Stirlingの公式

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1 Stirlig Stirlig Fourier Laplace 3 3. Gauss Stirlig 8 4. Stirlig Stirlig URL Ver... (( 6 3 Ver..22: Ver..23(89 : Ver..24: Ver..25(9 : Ver..26(94 : Pearso Ver..27(94 :. 9 Ver..28(96 : Ver..29(96 : Ver..29a:. 4 Ver..3(97 : Ver..3(98 : Taylor (.. Riema-Lebesgue ( Ver..32(98 :..

2 2 5 : Fourier 2 5. Gauss Riema-Lebesgue Fourier Fourier : Gauss Fourier Cauchy : Gauss Jacobia : Wallis Stirlig-Biet ( Stirlig-Biet ( : Pearso t F Maxwell-Boltzma ( Maxwell-Boltzma ( Poisso

3 3 : Tauber 74. Tauber Laplace Tauber Wallis x x 2 + x 4 x 8 + x 6 x 32 + x? Laplace-Stieltjes Laplace-Stieltjes Tauber : Taylor 9. Taylor Taylor Taylor : Ver.. 3. : Kullback-Leibler ( Saov, Saov, Boltzma (e βe i, Gibbs (, e βe i q i /Z. Stirlig! e 2π (. a b ( lim (a /b =.! = e ( 2π + ( 2 + O ( 2. Stirlig. Stirlig... Gauss,,, Fourier.

4 4..: Stirlig! A = e 2π ( /! A ( + /(2 ( /!.92 (7.78%.9989 (.% (2.73% (.28% (.83% (.32% (.28% ( (.8% ( , e 2π!, = 3 3%, = %. /(2, =.% : ( 2π + = e 2 Stirlig 2. 3 : e x/τ x α (x >, f α,τ (x = Γ(ατ α (x. α, τ > 4. α = >, τ = f (x = f, (x : f (x = e x x Γ( (x > Gergö Nemes, New aymptotic expasio for the Γ(z fuctio, 27. Nemes [( ] 2π! = + = 2 e ( 2π + e s > Γ(s = e x x s dx. Γ( =, Γ(s + = sγ(s, Γ( + =!. 4 α shape parameter, τ scale parameter. ατ ατ 2.

5 5 f (x X. X µ σ 2 5 : Γ( + µ = E[X ] = xf (x dx = =, Γ( E[X] 2 = x 2 Γ( + 2 f (x dx = = ( +, Γ( σ 2 = E[X 2 ] µ 2 =. Y = (X µ /σ = (X /, f ( y + = e ( y+ ( y + 6. y = Γ( f ( = e Γ( = e Γ( +. > Γ( + =!, / 2π Stirlig.,, R φ(x, ( x φ f (x dx = φ(y f ( y + dy φ(y e y2 /2 2π dy. φ(y δ(y ( y, f ( = e Γ( = e Γ( + 2π (. Stirlig. y.,.. 2 Stirlig.,. 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 a/b, E[g(Y ] = R g((x a/bf(x dx = g(ybf(by + a dy, Y bf(by + a. R

6 6 2., ( Fourier Fourier. Fourier, Stirlig Stirlig f (x = e x x /Γ( (x > ( Fourier F (t 8 : F (t = e itx f (x dx = Γ(, α Γ( e ( itx x dx = e αt t dt = α. Cauchy 9. Fourier, ( it. f (x = e x x Γ( = 2π e itx F (t dt = e itx dt (x >. 2π ( it Stirlig., t = u, f ( = e Γ( + = e iu ( iu/ 2π = log e it ( it dt = e iu 2π ( iu/ du. Stirlig, / 2π. : ( log iu iu, = ( iu u2 2 + o ( e iu ( iu/ e u2 /2. iu = u2 2 + o(. 7 obuo/pdf/prob/stir.pdf Cauchy. f(α, f( = f (α = (/αf(α,. α t = x/α. Fourier 5.

7 2.2. 7, f ( = e Γ( + = 2π e iu ( iu/ du e u2 /2 du = 2π 2π. α e u2 /α du = απ 2. Stirlig. 2.2 f (x = e x x X, Y = (X / (. Y f ( y + = e y ( y +., ( ( log e y + y Γ( = log = = e /2 Γ( ( + y y ( y y2 2 + o ( e y ( + y/ + y/ y = y2 2 + o(, e y ( + y/ e y2 /2, + y/., Stirlig : f ( y + = e y ( y + Γ( e y2 /2 2π (. Y Stirlig. Stirlig. Y f( : f ( y + = 2π 2π e it( y+ ( it dt = 2π e iuy e iuy e u2 /2 du = 2π e y2 /2 e it ( iu/ dt (., Lebesgue. 2 Gauss dx = π e x2 x = u/ α. Gauss. 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. Gauss,., z π. I, I 2.

8 8 2., Cauchy 3 e iuy e u2 /2 du = e (u+iy2 /2 y 2 /2 du = e y2 /2 e v2 /2 dv = e y2 /2 2π., Fourier, Stirlig. 2.3 Fourier,! = Γ( + = e x x dx. x = + y = ( + y/ y,! = e ( e y + y dy. c = {! e y e, h ( + y/ (y >, (y = (y., c = h (y dy. log h (y y = Taylor log h (y = y 2 /2 + o( (, lim h (y = e y2 /2. lim h (y dy = e y2 /2 dy 4, lim c = 2π. Stirlig! lim e 2π =. Fourier. Lebesgue h (y { e y ( + y (y, e y2 /2 (y. 3., e ity Taylor., f (y = yf(y, f( = 2π (. Cauchy u + iy (u > v >. 4 y h (y h (y = e y ( + y, y h (y e y2 /2, Lebesgue. Lebesgue, y M h.

