7.1. Lecture 8 & 9. f(x)dx =lim f(x)dx (7.1) I = f(x)dx (7.3) f(z), z (0 argz π), zf(z) 0. f(z)dz = I R := f(z)dz = f(re iθ )ire iθ dθ (7.

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Download "7.1. Lecture 8 & 9. f(x)dx =lim f(x)dx (7.1) I = f(x)dx (7.3) f(z), z (0 argz π), zf(z) 0. f(z)dz = I R := f(z)dz = f(re iθ )ire iθ dθ (7."

Jakub Kuczyński
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1 Lecture 8 & 9 7, r f(x) =lim f(x) (7.) r r f(x) =lim f(x) +lim f(x) (7.) r r r 7. f(z) I = f(x) (7.) f(z), z ( argz π), zf(z) [ R, R], : z = R Jordan C f(z). C f(z)dz = R R f(x) + f(z)dz =πi i Res z=zi f(z) (7.4). f(z) z i π I R := f(z)dz = f(re iθ )ire iθ dθ (7.5)

2 Figure 7.: C, ϵ>, R θ π θ Rf(Re iθ ) <ϵ I R ϵ π dθ = πϵ (7.6) lim I R = (7.7) R f(x) =πi i Res z=zi f(z) (7.8) 7.. x + = π (7.9) x =tanθ = dθ cos θ π x + = dθ = π π [ R, R], : z = R Jordan C f(z) = C, z +

3 C f(z)dz = Figure 7.: R R f(x) + f(z)dz =πir(i) (7.) z = i f(z) R(i) =(z i) z + = z=i i (7.) f(z)dz z + πr dz (R ) (7.) R lim f(z)dz = (7.) R x + =πi i = π (7.4) : 7.. (7.5) x 4 +, x 4 + = (7.6) x 4 + 4

4 Figure 7.: C :[ R, R] R, f(z) = +z 4. f(z) z = ω ±, ω ±, ω = e iπ/4 C f(z) dz πi + C f(z) dz πi f(z) dz πi = Res z=ω f(z)dz +Res z= ω f(z)dz (7.7) = R f(x) (R ) (7.8) πi R πi +x 4, z = Re iθ, z 4 + z 4, π f(z)dz R π f(z) dz = +R 4 e 4iθ dθ R dθ (R ) (7.9) R 4, Res z=ω f(z)dz = z ω lim z ω z 4 + = 4ω = ω 4 Res z= ω f(z)dz = z + ω lim z ω z 4 + = 4( ω ) = ω 4 = ω 4 +x 4 = +x 4 = πi ( 7.. a> ) ω4 ω + = πi ( e iπ ) iπ + e 4 = π (7.) (7.) a + x (7.) (a + x ) (7.) (7.) a (7.) 5

5 7. I = 6 a>, f(z) f(z) lim z f(z) =( argz π) f(x)e iax (7.4) [ R, R], : z = R Jordan C f(z), C f(z)e iaz dz = R R f(z)e iaz dz + f(z)e iaz dz =πi i Res z=zi f(z)e iaz (7.5) f(z) z i, z = Re iθ I R := f(z)e iaz dz = π f(re iθ )e iar cos θ ar sin θ ire iθ dθ (7.6) ϵ>, R, θ π θ f(re iθ ) <ϵ π I R ϵr e ar sin θ dθ =ϵr π/ Jordan e ar sin θ dθ (7.7) Jordan r> π π e r sin θ dθ < π r e r sin θ dθ < π r (7.8) (7.9) 6 Fourier.5 6

6 θ π, sinθ π θ, r> e r sin θ e r π θ π e r sin θ dθ π e r π θ dθ = π π r r e π θ = π r ( e r ) < π r [ π,π], θ = π ϕ π π e r sin θ dθ = π e r sin ϕ ( dϕ) = π e r sin ϕ dϕ (7.) ( ) Figure 7.4: Jordan I R Jordan I R ϵr π/ e arθ/π dθ =ϵr e ar ar/π < π a ϵ (7.) lim I R = (7.) R f(x)e iax =πi i Res z=zi (f(z)e iaz ) (7.) 7

