Stirlingの公式

Transcription

1 Stirling Stirling Fourier Lplce 3 3. Guss URL Ver... (( Ver..22: Ver..23(89 : Ver..24: Ver..25(9 : Ver..26(94 : Person Ver..27(94 :. 9 Ver..28(96 : Ver..29(96 : Ver..29:. 4 Ver..3(97 : Ver..3(98 : Tylor (.. Riemnn-Lebesgue ( Ver..32(98 : Ver..33(98 : 7.6. GitHub. 5 8 Ver..34(99 : mthtodon Ver..34(99 :. 7 2 Ver..35( : Dirichlet Ver..35:. Ver..35b:. 22 Ver..36: 8.7 Euler. 2 Ver.37: ( Ver..38(2 :. Wllis ( Ver..39(3 : ( Ver..4(3 :. 7 8 Ver..4(3 :.

2 2 4 Stirling 8 4. Stirling Stirling : Fourier 2 5. Guss Riemnn-Lebesgue Fourier Riemnn Dirichlet Riemnn Fourier : Guss Fourier Tylor Cuchy : Guss Jcobin : Fourier Wllis B(s, / Fresnel Dirichlet Stirling-Binet ( Stirling-Binet ( :

3 3 9.3 Person t F n Mxwell-Boltzmnn ( n Mxwell-Boltzmnn ( Poisson : Tuber 8. Tuber Lplce Tuber Wllis x x 2 + x 4 x 8 + x 6 x 32 + x? Lplce-Stieltjes Lplce-Stieltjes Tuber : Tylor 96. Tylor Tylor Tylor : Ver.. 3. : Kullbck-Leibler ( Snov, Snov, Boltzmnn (e βe i, Gibbs (, e βe i q i /Z. Stirling n! n n e n 2πn (n

4 4.. n b n (n lim n ( n /b n =. n! = n n e n ( 2πn + ( 2n + O (n n 2. Stirling. Stirling... Guss,,, Fourier....: Stirling n n! A n = n n e n 2πn ( /n! A n ( + /(2n ( /n!.92 (7.78%.9989 (.% (2.73% (.28% (.83% (.32% (.28% ( (.8% (3.4 7., n n e n 2πn n!, n = 3 3%, n = %. /(2n, n =.% : ( 2π + = e 2 Stirling 2. 3 : e x/τ x α (x >, f α,τ (x = Γ(ατ α (x Gergö Nemes, New ymptotic expnsion for the Γ(z function, 27. Nemes [( ] n 2πn n! = n + = n n 2n e n ( 2πn + n e 2n n n s > Γ(s = e x x s dx. Γ( =, Γ(s + = sγ(s, n Γ(n + = n!.

5 5 α, τ > 4. α = n >, τ = f n (x = f n, (x : f n (x = e x x n Γ(n (x >. f n (x X n. X n µ n σn 2 n 5 : Γ(n + µ n = E[X n ] = xf n (x dx = = n, Γ(n E[Xn] 2 = x 2 Γ(n + 2 f n (x dx = = (n + n, Γ(n σ 2 n = E[X 2 n] µ 2 n = n. Y n = (X n µ n /σ n = (X n n/ n, nfn ( ny + n = n e ( ny+n ( ny + n n 6. y = Γ(n nfn (n = n e n n n Γ(n = nn e n n Γ(n +. n > Γ(n + = n!, n / 2π Stirling.,, R φ(x, n φ ( x n n f n (x dx = φ(y nf n ( ny + n dy φ(y e y2 /2 2π dy. φ(y δ(y ( y, nfn (n = n e n n n Γ(n = nn e n n Γ(n + 2π (n. Stirling. y.,.. 4 α shpe prmeter, τ scle prmeter. ατ ατ 2. 5 f(x X, E[g(X] = g(xf(x dx R, µ = E[X], σ 2 = E[(X µ 2 ] = E[X 2 ] µ 2. 6 X f(x, Y Y = (X /b, E[g(Y ] = R g((x /bf(x dx = g(ybf(by + dy, Y bf(by +. R

6 Stirling.,., ( Fourier Fourier. Fourier, Stirling Stirling f n (x = e x x n /Γ(n (x > ( Fourier F n (t 8 : F n (t = e itx f n (x dx = Γ(n, α Γ(n e ( itx x n dx = e αt t n dt = α n. Cuchy 9. Fourier, ( it n. f n (x = e x x n Γ(n = 2π e itx F n (t dt = e itx dt (x >. 2π ( it n Stirling., t = nu, nfn (n = nn e n n Γ(n + = e iu n ( iu/ n n 2π n = n log e itn ( it dt = e iu n n 2π ( iu/ n du. n Stirling, n / 2π. : ( log iu iu n n = n ( iu n u2 2n + o ( n iu n = u2 2 + o(. 7 nobuo/pdf/prob/stir.pdf. 8 n n. 9 Cuchy. f(α, f( = f (α = (n/αf(α,. α t = x/α. Fourier 5.

7 2.2. 7, n e iu n ( iu/ n n e u2 /2., n nfn (n = nn e n n Γ(n + = e iu n 2π ( iu/ du e u2 /2 du = n n 2π 2π. α e u2 /α du = απ 2. Stirling. 2.2 f n (x = e x x n X n, Y n = (X n n/ n (. Y n nfn ( ny + n = n e ny n ( ny + n n Γ(n = e n n n /2 Γ(n e ny ( + y/ n n + y/ n., n ( ( log e ny + y n n = n log = n ( + y ny n ( y n y2 2n + o ( n ny = y2 2 + o(, n e ny ( + y/ n n e y2 /2, + y/ n., Stirling : nfn ( ny + n = n e ny n ( ny + n n Γ(n e y2 /2 2π (n. Y n Stirling. Stirling., Lebesgue. 2 Guss dx = π e x2 x = u/ α. Guss. I I 2 = +y 2 dx dy e (x2, I 2 z = e (x2 +y 2 z =. < z π log z, ( π log z dz = π[z log z z] = π. I = π. Guss,., z π. I, I 2.

8 8 2. Y n nf(n : nfn ( ny + n = n 2π 2π, Cuchy 3 e iuy e u2 /2 du = e it( ny+n ( it dt = n 2π e iuy e iuy e u2 /2 du = 2π e y2 /2 e (u+iy2 /2 y 2 /2 du = e y2 /2 e it n ( iu/ n n dt (n. e v2 /2 dv = e y2 /2 2π., Fourier, Stirling. 2.3 Fourier, n! = Γ(n + = e x x n dx. x = n + n y = n( + y/ n y, n! = n n e n ( n e n y + y n dy. n c n = n { n! e n y n n e n n, h ( + y/ n n (y > n, n(y = (y n., c n = h n(y dy. log h n (y y = Tylor log h n (y = y 2 /2 + o( (n, lim n h n (y = e y2 /2. lim h n (y dy = e y2 /2 dy n 4, lim n c n = 2π. Stirling n! lim n n n e n 2πn = 3., e ity Tylor., f (y = yf(y, f( = 2π (. Cuchy u + iy (u > v >. 4 y h n (y h (y = e y ( + y, y h n (y e y2 /2, Lebesgue. Lebesgue, y M h n.

9 Fourier. Lebesgue { e y ( + y (y, h n (y e y2 /2 (y. (ϕ(y. y > n, l n (y = log h n (y, n l n(y = + y/ n y n = + y/ n, l n(y = l n (y = ( + y/ n 2 <, 2/ n ( + y/ n 3 >, l n ( =, l n( =, l n( =. Tylor, y > n < θ <, l n (y = y2 2 + Ay3, A = 3! l n (θy = 3 n( + θy/ n 3 >. lim n l n (y = y 2 /2. lim n h n (y = e y2 /2. y, Ay 3 l n (y e y2 /2, h n (y e y2 /2. y, l (y = log(e y (+y l n (y (n. l ( = l n (. l (y = y/( + y, l n(y = y/( + y/ n, n + y + y/ n, l (y l n(y (y., y l (y l n (y. h n (y h (y = e y ( + y n X,..., X n Y n = X Xn 2 Y n n 2. n 2 shpe α = n/2 scle τ = 2. n 2 e y/2 y n/2 (y >, f n/2,2 (y = Γ(n/22 n/2 (y., n 2n., g(y e y/2 y n/2 Rn dy = g(x x Γ(n/22n/2 e (x2 2n + +x2 n/2 (2π n/2 dx dx n.,, n =. n = Γ(/2., x >

10 2. x = y g(x 2 e x2 /2 2π dx = 2 g(y e y/2 y /2 dy = 2π 2 g(y e y/2 y /2 dy. Γ(/22/2 Γ(/2 = π., n, 2 n 2n.., Stirling (. : n 2,, n Stirling., Stirling n X, X 2, X 3,.... µ = E[X k ] σ 2 = E[(X k µ 2 ] = E[X k ] 2 µ 2. Y n = (X + + X n nµ/ nσ 2 Y n. n Y n, (. X k (X k µ/σ. Y n. X k, X k φ(t = E[e itx k ], φ(t = t2 2 + o(t2. Y n = (X + + X n / n Y n, Y n : n ( E[e ity n ] = E[e itx k/ n n t ] = φ n = k= ( n ( t2 2n + o e t2 /2 n (n., Fourier 5, Y n 6 f n (y f n (y = ( n t e ity φ dt 2π n 5 φ(t/ n n Y n Fourier,,. 6 R.

11 2.6., n 7. 2π e ity e t2 /2 dt = e y2 /2 2π X n, g(x n n ( n E[g(X n ] = g(k p k q n k k k=. < p <, q = p, n, ( n k : ( n = k n! k!(n k!, (x + yn = n k= ( n x k y n k. k E[g(X n ] ( δ(x dx 9 : n ( n E[g(X n ] = g(xf n (x dx, f n (x = p k q n k δ(x k. k R, f n (x (, (.., g E[g(X] 2., g E[g(X], g E[g(X],. X φ X (t = E[e itx ]. R : φ X (t = E[e itx ] E [ e itx ] = E[] =, sup t R φ X (t + h φ(t = sup E[e itx (e ith ] E [ e ihx ] (h. t R Lebesgue. g(x g(x = 2π k= e itx ĝ(t dt 7. 8,. 9 ( δ(x dx f(x, g(xδ(x dx = g( R. 2.