9 (ϕ(y. y >, l (y = log h (y, l (y = + y/ y = + y/, l (y = l (y = ( + y/ 2 <, 2/ ( + y/ 3 >, l ( =, l ( =, l ( =. Taylor, y > < θ <, l (y = y2 2 + Ay3, A = 3! l (θy = 3 ( + θy/ 3 >. lim l (y = y 2 /2. lim h (y = e y2 /2. y, Ay 3 l (y e y2 /2, h (y e y2 /2. y, l (y = log(e y (+y l (y (. l ( = l (. l (y = y/( + y, l (y = y/( + y/, + y + y/, l (y l (y (y., y l (y l (y. h (y h (y = e y ( + y X,..., X Y = X X 2 Y 2. 2 shape α = /2 scale τ = 2. 2 e y/2 y /2 (y >, f /2,2 (y = Γ(/22 /2 (y., 2., g(y e y/2 y /2 R dy = g(x x Γ(/22/2 e (x x2 /2 (2π /2 dx dx.,, =. = Γ(/2., x > x = y g(x 2 e x2 /2 2π dx = 2 Γ(/2 = π. g(y e y/2 y /2 dy = 2π 2 g(y e y/2 y /2 dy. Γ(/22/2

10 2.,, 2 2.., Stirlig (. : 2,, Stirlig., Stirlig X, X 2, X 3,.... µ = E[X k ] σ 2 = E[(X k µ 2 ] = E[X k ] 2 µ 2. Y = (X + + X µ/ σ 2 Y. Y, (. X k (X k µ/σ. Y. X k, X k φ(t = E[e itx k ], φ(t = t2 2 + o(t2. Y = (X + + X / Y, Y : ( E[e ity ] = E[e itx k/ t ] = φ = k= ( ( t2 2 + o e t2 /2, Fourier 5, Y 6 f (y f (y = ( t e ity φ dt 2π, 7. 2π e ity e t2 /2 dt = e y2 /2 2π (. 5 φ(t/ Y Fourier,,. 6 R. 7.

11 X, g(x E[g(X ] = ( g(k p k q k k k=. < p <, q = p,, ( k : ( = k! k!( k!, (x + y = k= ( x k y k. k E[g(X ] ( δ(x a dx 9 : ( E[g(X ] = g(xf (x dx, f (x = p k q k δ(x k. k R, f (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 2., E[ ] E[g(X] = 2π ĝ(te[e itx ] dt = 2π ĝ(tφ X (t dt. 8,. 9 ( δ(x a dx f(x, g(xδ(x a dx = g(a R g(x ĝ(t g(x.

12 2 2., Y Y, φ Y φ Y, 22 g(y E[g(Y ] E[g(Y ] Y φ Y φ φ Y. φ Y φ, φ Y, Y Y 24.. ( φ X (t = E[e itx ] = e itk p k q k k k= ( = (pe it q k = (pe it + q k k=. µ = p σ 2 = pq. Y = X µ σ = X p pq, φ Y (t = E [ e ity ] = E [ e itp/ pq e itx / pq ] = e itp/ pq φ X (t/ pq = e itp/ pq ( pe it/ pq + q = ( pe itq/ pq + qe itp/ pq 25. X, X p = X (p+q p = qx p( X, φ Y (t = E [ e ity] = E [ e itqx/ pq e ] itp( X/ pq = = e itqk/ pq e itp( k/ pq k= k=. ( p k q k k ( (pe itq/ pq k ( qe itp/ pq k k = ( pe itq/ pq + qe itp/ pq pe itq/ pq = p + itpq pq qt2 2 + O qe itp/ pq = q itpq pq pt2 2 + O ( ( , g(y, g(t (. 24 Bocher. 25 p = q = /2 φ Y (t = (cos(t/.,

13 3 φ Y (t = ( ( t2 2 + O lim φ Y (t = e t2 /2, Y φ Y (t = E[e ity ] =, 26 g(y. lim k= e ity e y2 /2 2π dy = e t2 /2. lim E[g(Y ] = E[g(Y ] ( ( k p g p k q p /2 = g(y e y2 dy. pq k 2π g(y a y b, ( lim P a X p pq X. b e y2 /2 b = dy. a 2π 3 Laplace Stirlig ( Gauss (. Gauss Laplace. 3. Gauss Gauss Stirlig. log(e x x = log x x x = Taylor log x x = log (x (x 3 (x ,! = Γ( + = e x x dx (x 2 exp ( log dx = e e (x 2 /(2 dx = e 2π 2 26 a y b.

14 4 3. Laplace.! e 2π (. scilab. scilab. Stirlig e, 2π g (x = log(e x x = log x x x = Taylor 2. 3 y3 e y2 /α dy =. /( : k =,, 2,... e y2 /2 y 2k dy = 2 = 2 k 2 = 2 k 2 e y2 /2 (y 2 k dy = 2, e y2 /2 dy = e y2 /2 y 2 dy = 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 4 dy = 3 2π, e y2 /2 y 6 dy = 5 2π.. x + y = ( + y/! = Γ( + = e x x dx = e ( e y + y dy. ϕ (y : ( ϕ (y = log + y ( y = y2 2 + y3 3 y4 4 + o (. o(/. ( e y + y ( ( y = e y2 /2 3 exp ( 3 y4 4 + o ( y 3 = e y2 /2 + y3 3 y = e y2 /2 ( + y3 3 y4 4 + y6 8 + o 2 + o (. ( 27, Stirlig, Vol. 3 (979 No. 3, Wallis. Stirlig,, Stirlig, Vol. 36 (984 No. 2,

15 o(/ y. e y2 /2 < y <, : e y ( + y dx e y2 /2 ( y4 4 + y6 8 2π = 2π 3 2π = ( 2π O! = e ( 2π + ( 2 + O 2 ( 2 + O. ( dx + O ( 2 (. /(2 28 /(2,,! e 2π! Laplace Stirlig Gergö Nemes, Asymptotic expasios for itegrals, 22, M. Sc. Thesis, 4 pages. Example.2.., Stirlig /(2. : a > (a =, a e t t s dt = s a e x x s dx Γ(s s (. t = x/., a e t (α t s + α 2 t s 2 + dt = α Γ(s s + α 2Γ(s 2 s 2 + (. Stirlig /(2. f(x f(x = x log( + x (x >, y = ( + x,! = Γ( + = = e y y dy e x ( + x dx = + e e f(x dx. 28.