7 7.. Re(e iax )=cosax cos ax b + x = π b e ab (a>,b>) (7.4) ( ) f(z) = eiaz z + b (7.5) e iax (7.6) x + b C ib -R O R Figure 7.5: [ R, R] (R>b) : z = Re iθ ( θ π) C f(z). C f(z) z = ib Res z=ib f(z)dz = eiaz z + ib = e ab z=ib ib R e iax f(z)dz = b + x + f(z)dz =πires CR z=ib f(z)dz = πe ab b C R f(z) f(z)dz f(z) dz π R sin θ e R b Rdθ (7.7) (7.8) Rπ R b (7.9) R lim R f(z)dz = e iax = lim b + x R (7.4) R R e iax πe ab = b + x b (7.4) 8

8 7.4. x sin x +x = π e (7.4) x sin x xe ix =Im +x +x, xe ix [ R, R] R C +x R, xe ix +x + ze CR iz +z dz =πir(i) =πi e +z dz Re R sin θ R rdθ CR ze iz π Jordan (7.9) < R e R sin θ dθ R < R π (R ) (7.4) R R x sin x +x =Imπi e = π e : πi = e iax x ib { e ab (a >) (a<) (b >,a : ) (7.4) (7.4) b + (b ) lim b + πi e iax x ib = (step function) θ(x) { (x>) θ(x) := (x<) { (a>) (a<) (7.44) (7.45) 9

9 7.. e ixt θ(x) = lim dt (7.46) ϵ + πi t iϵ cos x (a >,b>) (7.47) (x + a )(x + b ) x = π (7.48) f(z) = +z z =, 7.6. r +x + C r Figure 7.6: dz z + + re πi πir(e πi )=πi z C r z=e πi dz z + r π r dz πi =πir(e z ) + =πi e πi (r )

10 re πi dz z + = r = e πi +x =πi e πi e πi ds s + r (z = se πi ) x + e πi = π (7.49) 7.. +x 5 (7.5) 7.6. sin x x = eix e ix ix sin x x = π (7.5) sin x x = lim R ϵ f(z) = eiz z R ϵ e ix e ix ix = lim R ϵ ( R + i ϵ ϵ R ) e ix (7.5) x eiz, z C r C ε Figure 7.7: ( r ϵ + + C r ϵ r ) e iz + dz = (7.5) C ϵ z

11 . C r Jordan e Cr iz z dz π r sin θ e ire iθ dθ re iθ π r (r ) (7.54) C ϵ Cϵ e iz z dz (7.5) π = = i π i iϵ(cos θ+i sin θ) e ϵe iθ iϵe iθ dθ e iϵ(cos θ+i sin θ) dθ iπ (ϵ ) (7.55) sin x iπ = (7.56) x sin x x = π (7.57) Cauchy f(x), x f(x) x δ P f(x) := lim f(x) + f(x) (7.58) δ + x +δ (principal value) a, f(z) f(x) P (7.59) x a Imz > z z k f(z) <M(k >,M > : )

12 z = a C C f(z) f(x) dz =P iπf(a) (7.6) C z a x a f(z) f(x) dz =P + iπf(a) (7.6) z a x a C f(z)dz R π f(re iθ ) dθ < R a R R a πm (R ) (7.6) Rk C ϵ z = a + ϵe iθ f(z)dz = f(a + ϵe iθ )idθ iπf(a) (ϵ ) (7.6) C ϵ π (7.6) Figure 7.8: C, C f(z) (7.6) = f(a) = iπ P f(x) (7.64) x a Ref(a) = π P Imf(x) (7.65) x a Imf(a) = π P Ref(x) (7.66) x a (dispersion formula)