12 , E[ ] E[g(X] = 2π ĝ(te[e itx ] dt = 2π ĝ(tφ X (t dt., Y n Y, φ Yn φ Y, 22 g(y E[g(Y n ] E[g(Y ] Y n φ Yn φ φ Y n. φ Yn φ, φ Y, Y n Y 24.. n ( n φ Xn (t = E[e itxn ] = e itk p k q n k k k= n ( n = (pe it n q n k = (pe it + q n k k=. µ n = np σ 2 n = npq. Y n = X n µ n σ n = X n np npq, φ Yn (t = E [ e ity n ] = E [ e itnp/ npq e itx n/ npq ] = e itnp/ npq φ Xn (t/ npq = e itnp/ npq ( pe it/ npq + q n = ( pe itq/ npq + qe itp/ npq n 25. X n, X n np = X n (p+q np = qx n p(n X n, φ Yn (t = E [ e ] ity n = E [ e itqx n/ npq e itp(n X n/ ] npq = = n e itqk/ npq e itp(n k/ npq k= n k= ( n p k q n k k ( n (pe itq/ npq k ( qe itp/ npq n k k = ( pe itq/ npq + qe itp/ npq n 2 g(x ĝ(t g(x , g(y, g(t (. 24 Bochner. 25 p = q = /2 φ Yn (t = (cos(t/ n n.

13 3. pe itq/ npq = p + itpq npq qt2 2n + O qe itp/ npq = q itpq npq pt2 2n + O φ Yn (t = ( n n ( n n ( n ( t2 2n + O n n lim φ Y n (t = e t2 /2 n, Y φ Y (t = E[e ity ] =, e ity e y2 /2 2π dy = e t2 /2., 26 g(y. lim n n g k= lim E[g(Y n] = E[g(Y ] n ( k np npq ( n k p k q n p /2 = g(y e y2 dy. 2π g(y y b, ( lim P X n np n npq X n. b e y2 /2 b = dy. 2π 3 Lplce Stirling ( Guss (. Guss Lplce. 3. Guss Guss Stirling. log(e x x n = n log x x x = n Tylor (x n2 (x n3 n log x x = n log n n + 2n 3n 2 (x n4 4n y b.

14 4 3. Lplce, n n! = Γ(n + = e x x n dx (x n2 exp (n log n n dx = n n e n e (x n2 /(2n dx = n n e n 2πn 2n. n! n n e n 2πn (n. scilb. scilb. Stirling n n e n, 2πn g n (x = log(e x x n = n log x x x = n Tylor 2. 3 y3 e y2 /α dy =. /(2n 27.. : k =,, 2,... e y2 /2 y 2k dy = 2 = 2 k 2 = 2 k 2 e y2 /2 (y 2 k dy = 2, /2 e y2 dy = /2 e y2 y 2 dy = 2π, e y2 /2 y 4 dy = 3 2π, e x (2x k 2 x /2 2 e x x k /2 dx = 2 k 2 Γ(k + /2 dx 3 (2k 2 k π = 3 (2k 2π. e y2 /2 y 6 dy = 5 2π.. x n + n y = n( + y/ n n! = Γ(n + = e x x n dx = n n e n ( n e n y + y n dy. n n ϕ n (y : ( ϕ n (y = n log + y ( n y = y2 n 2 + y3 3 n y4 4n + o n (n. o(/n n n. ( e n y + y n ( ( y = e y2 /2 3 exp n 3 n y4 4n + o n ( = e y2 /2 + y3 3 n y4 4n + ( y 3 2 ( o n n ( = e y2 /2 ( + y3 3 n y4 4n + y6 8n + o. n 27, Stirling, Vol. 3 (979 No. 3, Wllis. Stirling,, Stirling, Vol. 36 (984 No. 2,

15 o(/n n y. e y2 /2 < y <, : e n y n ( + y n n dx e y2 /2 ( y4 4n + y6 8n 2π = 2π 3 2π 4n + 5 8n = ( 2π + 2n + O n! = n n e n ( 2πn + ( 2n + O n 2 ( n 2 + O. ( dx + O ( n 2 (n. /(2n 28 /(2n, n, n! n n e n 2πn n n! 2n. n Lplce Stirling Gergö Nemes, Asymptotic expnsions for integrls, 22, M. Sc. Thesis, 4 pges. Exmple.2.., Stirling /(2n. : > ( =, e nt t s dt = n s n e x x s dx Γ(s n s (n. t = x/n., e nt (α t s + α 2 t s 2 + dt = α Γ(s n s + α 2Γ(s 2 n s 2 + (n. Stirling /(2n. f(x f(x = x log( + x (x >, y = n( + x, n! = Γ(n + = = e y y n dy e n nx n n ( + x n n dx = n n+ e n e nf(x dx. 28.

16 6 3. Lplce x > x < n! n n+ e n = e nf(x dx + e nf( x dx. f(x = t f( x = t,.. f(x = x log( + x f (x = + x = x + x x > f (x >, < x < f (x <. f(x x = f( =, x >, x <. x > < x < t = f(x x = x(t. x = x(t, x = αt /2 + βt + γt 3/2 + t = f(x = x log( + x = x2 2 x3 3 + x4 4 29, α, β, γ., t = (αβ α2 2 t + + α3 t 3/2 + (αγ + β α2 β + α 4 t 2 +. α, β, γ,. α = 2, β = 2 3, γ = 2 8 x = 2 t / t t3/2 + f(x = t. x >. x < t /2 t /2, x x. x = 2 t /2 2 3 t t3/2 f( x = t., dx dt = 2 2 t/2 ± t + 2 t3/2 ± 2 t.. 29 x < Tylor log( + x = x x 2 /2 + x 3 /3 x 4 /4 +.

17 f(x = t, n e nf(x dx = nt dx e dt dt ( 2 = e nt 2 t/ t + 2Γ(/2 = + 2Γ( + 2n /2 3 2π = 2n + 2 2π /2 3n + 24n + 3/2 2Γ(3/2 2n 3/ t3/2 + dt. Γ(/2 = π, Γ( =, Γ(3/2 = (/2Γ(/2 = π/2., f( x = t, n e nf( x dx = 2π 2n 2 2π /2 3n + 24n 3/2. 2, n, : ( n! 2π 2π n n+ e = n n + /2 2n + O (n. 3/2 n 5/2 : n! = n n e n 2πn ( + ( 2n + O n 2 (n. /(2n Lplce t = t + t2 t ( k t k + ( k t k + t e nt dt + t = Γ( n =! Γ(2 n 2 k Γ(k + + ( n k + ( k n! (k! + + ( k + ( k n2 n k e nt t k dt + t e nt t k dt. + t +., e nt dt k (k! = ( + t n k k=. n. +, +.

18 8 4. Stirling 4 Stirling Stirling : log n! (n + /2 log n + n log 2π (n., : log n! = n log n n + o(n (n. o(n n n. Stirling Stirling f(x f(k k+ f(x dx f(k +, k f( f(x, f( + f(2 + + f(n n f(x = log x n n f(x dx f( + f(2 + + f(n f(x dx f( + f(2 + + f(n. n f(x dx + f(n. log x dx = [x log x x] n = n log n n +, log + log log n = log n! n log n n + log n! n log x n + + log n. log n! n log n + n + log n. log n! = n log n n + O(log n = n log n n + o(n (n. O(log n log n. 3 o(n O(log n. O(log n log n.

19 Stirling, n bn ( ( n log = log(n! log(bn! log(( bn! bn. lim n = n log + n log n n + o(n bn log b bn log n + bn + o(n ( bn log( b ( bn log n + ( bn + o(n = n log + o(n. b b ( b b log ( /n n b log bn b b ( b b ( /n ( n = lim bn n (n! (bn!(( bn! /n = (n. b b ( b b. n bn n n (kn! k k. 988 : ( lim n 3nC n 2nC n /n. 3 3 /( /( = = 27 6., Stirling,.., lim n ( /n 2n = 22 n = 22. (o(n n n : ( 2n = 2 2n e o(n (n. n Wllis ( 8.4. ( 2n 22n (n n πn

20 2 4. Stirling. 968 : lim n n n 2nP n. ( 2 2 e. Stirling. : ((n! /n lim = e. n n log ((n!/n n = log(n! log n n = (n log + n log n n + o(n log n n = log + o( = log( e + o(. Stirling. 4.3 Stirling. c, log n! = n log n + log n n + c + o( 2 (n Stirling (/2 log n c.. n log x dx = [x log x x]n = n log n n + k =, 2, 3,..., n [k /2, k + /2] [, log k] [n /2, n] [, log n] log(n! + (/2 log n = log n! (/2 log n, (/2 log n., { (x, y x n, y log x }, n log x dx n,. log x α k, β k α k = k+/2 k log x dx log k, 2 β k = 2 log k k k /2 log x dx., log n! n 2 log n n log x dx = log k + n 2 log n log x dx k= = α + β 2 α 2 + β 3 + β n α n + β n. 3 c log 2π, Wllis e c = 2π.

21 2 n. log x, α, β 2, α 2, β 3, α 3,..., log x x,. n 32., c = +, n log n! = n 2 log n + log x dx + + o( = n log n + log n n + c + o(. 2 c = log 2π Wllis ( 8.4. n! = n n+/2 e n e c e o( Wllis π = lim n 2 2n (n! 2 (2n! n, 2 2n n 2n+ e 2n e 2c π = lim n 2 2n+/2 n 2n+ e 2n e = ec. c 2 e c = 2π. Wllis, Stirling, Stirling n! n n e n 2πn. 5 : Fourier, Fourier. f(x Fourier F (p F (p = e ipx f(x dx. f, f(x = e ipx F (p dp 2π. Fourier. 5. Guss >, f(x = e x2 /(2, F (p Fourier. F (p = e ipx e x2 /(2 dx = e p2 /(2 2π 32 n k= ( k k. (.

22 22 5. : Fourier 33. x, p,. 2, e ipx e p2 /(2 dp = e x2 /(2 2 π f(x = 2π e ipx F (p dp. f(x = e x2 /(2 Fourier. f(x Fourier f(x f(x µ Fourier., F (p f(x Fourier, f(x µ Fourier, e ipx f(x µ dx = e ip(x +µ f(x dx = e ipµ F (p e ipx e ipµ F (p dp = e ip(x µ F (p dp = f(x µ. 2π 2π, f(x µ = e (x µ2 /(2 Fourier. Fourier Fourier, f(x µ = e (x µ2 /(2 Fourier > ρ (x ρ (x = 2π e x2 /(2. ρ (x > ρ (x dx =., ρ (x µ Fourier. f(x f (x ρ f (x : f (x = ρ (x yf(y dy. f (x Fourier 35., f (x Fourier F (p, ( F (p = e ipx f (x dx = e ipx ρ (x y dx f(y dy 33 Cuchy, e ipx Tylor,. 34., Fourier. 35 f (x Fourier ρ (x µ f(µ,.