16 6 3. Laplace x > x <! + e = e f(x dx + e f( 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 < Taylor log( + x = x x 2 /2 + x 3 /3 x 4 /4 +.

17 f(x = t, e f(x dx = t dx e dt dt ( 2 = e t 2 t/ t + 2Γ(/2 = + 2Γ( + 2 /2 3 2π = π / /2 2Γ(3/2 2 3/ t3/2 + dt. Γ(/2 = π, Γ( =, Γ(3/2 = (/2Γ(/2 = π/2., f( x = t, e f( x dx = 2π 2 2 2π / /2. 2,, : (! 2π 2π + e = + /2 2 + O (. 3/2 5/2 :! = e 2π ( + ( 2 + O 2 (. /(2 Laplace t = t + t2 t ( k t k + ( k t k + t e t dt + t = Γ( =! Γ(2 2 k Γ(k + + ( k + ( k! (k! + + ( k + ( k 2 k e t t k dt + t e t t k dt. + t +., e t dt k (k! = ( + t k k=.. +, +.

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

19 Stirlig, a b ( ( a log = log(a! log(b! log((a b! b. lim = a log a + a log a + o( b log b b log + b + o( (a b log(a b (a b log + (a b + o( a a = log + o(. b b (a b a b log ( / a a b log b b b (a b a b ( / ( a = lim b (a! (b!((a b! / = (. a a b b (a b a b. a b (k! k k. 988 : ( lim 3C 2C /. 3 3 /( /( = = 27 6., Stirlig,.., lim ( / 2 = 22 = 22. (o( : ( 2 = 2 2 e o( (. Wallis ( 8.4. ( 2 22 ( π

20 2 4. Stirlig. 968 : lim 2P. ( 2 2 e. Stirlig. : ((a! / lim = a a e a. a log ((a!/ a = log(a! a log = (a log a + a log a + o( a log = a log a a + o( = log(a a e a + o(. Stirlig. 4.3 Stirlig. c, log! = log + log + c + o( 2 ( Stirlig (/2 log c.. log x dx = [x log x x] = log + k =, 2, 3,..., [k /2, k + /2] [, log k] [ /2, ] [, log ] log(! + (/2 log = log! (/2 log, (/2 log., { (x, y x, y log x }, log x dx,. 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! 2 log log x dx = log k + 2 log log x dx k= = α + β 2 α 2 + β 3 + β α + β. 3 c log 2π, Wallis e c = 2π.

21 2. log x, α, β 2, α 2, β 3, α 3,..., log x x,. 32. a, c = + a, log! = 2 log + log x dx + a + o( = log + log + c + o(. 2 c = log 2π Wallis ( 8.4.! = +/2 e e c e o( Wallis π = lim 2 2 (! 2 (2!, e 2 e 2c π = lim 2 2+/2 2+ e 2 e = ec. c 2 e c = 2π. Wallis, Stirlig, Stirlig! e 2π. 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. Gauss a >, f(x = e x2 /(2a, F (p Fourier. F (p = e ipx e x2 /(2a dx = e p2 /(2a 2aπ 32 a k= ( k a k. (.

22 22 5. : Fourier 33. x, a p, a. 2, e ipx e p2 /(2a dp = e x2 /(2a 2a π f(x = 2π e ipx F (p dp. f(x = e x2 /(2a 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 /(2a Fourier. Fourier Fourier, f(x µ = e (x µ2 /(2a Fourier a > ρ a (x ρ a (x = 2πa e x2 /(2a. ρ a (x > ρ a(x dx =., ρ a (x µ Fourier. f(x f a (x ρ a f a (x : f a (x = ρ a (x yf(y dy. f a (x Fourier 35., f a (x Fourier F a (p, ( F a (p = e ipx f a (x dx = e ipx ρ a (x y dx f(y dy 33 Cauchy, e ipx Taylor,. 34., Fourier. 35 f a (x Fourier ρ a (x µ f(µ,.

23 π e ipx F a (p dp = = ( ( e ipx e ipx ρ a (x y dx dp f(y dy 2π ρ a (x yf(y dy = f a (x. 2 ρ a (x µ Fourier. 36. F a (p = e ipx ρ a (x y dx = e ipy e ap2 /2 e ipy e ap2 /2 f(y dy = e ap2 /2 F (p e ipx F a (p dp = e ipx e ap2 /2 F (p dp. 2π 2π e ipx e ap2 /2 F (p dp = ρ a (x yf(y dy = f a (x. 2π F (p, Lebesgue, lim a 2π e ipx e ap2 /2 F (p dp = 2π e ipx F (p dp., f(x, a f a (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 a, a > ρ y x >δ a(x y dy ε/(2m. f a (x f(x = ρ a (x y(f(y f(x dy ε 2 + y x δ ρ a (x y f(y f(x dy ρ a (x y ε 2 dy + ε 2M M = ε. y x >δ lim a f a (x = f(x.. ρ a (x ym dy 36 Fourier Fourier. 37 ρ a (x a, ρ a (x Dirac (,.

24 24 5. : Fourier,, (996, xii+324, 3,8. Lebesgue Fourier., ,. 5.3 Riema-Lebesgue f(x R 38., Fourier f(p = e ipx f(x dx, p. lim e ipx f(x dx =. p Riema-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. Riema-Lebesgue L, Riema-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 = [a, b] I I I. α i I i i= α i Ii.,. f = i= α i Ii, I i = [a i, b i ], a i < b i R f(x dx = i= α i(b i a i. f (x R f m(x f (x dx (m,, x R f (x. (. f(x = lim f (x f(x ( x f. R f m(x dx R f (x dx R f m(x f (x dx (m, R f (x dx. f(x dx. f(x. R R f m(x dx R f (x dx R f m(x f (x dx (m,, R f (x dx, f(x dx <. 39 R Lebesgue. f f, φ f φ, f f, R f (x dx f(x dx. R. 4 R f R f L R f (x f(x dx. R

