( δ t τ ) ρ 2 + s 2 /c ds ρ2 + s 2 δ (t τ α) 2πc ρ/c c2 α 2 ρ dα 2, c(t τ) < ρ 2πc c 2 (t τ) 2 ρ 2, c(t τ) > ρ E = (t < τ) 5

152 8.23 y η, c(t τ) < x ξ = ρ E(x, y, t; ξ; η, τ) = c(t τ) > x ξ = ρ y η = s, c 2 (t τ) 2 (x ξ) 2 = a 2 a E(x, y, t; ξ; η, τ) = 2πc a a2 s ds 2 = [ ( sin s )] a = πc a 2c, c(t τ) < ρ =, c(t τ) > ρ 2c E = (t < τ) x t τ = x ξ /c t,2 (x, y) t τ = ρ/c t δ (x, y) 8.4 H = H + H I (8.24) H = ħ2 2m 2 Hϕ(r) = Eϕ(r) (E H)ϕ(r) = Lϕ(r) E Green G = L = (E H) (8.25) 52

153 E H I LG = (E H)G = I r (E H)G r = r I r = δ(r r ) (8.26) r (E H)G r = r (E H) r r r G dr Green G(r, r ) = r r G dr = ϕ (r )Gϕ(r )dr L 8.24 r (E H) r = r L r = Lr r = Lϕ (r)ϕ(r )dr = Lδ(r r ) = (E H)δ(r r ) = (E + ħ2 2m 2 r H I(r ))δ(r r ) f(r) = f(r )δ(r r )dr (E + ħ2 2m 2 r H I(r ))δ(r r )G(r r )dr = (E + ħ2 2m 2 r H I (r))g(r r ) 8.42 (E + ħ2 2m 2 r H I (r))g(r, r ) = δ(r r ) 8.5 Green n Green Green G L = (E H) = n (E H) n n = n = n n n (E H) n n n n E E n (8.27) 53

154 Green G(l, l ) = l G l = n < l n >< n l > E E n ϕ n () =, ϕ n (a) = P E n ϕ(x) = ˆ 2 n 2m ϕ(x) k n = 2mEn ħ 2 a k n = nπ < n x >= d 2 dx < n x > +k2 n < n x >= 2 2 a sin(k n x) = a sin(nπ a x) E n = 2m Green G(x, x ) = x G x = < x n >< n x > E E n n = ( 2 ) ( ( a sin nπ a x) nπ sin a x ) ( n E nħπ ) 2 = 2m a n ( nħπ a ) 2 ( ) ( ( ) nπ a cos a (x ) x E 2m ( nπ cos ( nħπ a ) 2 )) a (x + ) x x :, x ; 2, 2 a = n a 8.3: n=2 a=, a=2 54