23 π e ipx F (p dp = = ( ( e ipx e ipx ρ (x y dx dp f(y dy 2π ρ (x yf(y dy = f (x. 2 ρ (x µ Fourier. 36. F (p = e ipx ρ (x y dx = e ipy e p2 /2 e ipy e p2 /2 f(y dy = e p2 /2 F (p e ipx F (p dp = e ipx e p2 /2 F (p dp. 2π 2π e ipx e p2 /2 F (p dp = ρ (x yf(y dy = f (x. 2π F (p, Lebesgue, lim 2π e ipx e p2 /2 F (p dp = 2π e ipx F (p dp., f(x, f (x f(x, f(x Fourier 37., f x. M > f(y f(x M (y R. ε >. δ > y x δ f(y f(x ε/2. ρ, > ρ y x >δ (x y dy ε/(2m. f (x f(x = ρ (x y(f(y f(x dy ε 2 + y x δ ρ (x y f(y f(x dy ρ (x y ε 2 dy + ε 2M M = ε. y x >δ lim f (x = f(x.. ρ (x ym dy 36 Fourier Fourier. 37 ρ (x, ρ (x Dirc (,.

24 24 5. : Fourier,, (996, xii+324, 3,8. Lebesgue Fourier., ,. 5.3 Riemnn-Lebesgue f(x R 38., Fourier f(p = e ipx f(x dx, p. lim e ipx f(x dx =. p Riemnn-Lebesgue (. ˆf(p Lebesgue 39., e ihx f(x 2 f(x f(x, f(x + h f(x e ihx f(x dx R R e ix f(x dx = (h. ˆf. Riemnn-Lebesgue L, Riemnn-Lebesgue 4.. ε > R f, g, f(x g(x dx ε R lim p R e ipx g(x dx =, e ipx f(x dx e ipx g(x dx + e ipx (f(x g(x dx R R R e ipx g(x dx + f(x g(x dx e ipx g(x dx + ε R R 38 R f(x dx < R. R. I = [, b] I I I. α i I i n i= α i Ii.,. f = n i= α i Ii, I i = [ i, b i ], i < b i R f(x dx = n i= α i(b i i. f n (x R f m(x f n (x dx (m, n, x R f n (x. (. f(x = lim n f n (x f(x ( x f. R f m(x dx R f n(x dx R f m(x f n (x dx (m, n R f n(x dx n. f(x dx. f(x. R R f m(x dx R f n(x dx R f m(x f n (x dx (m, n, R f n(x dx, f(x dx <. 39 R Lebesgue. f n f, φ n f n φ, f n f, R f n(x dx n f(x dx. R. 4 R f n R f L n R f n(x f(x dx. R

25 5.4. Fourier Riemnn 25 lim sup p R e ipx f(x dx ε., R f f n f L, Riemnn-Lebesgue Riemnn-Lebesgue. I = [, b], I. I., Riemnn-Lebesgue I Riemnn-Lebesgue. : b I (p = e ipx dx = e ipb e ip ip, I (p ( p. Riemnn-Lebesgue. 5.4 Fourier Riemnn N >. R f Fourier f(p = s N (f(x = N e ipx f(p dp 2π N e ipy f(y dx, Fourier N. N : ( N s N (f(x = e ip(x y dp f(y dy 2π = = = = π N e in(x y e e in(x y f(y dy 2πi(x y sin(n(x y f(y dy. π(x y sin(n y (f(x + y + f(x y dy πy f(x + y + f(x y sin(ny dy. y 4 y x + y, sin(ny/y. δ >. y δ (f(x + y + f(x y/y. Riemnn-Lebesgue, lim N δ f(x + y + f(x y sin(ny dy =. y N s N (f(x N, π δ f(x + y + f(x y sin(ny dy y N,. Riemnn.

26 26 5. : Fourier 5.5 Dirichlet Riemnn-Lebesgue Riemnn f(x = e x2 /2 Dirichlet ( lim R R sin x x dx = π 2. f(x = e x2 /2., 6, f(p = e p 2 /2 2π lim s N(f(x = e ipx f(p dp = f(x. N 2π, Riemnn x =, δ > lim s N(f( = lim N N π lim N ( δ sin(n y y dy + δ /2 sin(ny 2e y2 y δ sin(ny e y2 /2 y dy = e 2 /2 =. dy = π 2. Riemnn-Lebesgue N. δ lim N sin(n y y dy = π 2. y = x/n, π 2 = lim N Nδ sin x x dx = lim R Dirichlet Riemnn Riemnn-Lebesgue e x2 /2 Fourier 4. Dirichlet x > x, ± R sin(±x lim dx = ± π ( >,. R x 2 : R sin(x π/2 ( >, lim dx = ( =, R x π/2 ( <. Dirichlet. R sin x x 5. (Dirichlet. Fourier Dirichlet, e x sin x x dx = dx. dt + t = π rctn ( ( 2 2

27 5.5. Dirichlet 27. e x /x : t >, 2 I := e tx sin x dx = = [ e tx cos x] x= t e tx ( cos x dx = t 2 e tx sin x dx = t 2 I e tx cos x dx = t e tx sin x dx = I = + t 2., x > : e x x = e tx dt. e tx (sin x dx (, I, e x sin x ( dx = e tx dt sin x dx x ( = e tx dt sin x dx dt = + t. 2 (. (t = tn θ ( Cuchy α < π/2, < ε < R, K : K = { re iθ ε r R, θ α }. C, εe iα Re iα C, Re iα Re iα C 2, Re iα εe iα C 3, εe iα εe iα C 4., ε, R, z dz e C z exp( te iα eiα dt te = exp( te iα dt iα t, z dz α z = exp( Re it d(reit α = exp( Re it i dt, Re it C 2 e z dz e C 3 z dz e C 4 z z α α α exp( te iα d(te iα d(εe it εe it = α α 2 t α. te iα i dt = 2iα. = exp( Re it R cos α e α exp( te iα dt t, exp( te iα exp( te iα = e t cos α it sin α e t cos α+it sin α = 2ie t cos α sin(t sin α

28 28 5. : Fourier, Cuchy, z dz = e z = 2i C t sin α = x e t cos α sin(t sin α dt t + 2iα. e t cos α sin(t sin α dt t = α. e x cot α sin x dx x = α. α = cot α = tn(π/2 α e x sin x dx x = π rctn. 2 (. Dirichlet Riemnn R f x R, δ > (f(x + y + f(x y/2 f(x y < y < δ 42, Fourier N x f(x : lim N s N(f(x = f(x.. Riemnn, δ >, N s N (f(x = π δ Dirichlet, N 2 f(x = lim N π δ s N (f(x f(x = 2 π sin(n y y δ f(x + y + f(x y sin(nx dy + o(. y dy f(x = 2 π δ sin(ny f(x y dy + o(. (f(x + y + f(x y/2 f(x sin(ny dy + o(. y [(f(x+y+f(x y/2 f(x]/y < y < δ Riemnn-Lebesgue, N.. 42 Dini.

29 5.6. Riemnn f x, δ >, [(f(x + y + f(x y/2 f(x]/y < y < δ. f x, lim N s N (f(x = f(x f x f(x = lim ε f(x ε, f(x + = lim ε f(x + ε, f(x = (f(x + + f(x /2. x f f(x ε f(x (x = lim, f f(x + ε f(x + (x + = lim ε ε ε ε., δ >, (f(x + y + f(x y/2 f(x = [ f(x + y f(x + y 2 y < y < δ.. lim s N(f(x = lim N N 2π N N ] f(x y f(x y e ipx f(p dp = f(x = f(x + + f(x > f (x : /(2 ( < x <, f (x = /(4 (x = ±, (x < < x. f (p = e ipx dx = e ip e ip 2 2ip = sin(p. p. Fourier N : s N (f (x = 2π = 2π N N N ixp sin(p e p dp = 2 2π N sin(( + xp + sin(( xp p cos(xp sin(p p N 2 Dirichlet 2π. Dirichlet x > π/2, x =, x < π/2, 2 Dirichlet x < π/2, x =, x > π/2. < x < π, x = ± π/2, x < < x.. lim s N(f = f (x N dp dp

30 3 5. : Fourier 5.7 Fourier, f R 2π, x 2π. f Fourier n (n Z n = 2π e iny f(y dy 2π. N, f Fourier N : s N (f(x = n= N N n= N n e inx. N : s N (f(x = ( 2π N e in(x y f(y dy 2π = 2π = 2π = 2π = 2π = 2π = π 2π 2π 2π 2π π π e i(n+(x y e in(x y f(y dy e i(x y e i(n+/2(x y e i(n+/2(x y f(y dy e i(x y/2 e i(x y/2 sin((n + /2(x y f(y dy sin((x y/2 sin((n + /2y f(x + y dy sin(y/2 sin((n + /2y (f(x + y + f(x y dy sin(y/2 y/2 sin((n + /2y sin(y/2 f(x + y + f(x y y 5 y x + y, sin(αx/ sin(βx, 6. lim t (t/ sin t =, 5.4, N. Dirichlet : 2π 2π sin((n + /2y sin(y/2, π s N (( = : s N (( = π N n= N sin((n + /2y sin(y/2 2π e iny dy = 2π dy = s N (( =. dy =. N δ n =. n= N dy.