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

26 26 5. : Fourier N,. Riema. f(x = e x2 /2 Dirichlet ( R si x lim R 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, Riema x =, δ > lim s δ /2 N(f( = lim si(ny 2e y2 dy = e 2 /2 =. N N π y lim N ( δ si(n y y dy + δ si(ny e y2 /2 y dy = π 2. Riema-Lebesgue N. δ si(n y lim dy = π N y 2. y = x/n, π Nδ 2 = lim si x R si x dx = lim N x R x dx. Dirichlet Riema Riema-Lebesgue e x2 /2 Fourier 4. Dirichlet x a > ax, ± : R si(±ax lim dx = ± π R x 2 R si(ax lim dx = R x (a >,. π/2 (a >, (a =, π/2 (a <. Dirichlet. R f x R, δ > 4. (f(x + y + f(x y/2 f(x y

27 5.4. Fourier 27 < y < δ 42, Fourier N x f(x : lim N s N(f(x = f(x.. Riema, δ >, N s N (f(x = π δ Dirichlet, N f(x + y + f(x y si(nx dy + o(. y 2 δ si(n y f(x = lim dy f(x = 2 δ si(ny f(x dy + o(. N π y π y s N (f(x f(x = 2 π δ (f(x + y + f(x y/2 f(x si(ny dy + o(. y [(f(x+y+f(x y/2 f(x]/y < y < δ Riema-Lebesgue, N 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.. ] f(x y f(x y lim s N N(f(x = lim e ipx f(x + + f(x f(p dp = f(x = N N 2π N 2 42 Dii.

28 28 5. : Fourier 5.3. a > f a (x : /(2a ( a < x < a, f a (x = /(4a (x = ±a, (x < a a < x. f a (p = a e ipx dx = e iap e iap 2a a 2iap = si(ap. ap. Fourier N : s N (f a (x = N 2π = 2πa N N ixp si(ap e ap dp = 2 N cos(xp si(ap dp 2πa p si((a + xp + si((a xp p N 2 Dirichlet 2πa. Dirichlet x > a π/2, x = a, x < a π/2, 2 Dirichlet x < a π/2, x = a, x > a π/2. a < x < a π, x = ±a π/2, x < a a < x.. lim s N(f a = f a (x N dp 5.5 Fourier, f R 2π, x 2π. f Fourier a ( Z a = 2π e iy f(y dy 2π. N, f Fourier N : s N (f(x = N a e ix. = N

29 5.5. Fourier 29 N : s N (f(x = ( 2π N e i(x y f(y dy 2π = 2π = 2π = 2π = 2π = 2π = π 2π 2π 2π 2π π π = N 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 si((n + /2(x y f(y dy si((x y/2 si((n + /2y f(x + y dy si(y/2 si((n + /2y (f(x + y + f(x y dy si(y/2 y/2 si((n + /2y si(y/2 f(x + y + f(x y y 5 y x + y, si(αx/ si(βx, 6. lim t (t/ si t =, 5.4, N. Dirichlet : 2π 2π si((n + /2y si(y/2, π s N (( = : s N (( = π N = N si((n + /2y si(y/2 2π e iy dy = 2π dy = s N (( =. dy =. N = N δ =. e iy = e iy 2π., f(x = π π si((n + /2y si(y/2 s N (f(x, s N (f(x f(x = 2 π π dy f(x = π π y/2 si((n + /2y si(y/2 y/2 si((n + /2y si(y/2 dy. 2f(x y (f(x + y + f(x y/2 f(x y si((n + /2y δ y < π Riema-Lebesgue, δ >, lim N π δ y/2 si((n + /2y si(y/2 (f(x + y + f(x y/2 f(x y dy =. dy. dy.

30 3 6. : Gauss Fourier, N, s N (f(x f(x = 2 π δ < y < δ y/2 si((n + /2y si(y/2 (f(x + y + f(x y/2 f(x y (f(x + y + f(x y/2 f(x y dy + o(. N s N (f(x f(x, lim N s N (f(x = f(x : Gauss 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 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. 2

31 , 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 t = ( e ipx e x2 /2 2π dx = e p2 /2 (, x, p x/ t, t p t > (. ( (. si(px e x2 /2 si(px dx =. e x2 /2 cos(px dx = e p2 /2 2π. cos(px Taylor-Maclauli.. /2 e x2 x 2 dx. e x2 /2 x 2 dx = = =,, 2,... ( e x2 /2 x 2 dx e x2 /2 (x 2 dx = (2 e x2 /2 x 2 dx = ( π = (2! 2π. 2! e x2 /2 x 2 2 dx = 2!.

32 32 7. : Gauss, e x2 /2 = e x2 /2 cos(px dx = ( (px2 (2! dx ( p 2 = e x2 /2 x 2 ( p 2 /2 dx = 2π = e p 2 /2 2π. (2!! = (. = 6.4 Cauchy,. Cauchy 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 : Gauss : I := e x2 dx = π. (. I 2 = e (x2 +y2 dx dy = π R 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. 43 z = e (x2 +y 2, r 2 = x 2 + y 2, πr 2 = π( log z.

33 x = r cos θ, y = r si θ, 2π I 2 = e (x2 +y2 dx dy = dθ R 2 e r2 r dr = 2π [ ] e r2 = π. 2 2 Jacobia r. dx dy = (cos θ dr r si θ dθ (si θ dr + r cos θ dθ = r dr dθ, K = { (r, θ r >, θ < 2π }, I 2 = e (x2 +y2 dx dy = e r2 r dr dθ = R 2 K 2π dθ 7.3 Jacobia e r2 r dr = π. Gauss. y = x ta θ y θ,, I 2 = 4 = 4 π/2 ( ( dy = dθ cos 2 θ, x2 + y 2 = x 2 ( + ta 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 = 4 exp ( x2 π/2 cos 2 θ ( exp x cos 2 θ dθ dx x2 cos 2 θ 3. y = xt y t : I 2 = 4 = 4 = 2 ( ( e (x2 +y 2 dy dx = 4 e (+t2 x 2 x dx dt = 4 dt + t 2 = 2[arcta t] = 2 π 2 = π. ( [ 2 e (+t2 x 2 x dt e (+t2 x 2 2( + t 2 ] x= 3, 6 arcta t /( + t t = ta θ dt/dθ = + ta 2 θ = + t 2, θ = arcta t dθ/dt = /( + t 2., arcta t = t dt/( + t2. x= dt dx x= x= dθ

34 34 7. : Gauss 7.4 Jacobia Gauss.,, Gauss.. ( 9., Gauss. 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( =, Γ( + =!. Gauss 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 Gauss. : B(p, q = 2 π/2 cos 2p θ si 2q θ dθ = t p dt ( + t = du p+q p ( + u /p. p+q x = cos 2 θ = t/( + t, t = u /p. 3 ( p = /2 t, 2 F. Γ(/2 Gauss, χ 2. ( 9. B(/2, /2 = π., Γ(pΓ(q = Γ(p + qb(p, q, Γ(/2 2 = B(/2, /2 = π. Gauss... A, x, y A 45..