13 Dirac delta (7.6) z = x + iϵ (ϵ >) ϵ (7.6) lim ϵ + lim ϵ + ( lim ϵ + δ(x a) := πi lim ϵ + f(x) =P x a + iϵ f(x) =P x a iϵ x a + iϵ x a iϵ ( ) x a + iϵ x a iϵ Dirac delta 7 (7.69) (7.67) (7.68) ϵ f(x) iπf(a) x a (7.67) f(x) + iπf(a) x a (7.68) ) f(x) = πif(a) (7.69) = lim ϵ + π ϵ (x a) + ϵ (7.7) δ(x a)f(x) = f(a) (7.7) x a + iϵ =P iπδ(x a) x a (7.7) x a iϵ =P + iπδ(x a) x a (7.7) log x (7.74) +x I r C r ϵ C ϵ f(z) = log z +z r f(x) + f(z)dz + f(z)dz + f(z)dz =πir(z = e iπ/ ) (7.75) C r re πi/ C ϵ 4

14 Figure 7.9: z = e iπ/, (z z )logz R(z )= lim = log z z z +z z = z=z log z dz = eπi/ +z re πi/ r πi/ πi = eπi/ 9 e πi (7.76) log r +πi/ dr e πi/ I πi dr +r eπi/ (7.77) +r C r f(z)dz = π/ r r I = e ± πi = ± i dr +r I e πi/ I πi eπi/ log(re iθ ) +r e iθ rieiθ dθ (7.78) dr +r =πiπi 9 e πi/ (7.79) ( e πi πi )I e πi I = π πi 9 e (7.8), (7.8) I + π I = π 9 I + π I = 7 π 9 (7.8) (7.8) 5

15 I = π 7, I = π 9 (7.8) log x (7.85) (x +) 7.8. x a +x = π, ( <a<) (7.86) sin aπ f(z) = z a +z z = 9 z a C argz =, z a, C (e πi z) a C - C ε C R C dr +r = π Figure 7.: I = π 7 (7.84) 9 c C z c := exp(c log z) log z Logz+πik (k Z), z c exp(c(logz+ πik)) 6

16 R x a R ϵ +x + e πia x a f(z)dz C r ϵ +x + f(z)dz C ϵ = πires z= f(z) =πie πia (7.87) C ϵ π f(z)dz ϵe πi+iθ a ϵdθ = ϵ a π ( <a<,ϵ ) (7.88) C ϵ ϵ ϵ C r π f(z)dz (re iθ ) a r a rdθ = πr ( <a<,r ) (7.89) C r r r. ( e πia x a ) +x =πie πia (7.9) x a +x = π sin πa (7.9) 7.5. x a ( <a<) (7.9) (x +) 7.5 sin θ, cosθ F (sin θ, cos θ) I = π C : z = e iθ ( θ π) F (sin θ, cos θ)dθ (7.9) sin θ = z z, cos θ = z + z i I = C F ( z z i, z + z ) dz iz dz = izdθ (7.94) 7

17 7.9. π dθ +asin θ = π, ( <a<) (7.95) a az +iz a = I = = C C +a z z i dz iz dz (7.96) az +iz a z = ± a i (7.97) a z ( ) z ( ). z C I =πir(z ) =πi az +i π = a (7.98) 7.6. π cos nθ dθ ( <a<,n=,,,...) (7.99) a cos θ + a 7.6 Fresnel 7.. cos x = sin x = π (7.) Fresnel, (Fresnel ) Gauss e x = π (7.) 8

18 f(z) =e z Figure 7.: Fresnel C R e x e x = π (R ) (7.) C z = Re iθ π/4 dz R e R cos θ dθ C e z = R R π/ π/ e R sin ϕ dϕ e R ϕ π dϕ = R R π ( e R ) (R ) (7.4) I = e x, I = π π I = dye x y = drr dθe r =π e r = π (7.) 9

19 C : z = re iπ/4 dz = C e z C +C +C e z dz = R e ri e iπ/4 dr = +i R (cos r i sin r )dr +i (cos r i sin r )dr (R ) (7.5) (cos r i sin r )dr = i π (7.6), (7.) 7.7 : 7.. b a b a (x a)(b x) = π (a <b) (7.7) (x a)(b x) = π 8 (a b) (a<b) (7.8) f(z) =((z a)(z b)) ±/ z =

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