31 3 e iy = e iny 2π., f(x = π π sin((n + /2y sin(y/2 s N (f(x, s N (f(x f(x = 2 π π dy f(x = π π y/2 sin((n + /2y sin(y/2 y/2 sin((n + /2y sin(y/2 2f(x y (f(x + y + f(x y/2 f(x y sin((n + /2y δ y < π Riemnn-Lebesgue, δ >, lim N π, N, δ s N (f(x f(x = 2 π δ < y < δ y/2 sin((n + /2y sin(y/2 y/2 sin((n + /2y sin(y/2 (f(x + y + f(x y/2 f(x y (f(x + y + f(x y/2 f(x y (f(x + y + f(x y/2 f(x y dy =. dy + o(. N s N (f(x f(x, lim N s N (f(x = f(x dy. dy. 6 : Guss Fourier t > :. e ipx e x2 /(2t 2πt dx = e tp2 /2. ( 6. u = u(t, x : u(t, x = e x2 /(2t 2πt. u = u(t, x : u t = 2 u xx, lim f(xu(t, x dx = f(. t

32 32 6. : Guss Fourier f(x. u = u(t, x. 5.2., U(t, p = e ipx u(t, x dx, t U(t, p = e ipx 2 u(t, x dx = 2 x e ipx x 2 lim U(t, p = lim e ipx u(t, x dx = e ip =. t t U(t, p = e tp2 /2. (. u(t, x dx = p2 U(t, p , U(t, p = ( ixe ipx u(t, x dx = it e ipx u(t, x dx p x ( = it x e ipx u(t, x dx = it ( ipe ipx u(t, x dx = tpu(t, p. 2 u x = (x/tu, 3. U(t, = u(t, x dx =. U(t, p = e tp2 /2. u(t, x. 6.3 Tylor t = ( e ipx e x2 /2 2π dx = e p2 /2 (, x, p x/ t, t p t > (. ( (. sin(px e x2 /2 sin(px dx =. e x2 /2 cos(px dx = e p2 /2 2π

33 6.4. Cuchy 33. cos(px Tylor-Mclulin.. /2 e x2 x 2n dx. ( e x2 /2 x 2n dx = e /2 x2 x 2n dx = n =,, 2,... e x2 /2 (x 2n dx = (2n e x2 /2 x 2n dx = (2n 5 3 2π = (2n! 2π. 2 n n! e x2 /2 x 2n 2 dx. 2 2n 4 2 = 2 n n!., e x2 /2 cos(px dx = e x2 /2 ( n (px2n (2n! dx n= ( p 2 n = e x2 /2 x 2n ( p 2 /2 n dx = 2π = e p 2 /2 2π. (2n! n! n= (. n= 6.4 Cuchy,. Cuchy p. e ipx e x2 /2 dx = e (x+ip2 /2 dx = e (x+ip2 /2 p 2 /2 dx = e p2 /2 e x2 /2 dx = 2π (. e (x+ip2 /2 dx = e p2 /2 2π. 7 : Guss : I := e x2 dx = π. (. I 2 = e (x2 +y2 dx dy = π R 2.

34 34 7. : Guss 7. 2 I 2 = e (x2 +y 2 dx dy z = e (x2 +y 2 z = R 2. z 43 π( log z < z. I 2 =. π( log z dz = π[z log z z] = π. 7.2 x = r cos θ, y = r sin θ, I 2 = e (x2 +y2 dx dy = R 2 2π dθ e r2 r dr = 2π [ ] e r2 = π. 2 2 Jcobin r. dx dy = (cos θ dr r sin θ dθ (sin θ dr + r cos θ dθ = r dr dθ, K = { (r, θ r >, θ < 2π }, I 2 = e (x2 +y2 dx dy = R 2 K e r2 r dr dθ = 2π dθ e r2 r dr = π. 7.3 Jcobin Guss. y = x tn θ y θ,, I 2 = 4 = 4 π/2 ( ( dy = dθ cos 2 θ, x2 + y 2 = x 2 ( + tn 2 θ = x2 cos 2 θ e (x2 +y 2 dy dx = 4 π/2 = 4 2 dθ = 4 π 2 2 = π. ( π/2 exp ( x2 x cos 2 θ cos 2 θ dx dθ = z = e (x2 +y 2, r 2 = x 2 + y 2, πr 2 = π( log z. exp ( x2 π/2 cos 2 θ ( exp x cos 2 θ dθ dx x2 cos 2 θ 2 x= x= dθ

35 y = xt y t : ( ( I 2 = 4 e (x2 +y 2 dy dx = 4 e (+t2 x 2 x dt dx ( [ ] x= = 4 e (+t2 x e 2 (+t2 x 2 x dx dt = 4 dt 2( + t 2 = 2 dt + t 2 = 2[rctn t] = 2 π 2 = π. 3, 6 rctn t /( + t x= 7.4 Jcobin Guss.,, Guss.. ( 9., Guss. s, p, q > ( s, p, q, Γ(s = e x x s dx B(p, q = x p ( x q dx Γ(s B(p, q 45. Γ(s + = sγ(s, Γ( =, n Γ(n + = n!. Guss I Γ(/2 : I = 2 e x2 dx = 2 t t /2 e dt = 2 e t t /2 dt = Γ(/2. 2 x = t. Γ(/2 2 = π Guss. : B(p, q = 2 π/2 cos 2p θ sin 2q θ dθ = t p dt ( + t = du p+q p ( + u /p. p+q 44 t = tn θ dt/dθ = + tn 2 θ = + t 2, θ = rctn t dθ/dt = /( + t 2., rctn t = t dt/( + t

36 36 7. : Guss x = cos 2 θ = t/( + t, t = u /p. 3 ( p = /2 t, 2 F. Γ(/2 Guss, χ 2. ( 9. B(/2, /2 = π., Γ(pΓ(q = Γ(p + qb(p, q, Γ(/2 2 = B(/2, /2 = π. Guss... A, x, y A, x, y A (x, y, ( Γ(pΓ(q = e (x+y x p y q dy dx ( = e z x p (z x q dz dx x ( = x<z (x, ze z x p (z x q dz dx ( = x<z (x, ze z x p (z x q dx dz ( z = e z x p (z x q dx dz ( = e z (zt p (z zt q z dt dz = e z z p+q dz t p ( t q dt = Γ(p + qb(p, q. 2 y = z x, 4, 6 x = zt. 7.5 Hirokzu Iwsw, Gussin Integrl Puzzles, The Mthemticl Intelligencer, Vol. 3, No. 3, 29, pp Steven R. Dunbr, Evlution of the Gussin Density Integrl, October 22, : ( exp (x dx = 2π e. 2 x 2 (

37 37 x >, y = x /x x > < y <. y R x /x = y x > x = 2 (y + y dx = 2 ( + y dy. y ( exp ( x 2 2 x dx = 2 = 2 = ( exp ( 2 ( exp 2 y2 2 e y2 /2 dy = 2π. x 2 dx x ( 3 y/ y y. ( x 2 = (x x 2 x 2 ( exp ( x 2 ( dx = e exp 2 x 2 (. + y y2 + 4 (x dx. x 2 dy 8 : Γ(/2 2 = B(/2, /2 = π : Γ(sΓ( s = B(s, s = π sin(πs.. sin z Γ(s sin z = z ( z2 sin(πs, i.e. = s ( s2, π 2 n 2 π n 2 n= n= Γ(s = lim s(s + (s + n [( = e γs s + s ] e s/n n n!n s n n=

38 38 8. : 46. γ Euler ( γ = lim n n log n., Γ(sΓ( s = Γ(s( sγ( s = s( s s n= [( + s ( s ] n n 2 : Γ(sΓ( s = B(s, s = t s + t dt. = sin(πs. π < s <, < ε < < R C : ε R. R. R ε. ε. C zs dz/( + z z s dz/( + z z = 2πi : z s dz + z = 2πieπis. C ε, R C zs dz/( + z t s dt/( + z e 2πis 47 : C z s dz + z = ( e2πis 2 B(s, s = t s dt + t. t s dt + t = 2πieπis e 2πis = 2πi e πis e πis = π sin(πs. t = u /s s du/( + u /s., : B( + s, s = sb(s, s = du πs = + u/s sin(πs.. R >, R, 2πs Re 2πis, C, C dz/( + z/s dz/( + z /s z = e πis se πis 2πi, R C dz/( + z/s du/( + u /s e 2πis 48. du 2πiseπis = = + u/s e 2πis 2πis e πis e πis = πs sin(πs. 46 Γ(sΓ( s = π/ sin(πs (, sin z. 47 z s e 2πis. 48 z /s z e 2πis, dz e 2πis.

39 πi., ( 5 (2 267., f(s (s > 3 : : f(s > (s >, : f(s + = sf(s (s >, : log f(s s >. 3 : n!n s f(s = f( lim n s(s + (s + n (s >. ( Γ(s 3 Γ( =, Guss n!n s Γ(s = lim n s(s + (s + n, 3 Γ(s.., (. s(s + (s + n ( n!n s = s + s ( + s ( 2 = s + s e s ( + s 2 s(s + (s + n lim n n!n s ( s log n e + s n ( e s 2 + s n e s n e s( n log n log n 2 n n Euler γ 49. n k= (+s/ke s/k n. z ( + ze z z = 2, ( + ze z = + O(z 2 (z. ( + s/ke s/k = + O(s 2 /k 2 (k. k= ( + s/ke s/k. : s(s + (s + n lim n n!n s = e γs s n= [( + s ] e s/n n 49 /x, + /2 + + /n log n n+ dx/x log n = log(n + log n /(n + n+ dx/x = log(n + log n, + /2 + + /n log n n..

40 4 8. : 5. /Γ(s Γ(s = eγs s n= [( + s ] e s/n n Weierstrss. s C. F (s : n!n s F (s = lim n s(s + (s + n. F (s + = lim n ns n!n s s + + n s(s + (s + n = sf (s, F ( = n! n (n +! =. ( f(s = f(f (s (s >, < s < f(s = f(f (s., f(s, 2 n < s <, f(n+s f(n, f(n, f(n + ( s ( s f(n f(n + f(n f(n + s f(n ( < s < (# f(n f(n. g(s < b < c g(b g( b g(c g( c g(c g(b c b 5. g(s = log f(s, (, b, c = (n, n + s, n +, log f(n + s log f(n log f(n log f(n +. s (, b, c = (n, n, n + s, log f(n log f(n log f(n + s log f(n. s 2 f(n + s (#. f(n + s (# f. f f(n + f(n = n, f(s + n = (s + n (s + sf(s, f(n = (n!f(. (# n n +, n s n!f( (n + s(n + s sf(s, f(n!n s s(s + (s + n f(s. 5 /Γ(s. /Γ(s, Γ(s s =,, 2,

41 (#, f(s f((n!n s s(s + (s + n = n + s f(n!n s n s(s + (s + n. f(n!n s s(s + (s + n f(s n + s f(n!n s n s(s + (s + n., (. 3 (,,. Γ(s = e x x s dx,. g(s = log Γ(s, g (s. f(s g(s = log f(s g (s : f(s = b e sϕ(x+ψ(x dx. ϕ(x, ψ(x, s. (, b = (,, ϕ(x = log x, ψ(x = x log x f(s = Γ(s 52., g(s = log f(s g = d f ds f = ff f 2. f 2 f 2 ff. f(s, f(sλ 2 + 2f (sλ + f (s = = b b e sϕ(x+ψ(x (λ 2 + 2ϕ(xλ + ϕ(x 2 dx e sϕ(x+ψ(x (λ + ϕ(x 2 dx. f 2 ff. Γ(s. Guss.. Guss. Guss. n s B(s, n +, n s B(s, n + = ns Γ(sΓ(n + Γ(s + n + = n s n! s(s + (s + n n s B(s, n + = n s x s ( x n dx = n t s ( t n n dt 52 (, b = (,, ψ(x = log x ϕ(x = t log( x f(s = B(s, t. B(s, t s. F (s = Γ(s + tb(s, t s. F (s + = sf (s, F ( = Γ(t F (s = Γ(sΓ(t..