35 , 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 Hirokazu Iwasawa, Gaussia Itegral Puzzles, The Mathematical Itelligecer, Vol. 3, No. 3, 29, pp Steve R. Dubar, Evaluatio of the Gaussia Desity Itegral, October 22, a : ( exp (x 2 + a2 dx = 2π e a. 2 x 2 ( x >, y = x a/x x > < y <. y R x a/x = y x > x = 2 (y + y 2 + 4a dx = 2 ( + y dy. y2 + 4a

36 36 8. :. ( exp ( x a 2 dx = 2 2 x = ( exp ( y 2 y2 + dy 2 y2 + 4a e y2 /2 dy = 2π. 2 y/ y 2 + 4a y. ( x a 2 = (x 2 + a2 + a 2 x 2 x 2 ( exp ( x a 2 ( dx = e a exp 2 x 2 (. (x 2 + a2 dx. x 2 8 : Γ(/2 2 = B(/2, /2 = π : Γ(sΓ( s = B(s, s = π si(πs.. si z Γ(s si z = z ( z2 = π 2 2, i.e. Γ(s = lim s(s + (s +! s si(πs = s ( s2 π 2 = [( = e γs s + s ] e s/ 46. γ Euler ( γ = lim log., Γ(sΓ( s = Γ(s( sγ( s = s( s s = = [( + s ( s ], = si(πs. π 46 Γ(sΓ( s = π/ si(πs (, si z.

37 : Γ(sΓ( s = B(s, s = t s + t dt. < s <, < ε < < R C : ε R. R. R ε. ε. C zs dz/( + z z s dz/( + z z = 2πi : C z s dz + z = 2πieπis. 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 si(πs. t = u /s s du/( + u /s., : B( + s, s = sb(s, s = du πs = + u/s si(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 si(πs. 2πi., ( 5 (2 267., z s e 2πis. 48 z /s z e 2πis, dz e 2πis.

38 38 8. : 8.2 f(s (s > 3 : : f(s > (s >, : f(s + = sf(s (s >, : log f(s s >. 3 :! s f(s = f( lim s(s + (s + (s >. ( Γ(s 3 Γ( =, Gauss! s Γ(s = lim s(s + (s +, 3 Γ(s.., (. s(s + (s + lim! s s(s + (s + (! s = s + s ( + s ( 2 = s + s e s ( + s 2 ( s log e + s ( e s 2 + s e s e s( log log 2 Euler γ 49. k= (+s/ke s/k. 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 + lim! s = e γs s = [( + s ] e s/ 5. /Γ(s Γ(s = eγs s = [( + s ] e s/ 49 /x, + /2 + + / log + dx/x log = log( + log /( + + dx/x = log( + log, + /2 + + / log.. 5 /Γ(s. /Γ(s, Γ(s s =,, 2,....

39 Weierstrass. s C. F (s :! s F (s = lim s(s + (s +. F (s + = lim s! s s + + s(s + (s + = sf (s, F ( =! ( +! =. ( f(s = f(f (s (s >, < s < f(s = f(f (s., f(s, 2 < s <, f(+s f(, f(, f( + ( s ( s f( f( + f( f( + s f( ( < s < (# f( f(. g(s a < b < c g(b g(a b a g(c g(a c a g(c g(b c b 5. g(s = log f(s, (a, b, c = (, + s, +, log f( + s log f( log f( log f( +. s (a, b, c = (,, + s, log f( log f( log f( + s log f(. s 2 f( + s (#. f( + s (# f. f f( + f( =, f(s + = (s + (s + sf(s, f( = (!f(. (# +, s!f( ( + s( + s sf(s, (#, f(! s s(s + (s + f(s. f(s f((! s s(s + (s + = + s f(! s s(s + (s +. f(! s s(s + (s + f(s + s f(! s s(s + (s +. 5.

40 4 8. :, (. 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 a e sϕ(x+ψ(x dx. ϕ(x, ψ(x, s. (a, 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 a b a 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. Gauss.. Gauss. Gauss. s B(s, +, s B(s, + = s Γ(sΓ( + Γ(s + + = s! s(s + (s + s B(s, + = s x s ( x dx = 2 x = t/.,, s! ( s(s + (s + = t s t dt t s ( t dt t s e t dt = Γ(s.. (# f(s = Γ(s, < s < Γ(s + + s Γ( + (. 52 (a, 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..

41 8.3. 4, s >. s! s(s + (s + = s Γ(sΓ( + Γ(s + + Γ(s (.,,,, Gauss (. : : lim s B(s, + = Γ(s. Γ(s = lim s s! B(s, + = lim s(s + (s + = e γs s γ Euler. = [( + s ] e s/. 8.3 si z.,,, Γ(sΓ( s = B(s, s = Γ(sΓ( s = Γ(s( sγ( s = s si(πs = πs ( s2 = 2 π si(πs. =, si z = z ( s2 2. ( z2 = π 2 2, si(πs = π/(γ(s( sγ( s 53., si z cot z cot z = z + ( z π + z + π =,. 235.,, Fourier , Fourier 5.5..