42 42 8. : 2 x = t/n., n, n s n! n ( s(s + (s + n = t s n t n dt t s e t dt = Γ(s.. (# f(s = Γ(s, < s < Γ(s + n + n s Γ(n + (n., s >. n s n! s(s + (s + n = ns Γ(sΓ(n + Γ(s + n + Γ(s (n.,,,, Guss (. : : lim n ns B(s, n + = Γ(s. Γ(s = lim n s n s n! B(s, n + = lim n n s(s + (s + n = e γs s γ Euler. n= [( + s ] e s/n. n sin z.,,, Γ(sΓ( s = B(s, s = Γ(sΓ( s = Γ(s( sγ( s = s sin(πs = πs ( s2 n= n 2 π sin(πs. n=, sin z = z ( s2 n 2. ( z2 n= π 2 n 2.

43 , sin(πs = π/(γ(s( sγ( s 53., sin z cot z cot z = z + n= ( z nπ + z + nπ, Fourier,, Fourier 54. x cos(tx π x π Fourier, cot(πt 55. e itx Fourier, e itx Fourier e itx = lim n = π e inx e itx dx = 2π π 2π = ( n (e iπt e iπt 2πi(t n N n= N = sin(πt π = sin(πt π [ e inx e itx i(t n = ( n sin(πt π ] x=π x= π t n N n e inx = sin(πt N ( n e inx lim π N t n n= N [ t + ( ] e ( n inx t n + e inx t + n n= [ t + ( 2t cos(nx ( n + i t 2 n 2. cos(tx Fourier [ cos(tx = sin(πt ] π t + n 2t cos(nx ( t 2 n 2 n= n= ] 2n sin(nx t 2 n 2., π cos(tx sin(πt = t + n 2t cos(nx ( t 2 n 2 n= , Fourier x cos(tx π x π 2π R f t (x Fourier. cos(tx x < 2π 2π.

44 44 8. : x, π, π sin(πt = t + ( n 2t t 2 n = 2 t + ( ( n t n + t + n n= n= π cot(πt = t + 2t t 2 n = 2 t + ( t n + t + n n= n= 56. sin(πt π cot(πt, d sin(πt log dt πt = n= t = t = s, log sin(πs πs = n= ( t n + = t + n n= ( ( log s ( + log + s n n ( /n t/n + /n. + t/n = log ( s2 n= n 2, 57 sin(πs = πs ( s2 n= n 2. sin, /(Γ(sΓ( s sin(πs Γ(sΓ( s = π sin(πs. Γ(pΓ(q = Γ(p + qb(p, q, : π sin(πs = B(s, s = x s ( x s dx = t s dt + t = du s + u. /s n e inx = (e ix n e ix = cos x + i sin x, : sin(nx = ( n ( k (cos x n 2k (sin x 2k+. 2k + k<n/2 56 coth z = i cot( iz, coth(πt = iπ cot( πit = t + 57 sinh z = i sin( iz, sinh(πs = πs ( + s2 n= n 2. n= 2t t 2 + n 2.

45 (cos x 2 = (sin x 2, n sin(nx sin x n., m, n = 2m +, sin(nx = sin((2m + x. sin((2m + x sin x 2m +, 2π/(2m +, x k = kπ, k =, ±,..., ±m 2m +, π/2 < x m < x m+ < < x m < x m < π/2, sin x k. sin x 2m + sin((2m + x (2m + sin x k. C, sin((2m + x = C sin x m k= [( sin x sin x, x, 2m + = C m k= [( sin ( kπ sin x + sin 2m + ( kπ sin 2m + ] kπ. 2m + ] kπ. 2m +, sin((2m + x 2m + = sin x [( m sin x sin k= kπ 2m+ ( + sin x sin kπ 2m+ ]. x = πs/(2m +, (2m + /π [( ( sin(πs m πs πs sin sin 2m+ = + π sin sin sin πs 2m+ π 2m+ k= πk 2m+ 2m+ πk 2m+ m : sin(πs π : t = s k= [( s ( + s ]. k k ]. sin(t t, sin(t sin(bt b..

46 46 8. : 8.4 Wllis Wllis : 2 2n (n! 2 lim n (2n! n = π, i.e. ( 2n 22n. n πn Wllis. Wllis Guss s = /2 : π = Γ(/2 = lim n n /2 n! (/2(/2 + (/2 + n = lim n 2 n+ n /2 n! 3 (2n + = lim n = lim n 2 2n+ n /2 (n! 2 (2n +! 2 n+ n /2 n! 2 n n! 3 (2n (2n 2 2n (n! 2 2n /2 = lim n (2n! 2n + = lim 2 2n (n! 2 n (2n! n. Wllis : 2n 2n (2n (2n + = π 2. n= Wllis, Wllis. Wllis sin s = /2 :, sin(πs = ( π = sin = π 2 2 π Γ(sΓ( s = πs n= ( = π (2n 2 2 n= ( s2 n= n 2. (2n (2n +. 2n 2n Wllis,, sin. 7.4 B(p, q = 2., 2 π/2 π/2 (cos θ 2p (sin θ 2q dθ (sin θ 2q dθ = B(/2, q. B(p, q = Γ(pΓ(q/Γ(p + q Γ( =, Γ(/2 = π, Γ(s + = sγ(s, n, 2 2 π/2 π/2 (sin θ 2n dθ = B(/2, n + /2 = π (sin θ 2n+ dθ = B(/2, n + = 2 3 (2n, 2 4 (2n 2 4 (2n 3 5 (2n +.

47 8.5. B(s, /2 47 Wllis, n, : lim n ns B(s, n + = Γ(s. n,., (n + s lim n (n + b = s B(s, n + lim n B(s, n + b =. sin 2 lim n π 2 4 (2n 2 4 (2n 3 (2n 3 5 (2n + =., Wllis : k= (2k(2k (2k (2k + = π B(s, /2 mthtodon ( /2 n : ( /2 ( /2( 3/2 ( (2n /2 = = ( n 3 (2n n n! 2 n n! ( = ( n (2n (2n = ( n (2n! 2n ( n 2 n n! 2 4 (2n 2 n n! 2 n n! = n 2. 2n ( x /2 = ( /2 ( x n = n n= n= n= ( 2n x n n 2. 2n, B(s, /2 : ( 2n ( 2n 2 B(s, /2 = x s ( x /2 dx = 2 2n x n+s 2n dx = n n n + s. s = /2, n= ( 2n 2 2n n n + /2 = B(/2, /2 = π. n=

48 48 8. : 2, n= ( 2n 2 2n n 2n + = 2 B(/2, /2 = π 2. : ( 2n 2 2n = B(, /2 = 2, n n + n= ( 2n 2 2n n 2n + 3 = 2 B(3/2, /2 = π 4. n= 8.6 Fresnel Dirichlet mthtodon Fresnel π cos t 2 dt = sin t 2 dt = 2. lim R R R, t = x cos x sin x dx = dx = x x. π 2, Γ(/2 = π e πi/4 = ( + i/ 2,. e ix x /2 dx = e πi/4 Γ(/2 e ix x s dx = e πis/2 Γ(s (., Γ(z = z s e z dz z = ix = e πi/2 x 9. Cuchy (. ( x s sin x dx = Γ(s sin πs 2. s, Dirichlet ( 5.5 sin x x dx = π 2

49 8.7. Stirling-Binet ( 49. πs lim Γ(s sin s 2 = lim s Γ(s + sin(πs/2 s., Fresnel Dirichlet s = /2 π s =. = π Stirling-Binet ( E. T. Whittker nd G. N. Wtson, A course of modern nlysis (927.. (digmm, ψ(s : ψ(s = d ds log Γ(s = Γ (s Γ(s. ψ (s (trigmm. (Weierstrss, log Γ(s = γs log s n= [ ( log + s s ]. n n γ Euler. : ψ(s = d ds log Γ(s = γ s ψ (s = s 2 + n= n= [ n + s ]. n (n + s = 2 (n + s. 2,,., log Γ(s Stirling-Binet.,, log Γ(z., Euler ( e t e t γ = dt e t t n=

50 5 8. :. Euler /k = xk dx log n = n du/u, [ n ] [ n γ = ] n lim n k log n 2 = lim x k du dx n k= k= u [ 3 x n n ] [ = lim n x dx du 4 ( y n n ] du = lim dy u n y u [ n 5 ( u/n n n ] du = lim du n u u [ 6 ( u/n n n ] ( u/n n = lim du du n u u 7 = 8 = lim δ δ e u du u [ du u [ 9 = lim δ δ [ = lim δ 3 = lim δ δ 4 = du u du u ( e t δ δ δ e u u du ] u du ] e u u du ] e u u du dt t dt. e u e t e t ( e t e t e t t e u = lim δ 2 = lim δ u du [ du u [ δ δ du u e t dt e t e u δ e t δ ] u du ] dt t Euler, 2, 3 + x + + x n = ( x n /( x. 4 y = x, 5 y = u/n, 6 n. 6 n δ δ. 9. < = e δ < δ. δ δ du/u = log(δ/( e δ. 2, u = e t, 2 u = t., Euler ψ(s = d ( e t ds log Γ(s = e st dt. t e t (Guss., Euler c s = Γ(s e ct t s dt (Re c > ($