42 42 8. : x cos(tx π x π Fourier, cot(πt 55. e itx Fourier, e itx Fourier e itx = lim a = π e ix e itx dx = 2π π 2π = ( (e iπt e iπt 2πi(t N = N = si(πt π = si(πt π [ e ix e itx i(t = ( si(πt π ] x=π x= π t N a e ix = si(πt N ( e ix lim π N t = N [ t + ( ] e ( ix t + e ix t + = [ t + ( 2t cos(x ( + i t 2 2. cos(tx Fourier [ cos(tx = si(πt ] π t + 2t cos(x ( t 2 2., = = ] 2 si(x t 2 2 π cot(tx = π cos(πt si(πt = t + 2t cos(x ( t 2 2 = x π, π cot(πt = t + = 2t t 2 = 2 t + ( t + t + = 56. si(πt π cot(πt, d si(πt log dt πt = = t = t = s, log si(πs πs = = ( t + = t + = ( ( log s ( + log + s ( / t/ + /. + t/ = log ( s2 = 2 55 x cos(tx π x π 2π R f t (x Fourier. cos(tx x < 2π 2π. 56 coth z = i cot( iz, coth(πt = iπ cot( πit = t + 2t t =

43 8.4. Wallis 43, 57 si(πs = πs ( s2 = si, /(Γ(sΓ( s si(πs π Γ(sΓ( s = si(πs. Γ(pΓ(q = Γ(p + qb(p, q, : π si(πs = B(s, s = x s ( x s dx = 2. t s dt + t = s du + u /s Wallis Wallis : 2 2 (! 2 lim (2! = π, i.e. ( π Wallis. Wallis Gauss s = /2 : π = Γ(/2 = lim /2! (/2(/2 + (/2 + = lim 2 + /2! 3 (2 + = lim = lim 2 2+ /2 (! 2 (2 +! 2 + /2! 2! 3 ( (2 2 2 (! 2 2 /2 = lim (2! 2 + = lim 2 2 (! 2 (2!. Wallis : 2 2 (2 (2 + = π 2. = s = /2 :, si(πs = ( π = si = π 2 2 π Γ(sΓ( s = πs = 57 sih z = i si( iz, sih(πs = πs ( = π (2 2 2 ( + s2 = 2. = ( s2 = 2. (2 (

44 44 8. : 8.5 Stirlig-Biet ( E. T. Whittaker ad G. N. Watso, A course of moder aalysis (927.. (digamma, ψ(s : ψ(s = d ds log Γ(s = Γ (s Γ(s. ψ (s (trigamma. (Weierstrass, log Γ(s = γs log s = [ ( log + s s ]. γ Euler. : ψ(s = d ds log Γ(s = γ s ψ (s = s 2 + = = [ + s ]. ( + s = 2 ( + s. 2,,., log Γ(s Stirlig-Biet.,, log Γ(z., Euler ( e t e t γ = dt e t t =

45 8.5. Stirlig-Biet ( 45. Euler /k = xk dx log = du/u, [ ] [ ] γ = lim k log = lim x k du dx k= k= u [ x ] [ = lim x dx du ( y ] du = lim dy u y u [ ( u/ ] du = lim du u u [ ( u/ ] ( u/ = lim du du u u e u e u = du u u du [ du ] [ = lim δ δ u e u δ u du du δ = lim δ u du u [ du ] [ = lim δ u e u δ u du e t = lim dt δ δ e t ( e t e t = lim dt δ δ e t t ( e t e t = dt. e t t e u δ e t δ ] u du ] dt t Euler, 2, 3 + x + + x = ( x /( x. 4 y = x, 5 y = t/, < = e δ < δ. δ δ du/u = log(δ/( e δ. u = e t, 2 u = t., Euler ψ(s = d ( e t ds log Γ(s = e st dt. t e t (Gauss., Euler c s = Γ(s e ct t s dt (Re c > ($

46 46 8. : s = c = s, s +,, 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 dt lim e st dt + e t + e 2t + + e ( t = e t e t e t, = (e t e (+st dt e t e t e st + e (s+t dt e t e st e t e t dt e st + e (s+t + e (s+2t + + e (s+ t = e st e (s+t 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

47 8.5. Stirlig-Biet ( 47 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.

48 48 8. :, 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. Stirlig 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 /(2. ( + ( 2z + O z 2 (z

49 8.6. Stirlig-Biet ( Stirlig-Biet (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. Whittaker ad G. N. Watso, A course of moder aalysis ( Biet s first expressio for log Γ(z i terms of a ifiite itegral Biet s secod expressio., arcta(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 shape α >, scale 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.

50 5 9. : : 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 shape α X, α Y, scale τ, τ, X + Y shape α X + α Y, scale τ. 2 2 (χ 2., shape /2, scale 2 2 (χ 2 : f 2,/2 (x dx = e x/2 x /2 2 /2 Γ(/2 dx = e x/2 (x/2 /2 dx Γ(/2 x ( 2. X, X 2,...,. Y = X X :. E[f(Y ] = cost. f(ye y/2 y /2 dy. E[f(Y ] = E[f(X X] 2 = f(x 2 (2π /2 + + x 2 e (x2 + +x2 /2 dx dx R = A (2π /2 = A 2(2π /2 = A 2(2π /2 f(r 2 e r2 /2 r dr f(ye y/2 y ( /2 y /2 dy f(ye y/2 y /2 dy. 3 r = x x 2, R r dr, A. 4 r = y /2, dr = (/2y /2 dy.

51 9.3. Pearso , A, A = 2π/2 Γ(/ Pearso 2 K = (K,..., K r., p i >, r i= p i =, k,..., k r, K = (k,..., k r, k i r i= k i = P (K = (k,..., k r =,.! k! k r! pk p k r r 9.3 (. 6 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 = p i : µ i = E[K i ] = k + +k r = 3. K i σ 2 i = p i ( p i : E[K i (K i ] =! k! k r! pk p kr r k i = p i (p + + p r = p i. k + +k r=! k! k r! pk p kr r k i (k i = ( p 2 i (p + + p r 2 = ( p 2 i, σ 2 i = E[K 2 i ] µ 2 i = E[K i (K i ] + µ i µ 2 i 2. = ( p 2 i + p i 2 p 2 i = p i ( p i

52 52 9. : i j K i K j σ ij = σ ji = p i p j : σ ij = E[K i K j ] µ i µ j = k + +k r = = ( p i p j 2 p i p j = p i p j. 3. X = (X,..., X r X i = K i p i pi! k! k r! pk p k r r k i k j µ i µ j, X i, p ii = p i( p i p i = p i = p i pi, i j X i X j p ij = p ji = p ip j p i pj = p i pj. X = (X,..., X r P = [p ij ] P = E + aa T, a =. E, a T a. r i= p i =, a. v R r, P v = v a, v a a v (r = 3. Euclid,. P a, P 2 = P, P r (Pearso 2. K = (K,..., K r Pearso 2 : Y = r Xi 2 = i= p. pr r (K i p i Pearso 2 r 2 ( 6. i= 59 Pearso 2 2 (. 6 Pearso 2.,, Pearso 2 2. p i

53 9.3. Pearso , 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 = diag(,...,,,...,. }{{} 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 = diag(,...,,,...,. }{{} s 6..

54 54 9. : (&, Z,..., Z s, Z s+,..., Z r ( Z s+ = = Z r =. 9. r Zi 2 = Z Zs 2 (almost 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 = diag(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 > (Beta distributio of the secod kid Beta prime distributio : 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

55 9.4. t 55 >, α = /2, β = /2, γ =, t., t : ( (+/2 g (t dt = c + t2 dt. c = /2 B(/2, /2 = Γ(( + /2 π Γ(/2. t µ, σ 2 µ = ( >, σ 2 = 2 ( > 2.. t. t Cauchy,. 2 t,. 9.8 ( 2 t. Z, Y, Z, Y 2. T = Z Y/ t.. : E[f(T ] = cost.. a = /( 2π 2 /2 Γ(/2, [ ( ] ( Z ( E[f(T ] = E f = a Y/ ( = a f = a ( = a ( ( (+/2 f(t + t2 dt. f z y/ e (y+z2 /2 y /2 dz f(te (+t2 /y/2 y (+/2 dt ( f(t e (+t2 /y/2 y (+/2 dy = 2(+/2 Γ(( + /2a z y/ e z2 /2 e y/2 y /2 dz dy dy dt ( (+/2 f(t + t2 dt., z = t y/ (z 2 = yt 2 /, dz = y /2 dt/. 6 : e αy y s dy = e x ( x α s dx α = α s Γ(s (α, s >. dy

56 56 9. :,, 2 (+/2 Γ(( + /2a = 2(+/2 Γ(( + /2 2π 2 /2 Γ(/2 = Γ(( + /2 π Γ(/2 = c ( t. X, X 2,..., µ, σ 2, M = k= X k, U 2 = k= (X k M 2, T = M µ U / (, M U, M µ, σ 2 /, ( U 2 /σ 2 2, T t. (U. 9. (T., E[M ] = µ, E[U] 2 = σ 2. µ, σ 2 (populatio mea, (populatio variat, M, U 2 (sample mea, (ubiased variat., M µ, σ 2 /. T = M µ U / Z = M µ σ/., Z σ U,, t. σ 2 Z, Z. σ 2 U 2, t. 9. ( t. ( (+/2 ( /2 ( ( /2 + t2 = + t2 t2 + O e t2 /2 2, t. t X k X k µ, µ =. µ =., X, X 2,..., σ 2,. ( M, U. (U.

57 9.4. t 57 Y = M = (X + + X /. (X,..., X Y (Y,..., Y 62. Xk 2 = k= k= Y 2 k 63, (X k M 2 = k= = (Xk 2 2M X k + M 2 = k= Xk 2 M 2 = k= k= Xk 2 2M k= Yk 2 Y 2 = Yk 2. k= X k + M 2 k=, E[Yk 2] = σ2, [ ] [ ] E (X k M 2 = E = ( σ 2 k= 64. E[U 2 ] = σ 2. X k., X k, σ 2., Y k,, σ 2. k= Y 2 k U 2 = k= (X k M 2 = k= Y 2 k, M = k= X k = Y, 9., U 2 σ 2 = σ k= Y 2 k M µ σ/ ( U 2 /σ 2 = M µ U / t. 9.2 ( t ( 9.9. X s,k (s =,..., r, k =,..., s, s X s,k µ s, σ 2 s 62 Y (,,..., /,. (Y,..., Y Y. Y k Y k.

58 58 9. :. Z = s M s = s k= r (M s µ s = s= X s,k, U 2 s = s r, Y = σs 2 s= s r s, T = s= r s= s k= s U 2 σs 2 s = Z Y/( r, T r t 65. (X s,k M s 2, r s σ 2 s= s k= (X s,k M s 2,. 9.9, M s Us 2, M s µ s, σs/ 2 s, ( s Us 2 /σs 2 s 2. Z Y., s s= (M s µ s, r s= (σ2 s/ s, Z. 2, Y r 2., 9.8, T r t. s = 2, µ = µ, µ 2 = µ, σ = σ X s,k (s =, 2, k =,..., s, µ, σ. M s = T = s k= X s,k, Y = M M 2 + Y s (X s,k M s 2, s= k=, T t X, X 2,..., σ X,..., X X U 2 X = k= X k, U 2 = (X k X 2 65 σ 2 = = σr 2 = σ 2 T σ, Z σ µ X k X k µ. k=

59 : (X k X 2 = k= Xk 2 X 2. k= Y,..., Y : Y = Y k = X k = X, k= ( k X j kx k+ k(k + j= (k =, 2,...,. A = [a ij ] / 2 2 A = ( / 6 / 2... / ( +, Y j Y j = a ij X i i=. A 68. i= a kia li = δ kl i= Y 2 i = i,k,l, E[X i X j ] = σ 2 δ ij, a ki a li X k X l = k,l δ kl X k X l = Xk. 2 k= E[Y k Y l ] = j,j a ki a lj E[X i X j ] = σ 2 i,j a ki a li δ ij = σ 2 a ki a li = σ 2 δ kl. i=, Xk X 2 = k= Y k Y 2 = Y k. k= k= ( U 2 = Xk 2 X 2 = k= k= Y 2 k. 68t AA. A.