51 8.7. Stirling-Binet ( 5 s = c = s, s + n, n, ψ(s = d log Γ(s ds ( e t e t = dt e t t ( e t = e t t e t ( e t = e st t e t ( e t = e st dt t e t 3 dt + lim n dt lim n e st dt + e t + e 2t + + e (n t = e t e nt e t, n= (e nt e (n+st dt e t e nt e st + e (s+nt dt e t e st e t e nt dt e st + e (s+t + e (s+2t + + e (s+n t = e st e (s+nt e t.. log Γ(z. e t e zt dt = log z ( t. f(z, f( =, f (z = e zt dt = /z f(z = log z. s = z +, 2 e t : ψ(z + = d ( e t dz log Γ(z + = e zt dt. t e t, ( e zt ψ(z + = log z + e zt dt = log z t e t ( t + e zt dt. e t f(t = /(e t f( t = f(t /2 + f(t. /2 /t + f(t t =, 2 t + e t = t 2 t t O(t7 (. /t < t <. ψ(z +, 2 e zt dt = 2z

52 52 8. : ψ(z + = d log Γ(z + dz = log z + 2z ( 2 t + e zt dt. e t log Γ(2 = log =, z, log Γ(z + = z log z z + + ( 2 log z + 2 t + e zt e t dt. e t t log Γ(z + = log z + log Γ(z, log Γ(z = z log z z + 2 log z + I(z = ( 2 t + e zt e t dt. e t t ( 2 t + zt dt e e t t. Γ(/2 = π, log π = ( 2 + I I(. 2, I( t t/2, ( I( = 2 2 t + t/2 dt e e t/2 t I ( I( = 2 ( t et/2 t/2 dt e e t t = ( e t/2 t dt e t t, I(, ( ( e t/2 I = 2 t e t + e t 2 e t t ( e t/2 e t = e t dt t 2 t = dt e t t ( e t/2 e t + e t t 2 e t dt. 2t d e t/2 e t dt t e t/2 /2 e t t = e t/2 e t t 2 e t 2t = 2 + e t/2 /2 e t, t e t/2 e t I(/2 ( : I ( [ e t/2 e t = 2 t ] t= t= + 2 t e t e t/2 dt = t log 2 = 2 log 2.

53 8.7. Stirling-Binet ( 53, I( = ( 2 I + log π. = + log 2π. 2, log Γ(z = z log z z + 2 log z + I(z I( = z log z z + log 2π z + I(z. log Γ(z + = log z + log Γ(z, I(z log Γ(z + = z log z z + log 2πz + I(z. I(z = ( 2 t + zt dt e e t t. /t < t <. M Re z > I(z : I(z M e (Re zt dt = M Re z. z >, log Γ(z + = z log z z + log 2πz + O ( z (z. Stirling Γ(z + = z z e z 2πz ( + O(/z (z. (# ($, ( I(z = e zt 2 t t O(t6 dt = Γ( 2z Γ(3 72z + Γ(5 ( 3 324z + O 5 z 7 = 2z 36z + ( 3 26z + O (z. 5 z 7 Γ(z + = z z e z 2πz exp ( 2z 36z + ( 3 26z + O 5 z 7 (z. Γ(z + = z z e z 2πz /(2n. ( + ( 2z + O z 2 (z

54 54 9. : 8.8 Stirling-Binet (2 : log Γ(z + = z log z z + log 2πz + I(z, ( I(z = 2 t + zt dt e e t t, 2 t + e t = t 2 t t O(t7. E. T. Whittker nd G. N. Wtson, A course of modern nlysis ( Binet s first expression for log Γ(z in terms of n infinite integrl Binet s second expression., rctn(t/z I(z = 2 dt e 2πt. 9 : 9. µ, σ : f µ,σ (x dx = e (x µ2 /(2σ 2 2πσ 2,. dx. X, Y µ X, µ Y, σx 2, σ2 Y, X + Y µ X + µ Y, σx 2 + σ2 Y shpe α >, scle τ > : f α,τ (x dx = e x/τ x α Γ(ατ dx = e x/τ (x/τ α dx (x >. α Γ(α x x = ατ, ατ 2, α x = (α τ.

55 : φ τ,α (t = τ α Γ(α e itx e x/τ x α dx = ( iτt α. φ τ,α(t = iατ( iτt φ τ,α (t. : τ α Γ(αφ τ,α(t = ie itx e x/τ x α i dx = it τ x (eitx e x/τ x α dx = iτ iτt = iατ iτt x (eitx e x/τ x α dx = e itx e x/τ x α dx = iτ iτt iατ iτt τ α Γ(αφ τ,α (t. e itx e x/τ x xα dx, 4.. X, Y shpe α X, α Y, scle τ, τ, X + Y shpe α X + α Y, scle τ. 2 2 (χ 2., shpe n/2, scle 2 n 2 (χ 2 : f 2,n/2 (x dx = e x/2 x n/2 2 n/2 Γ(n/2 dx = e x/2 (x/2 n/2 dx Γ(n/2 x. 2 n. 9. ( 2. X, X 2,...,. Y = X Xn 2 n 2.. :. E[f(Y ] = const. f(ye y/2 y n/2 dy. E[f(Y ] = E[f(X Xn] 2 = f(x 2 (2π n/2 + + x 2 ne (x2 + +x2 n /2 dx dx n R n = A n (2π n/2 = A n 2(2π n/2 = A n 2(2π n/2 f(r 2 e r2 /2 r n dr f(ye y/2 y (n /2 y /2 dy f(ye y/2 y n/2 dy. 3 r = x x 2 n, Rn n r n dr, n A n. 4 r = y /2, dr = (/2y /2 dy.

56 56 9. : 9.2., n A n, A n = 2πn/2 Γ(n/ Person 2 K = (K,..., K r., p i >, r i= p i =, k,..., k r, K = (k,..., k r, k i r i= k i = n P (K = (k,..., k r =,. n! k! k r! pk p k r r 9.3 (. 6 n i K i, K = (K,..., K 6 r = 6, p i = / k + +k r =m m! k! k r! xk x k r r = (x + + x r m.. m. K i µ i = np i : µ i = E[K i ] = k + +k r =n 3. K i σ 2 i = np i ( p i : E[K i (K i ] = n! k! k r! pk p kr r k i = np i (p + + p r n = np i. k + +k r=n n! k! k r! pk p kr r k i (k i = n(n p 2 i (p + + p r n 2 = n(n p 2 i, σ 2 i = E[K 2 i ] µ 2 i = E[K i (K i ] + µ i µ 2 i 2. = n(n p 2 i + np i n 2 p 2 i = np i ( p i

57 9.3. Person 2 57 i j K i K j σ ij = σ ji = np i p j : σ ij = E[K i K j ] µ i µ j = k + +k r =n = n(n p i p j n 2 p i p j = np i p j. 3. X = (X,..., X r X i = K i np i npi n! k! k r! pk p k r r k i k j µ i µ j, X i, p ii = np i( p i np i = p i = p i pi, i j X i X j p ij = p ji = np ip j n p i pj = p i pj. X = (X,..., X r P = [p ij ] P = E + T, =. E, T. r i= p i =,. v R r, P v = v, v v (r = 3. Euclid,. P, P 2 = P, P r (Person 2. K = (K,..., K r Person 2 : Y = r Xi 2 = i= p. pr r (K i np i Person 2 n r 2 ( 6. i= 59 Person 2 n 2 (. 6 Person 2., n, Person 2 2. np i

58 58 9. :. 6, X = (X,..., X r, P (., X = (X,..., X r, P, Y = r i= X 2 i r X = (X,..., X r, P, P 2 = P P s, Y = r i= X2 i s 2.. P. P 2 = P P, P. U, U T P U = U P U = dig(,...,,,...,. }{{} s P, U (i, j p ij, u ij, Z i = r u ji X j j=., X = (X,..., X r Z = (Z,..., Z r Y =, U, r Xi 2 = i= r i= Z 2 i { r r ( i = l s, E[Z i Z l ] = u ji E[X j X k ]u kl = u ji p jk u kl = (. j,k= j,k= (& E[ ] : ] ] [Z Z r = [X X r U, Z ] X ] E. [Z Z r = U T E. [X X r U Z r X r = U T P U = dig(,...,,,...,. }{{} s 6. n.

59 9.4. t 59 (&, Z,..., Z s, Z s+,..., Z r ( Z s+ = = Z r =. 9. r Zi 2 = Z Zs 2 (lmost sure i= s (. r A, R r X = (X,..., X r, A, : E [ e i t,x ] = exp ( 2 t, At (t R r. (, R r Euclid. A. A = X (,...,. σ >,..., σ s >, A = dig(σ, 2..., σs, 2,...,, X,..., X r, i =,..., s X i, σi 2, i = s +,..., r X i.. A. A, R r f(x, ( E[f(X] = f(x exp det(2πa R 2 x, A x dx r. dx R r Lebesgue. ( A. 9.4 t α, β > (Bet distribution of the second kind Bet prime distribution : f α,β (x dx = x α dx (x >. B(α, β ( + x α+β β > α/(β, β > 2 (α(α+β /((β 2(β 2. 2 x = t 2 /γ (γ >, < t <, : ( t 2 t f α,β γ γ dt = t 2α dt γ α B(α, β ( + t 2 /γ α+β

60 6 9. : n >, α = /2, β = n/2, γ = n, n t., n t : ( (n+/2 g n (t dt = c n + t2 dt. n c n = n /2 B(/2, n/2 = Γ((n + /2 nπ Γ(n/2. n t µ n, σ 2 n µ n = (n >, σ 2 n = n n 2 (n > 2.. t. t Cuchy,. 2 t,. 9.8 ( 2 t. Z, Y, Z, Y n 2. T = Z Y/n n t.. : E[f(T ] = const.. n = /( 2π 2 n/2 Γ(n/2, [ ( ] ( Z ( E[f(T ] = E f = n Y/n ( = n f = ( n n = n n ( ( (n+/2 f(t + t2 dt. n f z y/n e (y+z2 /2 y n/2 dz f(te (+t2 /ny/2 y (n+/2 dt ( f(t e (+t2 /ny/2 y (n+/2 dy = 2(n+/2 Γ((n + /2 n n z y/n e z2 /2 e y/2 y n/2 dz dy dy dt ( (n+/2 f(t + t2 dt. n, z = t y/n (z 2 = yt 2 /n, dz = y /2 dt/ n. 6 : e αy y s dy = e x ( x α s dx α = α s Γ(s (α, s >. dy

61 9.4. t 6,, 2 (n+/2 Γ((n + /2 n n = 2(n+/2 Γ((n + /2 n 2π 2 n/2 Γ(n/2 = Γ((n + /2 nπ Γ(n/2 = c n ( t. X, X 2,..., µ, σ 2, M n = n n k= X k, U 2 n = n n k= (X k M n 2, T n = M n µ U n / n (, M n U n, M n µ, σ 2 /n, (n U 2 n/σ 2 n 2, T n n t. (U n. 9. (T n., E[M n ] = µ, E[Un] 2 = σ 2. µ, σ 2 (popultion men, (popultion vrince, M n, Un 2 (smple men, (unbised vrince., M n µ, σ 2 /n. T n = M n µ U n / n Z n = M n µ σ/ n., Z n σ U n,, n t. σ 2 Z n, Z n. σ 2 Un 2, t. 9. ( t. n ( (n+/2 ( /2 ( ( n/2 + t2 = + t2 t2 n n n + O e t2 /2 n 2, t. t X k X k µ, µ =. µ =., X, X 2,..., σ 2,. ( M n, U n. (U n.

62 62 9. : Y n = n M n = (X + + X n / n. (X,..., X n Y n (Y,..., Y n 62. n Xk 2 = k= n k= Y 2 k 63, n (X k M n 2 = k= = n (Xk 2 2M n X k + Mn 2 = k= n Xk 2 nmn 2 = k= k= n Xk 2 2M n k= n n Yk 2 Yn 2 = Yk 2. k= n X k + nmn 2 k=, E[Yk 2] = σ2, [ n ] [ n ] E (X k M n 2 = E = (n σ 2 k= 64. E[U 2 n] = σ 2. X k., X k, σ 2., Y k,, σ 2. k= Y 2 k U 2 n = n n k= (X k M n 2 = n n k= Y 2 k, M n = n n k= X k = n Y n, 9., n U 2 σ 2 n = n σ 2 n k= Y 2 k M n µ σ/ n (n U 2 n /σ 2 n = M n µ U n / n n t. 9.2 ( t ( 9.9. X s,k (s =,..., r, k =,..., n s, s X s,k µ s, σ 2 s 62 Y n (,,..., / n,. (Y,..., Y n Y n. Y k Y k.

63 Z = n s M s = n s k= r (M s µ s n = s= X s,k, U 2 s = n s r, Y = σs 2 n s= s r n s, T = s= r s= n s k= n s U 2 σs 2 s = Z Y/(n r, T n r t 65. (X s,k M s 2, r n s σ 2 s= s k= (X s,k M s 2,. 9.9, M s Us 2, M s µ s, σs/n 2 s, (n s Us 2 /σs 2 n s 2. Z Y., n s s= (M s µ s, r s= (σ2 s/n s, Z. 2, Y n r 2., 9.8, T n r t. s = 2, µ = µ, µ 2 = µ, σ = σ X s,k (s =, 2, k =,..., n s, µ, σ. M s = n T = n s k= X s,k, Y = M M 2 + Y n n 2 n + n n s (X s,k M s 2, s= k=, T n + n 2 2 t X shpe α >, scle τ >, Y = X. θ = τ, : f α,θ (y dy = e θ/y y α Γ(αθ α 65 σ 2 = = σ 2 r = σ 2 T σ, Z σ dy.

64 64 9. : θ scle. : e θ/y y α dy = Γ(αθ α. Z Y shpe α = n/2, scle θ = n/2, T = Y Z n t., shpe α = n/2, scle θ = n/2 Y, T n t. : ( E[f(T ] = const. f ( y z e z2 /2 e n/(2y y n/2 dz dy = const. = const. = const. ( n/2 dt dy y e (+t2 /n(/(2y y (n+/2 dy dt f(te t2 /(2y e n/(2y y ( f(t ( (n+/2 f(t + t2 dt. n 2 z = t/ y, X, X 2,..., σ n X,..., X n X n U 2 n X n = n n k= X k, U 2 n = n n (X k X n 2 k=. : n (X k X n 2 = k= n Xk 2 nx 2 n. k= Y,..., Y n : Y n = n Y k = n X k = n X n, k= ( k X j kx k+ k(k + j= (k =, 2,..., n. 67 µ X k X k µ.

65 n n A = [ ij ] / 2 2 A = (n / 6 / 2... / n(n + n, Y j Y j = n ij X i i=. A 68. n i= ki li = δ kl n i= Y 2 i = i,k,l, E[X i X j ] = σ 2 δ ij, ki li X k X l = k,l δ kl X k X l = n Xk. 2 k= E[Y k Y l ] = j,j ki lj E[X i X j ] = σ 2 i,j ki li δ ij = σ 2 n ki li = σ 2 δ kl. i=, n Xk n nx 2 n = k= n n Y k Yn 2 = Y k. k= k= ( n Un 2 = Xk 2 nx 2 = n n k= n k= Y 2 k. n n. E[Un] 2 = n E[Yk 2 ] = n n (n σ2 = σ 2. k= X k, σ 2, Y k, σ 2. X n = Y n / n, σ 2 /n, (n Un/σ 2 2 = n k= Y k 2/σ2 n,. 68t AA. A.

66 66 9. : 9.7 F α, β > (Bet distribution of the first kind : f α,β (x dx = B(α, β xα ( x β dx ( < x <. x = α/(α + β, (αβ/((α + β 2 (α + β +, α, β > x = (α /(α + β 2. x x/( + x (x > ( x f α,β + x dx ( + x 2 = B(α, β x α dx (x > ( + x α+β.. x/( + x = /( + x., m, n >, x mx/n (x >,, ( mx/n f α,β + mx/n (m/n dx ( + mx/n 2 = B(α, β., α = m/2, β = n/2, : (mx/n α dx ( + mx/n α+β x (x > g m,n (x dx = B(m/2, n/2 (mx/n m/2 dx ( + mx/n (m+n/2 x (x >. m, n F. m, n F µ m,n, σ 2 m,n, µ m,n = n n 2 (n > 2, σ 2 m,n = 2n2 (m + n 2 m(n 2 2 (n 4 (n > 4. X m, n F, (mx/n/( + mx/n m/2, n/2, mx/n m/2, n/ ( 2 F. Y, Z m, n 2, X = Y/m Z/n m, n F., Y k, Z l,, m, n F. X = ( m k= Y k 2 /m ( n l= Z2 l /n

67 9.7. F E[f(X] = const. (mx/n m/2 dx f(x ( + mx/n (m+n/2 x. = [2 (m+n/2 Γ(m/2Γ(n/2], [ ( ] Y/m ( ( y/m E[f(X] = E f = f e (y+z/2 y m/2 z n/2 dy dz Z/n z/n ( ( m m/2 = f(xe (+mx/nz/2 n xz z n/2 m n z dx dz ( (mx m/2 dx = f(x e (+mx/nz/2 z (m+n/2 dz n x ( m + n ( mx m/2 ( = 2 (m+n/2 Γ f(x + mx (m+n/2 dx 2 n n x 3 y/m = (z/nx (y = (mx/nz, dy = (m/nz dx. 5 : ( s t e αz z s dy = e t dt α α = α s Γ(s (α, s >.. ( m + n 2 (m+n/2 Γ = 2(m+n/2 Γ((m + n/2 2 2 (m+n/2 Γ(m/2Γ(n/2 = B(m/2, n/2.. x = t 2 /n, α = /2, β = n/2 n t. T n t, T 2, n F, T 2 n, F. T F.. F : g m,n (x dx = x m/2 (m/nm/2 B(m/2, n/2 ( + mx/n (m+n/2 dx. m =, g,n (x dx = x /2 n B(/2, n/2 ( + x/n (n+/2 dx. x = t 2, < t < g,n (t 2 t dt = dt n B(/2, n/2 ( + t 2 /n (n+/2. t g n (t dt.

68 68 9. : 9.8. Γ(s = e x x s dx, B(p, q = Γ(pΓ(q = Γ(p + qb(p, q t p ( t q dt = u α du ( + u α+β. B(p, q,. f(x, ye (x+y x p y q dx dy = dt x = zt, y = z( t dz f(zt, z( te z z p+q t p ( t q., f(x, y = f(x/y ( x f e (x+y x p y q dx dy y ( t = dt dz f t = Γ(p + q f e z z p+q t p ( t q ( t t p ( t q dt. t f(t/( t. f(x/y = Γ(pΓ(q = Γ(p+qB(p, q. t/( t = u t = u/( + u ( x f y e (x+y x p y q dx dy = Γ(p + q = Γ(p + q ( u f(u + u p ( q du + u ( + u 2 f(u up du ( + u p+q f(u. F,, F., f(x, ye (x+y x p y q dx dy = du x = uy x u dy f(uy, ye (+uy y p+q u α.

69 ( + uy = z y = z/( + u y z, f(x, ye (x+y x p y q dx dy ( uz = du dz f + u, z e z z p+q u p du + u ( + u. p+q f(x, y = f(x/y, ( x f e (x+y x p y q dx dy = Γ(p + q y f(u up du ( + u p+q.,, F f.,.., 2, t, F. : x : y = t : ( t = u : x, y, t, u 69. x x 2 Γ(s = 2 e x2 x 2s dx, x = r cos θ, y = r sin θ y = x tn θ. 4 x = r cos θ, y = r sin θ, g(x, ye (x2 +y 2 x 2p y 2q dx dy = 4 π/2 dθ g(x, y = g(y/x ( y 4 g e x (x2 +y 2 x 2p y 2q dx dy g(y/x =.. = Γ(p + q 2 B(p, q = 2 dr g(r cos θ, r sin θ e r2 r 2(p+q (cos θ 2p (sin θ 2q π/2 π/2 g(tn θ (cos θ 2p (sin θ 2q dθ. (cos θ 2p (sin θ 2q dθ 69, (26 7

70 7 9. : 4 = 4 y = x tn θ y θ, g(x, ye (x2 +y 2 x 2p y 2q dx dy π/2 dθ dx g(x, x tn θ e (+tn2 θx 2 x 2(p+q (tn θ 2q ( + tn 2 θ. x = r/ + tn 2 θ x r, 4 = 4 π/2 g(x, ye (x2 +y 2 x 2p y 2q dx dy ( r dθ dr g + tn 2 θ, g(x, y = g(y/x 4 r tn θ + tn 2 θ ( y g e x (x2 +y 2 x 2p y 2q dx dy = Γ(p + q 2 π/2 g(tn θ e r2 r 2(p+q (tn θ 2q ( + tn 2 θ p+q (tn θ 2q dθ. ( + tn 2 θ p+q. (tn θ 2q ( + tn 2 θ p+q = (cos θ2p (sin θ 2q. tn θ = sin θ/ cos θ, cos 2 θ = + tn 2 θ, sin2 θ = tn2 θ + tn 2 θ. t = u/( + u. : x : y = cos θ : sin θ = : tn θ., x, y 7, x : y = t : ( t = u : (t = sin 2 θ, t = cos 2 θ, u = tn 2 θ. 9.9 n Mxwell-Boltzmnn ( X i, R n = X Xn, 2 Z (n i = X i /R n. (Z (n,..., Z n (n n 7. Z (n i g n (z dz = c n ( z 2 (n 3/2 dz ( < z <, ( c n = ( z 2 (n 3/2 dz = B 2, n ( n = 2 n 2 B, n f(x, y = f(y/x. 7.

71 9.9. n Mxwell-Boltzmnn ( 7.,. n 2 S n 2 = { (x 2,..., x n x x 2 n = } dω, r = x x 2 n, x 2,..., x n r n 2, r n 2 r n 2, dx dx 2 dx n = r n 2 dx dr dω., r r = x x 2 n, r = r 2 x 2, r / r = r/r, dx dx 2 dx n = r(r 2 x 2 (n 3/2 dx dr dω. x z = x /r, dx dx 2 dx n = r n ( z 2 (n 3/2 dz dr dω., R n ρ(r, g(zρ(r dx dx n = g(z( z 2 (n 3/2 dz R n 2 c n, c n = ( z 2 (n 3/2 dz r n ρ(r dr dω. S n 2 c n 2. z = t /2, dz = t /2 dt/2 : ( c n = 2 ( z 2 (n 3/2 dz = t /2 ( t (n 3/2 dt = B 2, n. 2 2 ( z 2 = ( + z( z, z = 2t, dz = 2 dt : ( n c n = 2 (n 3/2 t (n 3/2 2 (n 3/2 ( t (n 3/2 2 dt = 2 n 2 B, n. 2 2., dupliction formul. (n /2 s c n :, Γ(/2 = π, ( z 2 s dz = B(/2, s = 2 2s B(s, s. Γ(/2Γ(s Γ(s + /2 = 22s Γ(s 2. Γ(2s Γ(2s = 22s π Γ(sΓ(s + /2.

72 72 9. : (Legendre s dupliction formul 72. Z (n i 73 : g 2 (z dz = π dz z 2 ( < z <., /2. z = sin θ, dθ ( π/2 θ π/2 (. π /2 + θ/π = /2 + (rcsin z/π ( z. 74. g 3 (z dz = dz ( z., / g 4 (z dz = 2 π z2 dz ( z., / z = cos θ, sin 2 2 π sin2 θ dθ ( θ π Legendre s dupliction formul n Guss s multipliction theorem : Γ(ns = nns /2 Γ(sΓ(s + /nγ(s + 2/n Γ(s + (n /n. (2π(n /2 Γ(3s = 3 3s /2 Γ(sΓ(s + /3Γ(s + 2/3/(2π ,..,.,..,,. 75 Wigner. N M i e M 2 ii /2 dm ii i<j e M 2 ij /2 dm ij, M., /4, N /4 Wigner..,,, (, Tte. Tte p p + 2 p sin (28. sin 2,, Dedekind sin2-, 6, (25.. sin 2. SU(2 (Hr SU(2 sin 2.. A SU(2 tr(a/2. ( GL r (C Lie (. A SU(2 tr(a/2, SU = S 3 R 4, S 3 R 4. SU(2 sin 2. Tte p SU(2

73 9.9. n Mxwell-Boltzmnn ( 73 n 4 g n (z. /n. Z (n i. c n /c n+2 = (n /n = /n., Z (n i /n : c n z 2 ( z 2 (n 3/2 dz = c n (c n c n+2 = c n = c n+ n. z 2 ( z 2. Y (n i = n Z (n i,, ( y dy g n n = ( (n 3/2 y2 dy n n cn n. n, ν = (n /2, (n 3/2 3/2 ( n/2 ( y2 = ( y2 y2 /2 e y2 /2 n n n/2 n cn = 2ν + 2 2ν B(ν, ν 2ν 2 2ν 2 πν = 2π ν 2 2ν 77. Wllis B(ν, ν = Γ(ν2 Γ(2ν = 2ν Γ(ν + 2 ν 2 Γ(2ν + = 2 ( 2ν 2 πν ν ν ν 2 2ν 78., Y (n i : n lim n ( y g n n = lim n n ( y 2 /n (n 3/2 n 2 n 2 B( n, n = e y2/ π, y g(y, Cn /2 g(y i dω n g(y e y2 dy n S n R 2π (n., n S n = { (y,..., y n R n y y 2 n = n } n n, C n, dω n. Mxwell-Boltzmnn. 3 S 3 = SU(2. 77 n c n = ( y2 /n (n 3/2 dy, lim n ( y 2 /n (n 3/2 = e y2 /2 lim n n cn = 2π.,. 78, Wllis.

74 74 9. : 9. n Mxwell-Boltzmnn (2 n n x i., n n m m... n m S n m = { (x m+,..., x n x 2 m+ + + x 2 n = } dω, r = x 2 m+ + + x 2 n, x m+,..., x n r n m dx dx n = r n m dx dx m dr dω., r r = x x 2 n, r = r 2 x 2 x 2 m r / r = r/r, dx dx n = r(r 2 x 2 (n m 2/2 dx dx m dr dω. x i (i =,..., m z i = x /r (i =,..., m, dx dx n = r n ( z 2 z 2 m (n m 2/2 dz dr dω., ρ(r, g(z,..., z m ρ(r dx dx n R n = c (n m g(z,..., z m ( z 2 zm 2 (n m 2/2 dz dz m. z 2 + +z2 m< ( c (n m = r n ρ(r dr dω S n m. m = c (n =., ρ(r = e r2 /2 /(2π n/2, n ( dω = (2π n/2 S n r n e r2 /2 dr = 2n/2 π n/2 2 n/2 Γ(n/2 = 2πn/2 Γ(n/2 = (dω n S n. : r s e r2 /2 dr = e t (2t (s 2/2 dt = 2 s/2 Γ(s/2. nπn/2 Γ(n/2 + r 2 /2 = t, r dr = dt, r s dr = r s 2 r dr., r n ρ(r dr n., c (n dω S (n m = n = dω S (n m. n m c (n m.

75 9.. n Mxwell-Boltzmnn (2 75 c (n m 79 : c (n m = ( z 2 zm 2 (n m 2/2 dz dz m. z 2+ +z2 m< = t /2 t /2 ( t t m (n m 2/2 dt dt m t i >, m i= t i< = Γ(/2m Γ((n m/2. Γ(n/2 m 2 z i = t i, : p i >, Γ(p Γ(p m+ Γ(p + + p m+ = t p t p m m ( t t m pm+ dt dt m. t i >, m i= t i<. B(p,..., p m+, B(p,..., p m+ = B(p,..., p m, p m + p m+ B(p m, p m+ (B,., t m = ( t t m u t m u B(p,..., p m, p m+ = t i >, m i= t i< dt dt m du t p t p m m ( t t m pm+p m+ u pm ( u p m+. (B. (, (Z (n,..., Z (n m g n (z,..., z m dz dz m = c (n m ( z 2 zm 2 (n m 2/2 dz dz m., σ >, (Y (n,..., Y m (n = n σ (Z (n,..., Z m (n ( nσ 2 (n m 2/2 m yi 2 dy dy m i= 8., lim n ( nσ 2 m i= y 2 i n m 2 2 = exp ( 2σ 2 m i= y 2 i 79 n = m + 2 c (m+2 m = π m/2 /Γ(m/2 + m.. 8 Y (n i

76 76 9. : (Y (n,..., Y m (n n m 8., C n ( g(y n σ,..., y m dω n n σ S n ( g(y (2σ 2 m/2,..., y m exp m y 2 R 2σ m 2 i dy dy m., n σ S n = { (y,..., y n R n y y 2 n = nσ 2 } n σ n, C n ( n σ, dω n. Mxwell-Boltzmnn, σ 2 Boltzmnn kt. i= 9. < p <. n. B p,n n p ( n P (B p,n = k = p k ( p n k (k =,, 2,..., n k. np np( p, E[e itb p,n ] = (pe it + q n. n., p n, (B p,n np/ np( p.. Γ(s + = s!, ( s t = s!/(t!(s t! B(α, β = (α + β! (α!(β! ( α + β 2 = (α + β α, α, β > ( α + β 2 f α,β (p dp = (α + β p α ( p β dp ( < p < α. α/(α + β, (αβ/((α + β 2 (α + β +, α, β > p = (α /(α + β 2. α + β 2 = n, α = k, ( n f k+,n k+ (p dp = (n + p k ( p n k dp ( < p < k, p = (k + /(n + 2, ((k + (n k + /((n (n + 2, p = k/n 82., A B n k, A B α = k +, β = n k Y (n i n, n. 82 k np (n, p, n p,. 83.

77 9.2. Poisson Poisson Poisson. < µ < n Z, N n n, p = µ/n : P (N n = k = = ( µ k ( n µ ( n k n n k ( µ n µ k ( µ ( k n k! n n ( k n, n ( P (N n = k = µ n µ k ( µ ( k ( k µ µk e n k! n n n k!. N k =,, 2,... µ µk P (N = k = e k!, N µ Poisson. N n n Poisson.. µ. N µ/n. N n. µ N, N n Poisson. Poisson. µ Poisson µ : (N = E[N] = E[N(N ] = µ 2 e µ k= k=2 µ µk ke k! = µe µ µ k 2 (k 2! = µ2, E[N 2 ] = E[N(N ] + E[N] = µ 2 + µ, k= (N = E[(N µ 2 ] = E[N 2 ] µ 2 = µ. Poisson : E[e itn ] = e µ k= µ k (k! = µ, itk µk e k! = exp(µ(eit = ( exp(e it µ., Poisson µ.,, µ, X = N µ µ