60 6 9. :. E[U] 2 = E[Yk 2 ] = ( σ2 = σ 2. k= X k, σ 2, Y k, σ 2. X = Y /, σ 2 /, ( U/σ 2 2 = k= Y k 2/σ2,. 9.6 F α, β > (Beta distributio of the first kid : 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, >, x mx/ (x >,, ( mx/ f α,β + mx/ (m/ dx ( + mx/ 2 = B(α, β., α = m/2, β = /2, : (mx/ α dx ( + mx/ α+β x (x > g m, (x dx = B(m/2, /2 (mx/ m/2 dx ( + mx/ (m+/2 x (x >. m, F. m, F µ m,, σ 2 m,, µ m, = 2 ( > 2, σ 2 m, = 22 (m + 2 m( 2 2 ( 4 ( > 4. X m, F, (mx//( + mx/ m/2, /2, mx/ m/2, /2.

61 9.6. F ( 2 F. Y, Z m, 2, X = Y/m Z/ m, F., Y k, Z l,, m, F. X = ( m k= Y k 2 /m ( l= Z2 l /. 9.. E[f(X] = cost. (mx/ m/2 dx f(x ( + mx/ (m+/2 x. a = [2 (m+/2 Γ(m/2Γ(/2], [ ( ] Y/m ( ( y/m E[f(X] = E f = a f e (y+z/2 y m/2 z /2 dy dz Z/ z/ ( ( m m/2 = a f(xe (+mx/z/2 xz z /2 m z dx dz ( (mx m/2 dx = a f(x e (+mx/z/2 z (m+/2 dz x ( m + ( mx m/2 ( = 2 (m+/2 Γ a f(x + mx (m+/2 dx 2 x 3 y/m = (z/x (y = (mx/z, dy = (m/z dx. 5 : ( s t e αz z s dy = e t dt α α = α s Γ(s (α, s >.. ( m + 2 (m+/2 Γ a = 2(m+/2 Γ((m + /2 2 2 (m+/2 Γ(m/2Γ(/2 = B(m/2, /2.. x = t 2 /, α = /2, β = /2 t. T t, T 2, F, T 2, F. T F.. F : g m, (x dx = x m/2 (m/m/2 B(m/2, /2 ( + mx/ (m+/2 dx.

62 62 9. : m =, g, (x dx = x /2 B(/2, /2 ( + x/ (+/2 dx. x = t 2, < t < g, (t 2 dt t dt = B(/2, /2 ( + t 2 / (+/2. t g (t dt 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

63 f(u. F,, F., x = uy x u f(x, ye (x+y x p y q dx dy = du dy f(uy, ye (+uy y p+q u α. ( + 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 x 2s dx e x2, x = r cos θ, y = r si θ y = x ta θ.. 69, (26 7

64 64 9. : x = r cos θ, y = r si θ, 4 g(x, ye (x2 +y 2 x 2p y 2q dx dy = 4 π/2 dθ g(x, y = g(y/x 4 g(y/x =. dr g(r cos θ, r si θ e r2 r 2(p+q (cos θ 2p (si θ 2q ( y g e x (x2 +y 2 x 2p y 2q dx dy = Γ(p + q 2 B(p, q = 2 π/2 π/2 g(ta θ (cos θ 2p (si θ 2q dθ. (cos θ 2p (si θ 2q dθ y = x ta θ y θ, 4 = 4 g(x, ye (x2 +y 2 x 2p y 2q dx dy π/2 dθ dx g(x, x ta θ e (+ta2 θx 2 x 2(p+q (ta θ 2q ( + ta 2 θ. x = r/ + ta 2 θ x r, 4 = 4 π/2 g(x, ye (x2 +y 2 x 2p y 2q dx dy ( r dθ dr g + ta 2 θ, g(x, y = g(y/x 4 r ta θ + ta 2 θ ( y g e x (x2 +y 2 x 2p y 2q dx dy = Γ(p + q 2 π/2 g(ta θ e r2 r 2(p+q (ta θ 2q ( + ta 2 θ p+q (ta θ 2q dθ. ( + ta 2 θ p+q. (ta θ 2q ( + ta 2 θ p+q = (cos θ2p (si θ 2q. ta θ = si θ/ cos θ, cos 2 θ = + ta 2 θ, si2 θ = ta2 θ + ta 2 θ

65 9.8. Maxwell-Boltzma ( 65. t = u/( + u. : x : y = cos θ : si θ = : ta θ., x, y 7, x : y = t : ( t = u : (t = si 2 θ, t = cos 2 θ, u = ta 2 θ. 9.8 Maxwell-Boltzma ( X i, R = X X, 2 Z ( i = X i /R. (Z (,..., Z ( 7. Z ( i g (z dz = c ( z 2 ( 3/2 dz ( < z <, ( c = ( z 2 ( 3/2 dz = B 2, ( = 2 2 B, ,. 2 S 2 = { (x 2,..., x x x 2 = } dω, r = x x 2, x 2,..., x r 2, r 2 r 2, dx dx 2 dx = r 2 dx dr dω., r r = x x 2, r = r 2 x 2, r / r = r/r, dx dx 2 dx = r(r 2 x 2 ( 3/2 dx dr dω. x z = x /r, dx dx 2 dx = r ( z 2 ( 3/2 dz dr dω., R ρ(r, g(zρ(r dx dx = g(z( z 2 ( 3/2 dz R 2 c, c = ( z 2 ( 3/2 dz r ρ(r dr dω. S 2 c 2. z = t /2, dz = t /2 dt/2 : ( c = 2 ( z 2 ( 3/2 dz = t /2 ( t ( 3/2 dt = B 2,. 2 7 f(x, y = f(y/x. 7.

66 66 9. : 2 ( z 2 = ( + z( z, z = 2t, dz = 2 dt : ( c = 2 ( 3/2 t ( 3/2 2 ( 3/2 ( t ( 3/2 2 dt = 2 2 B,. 2 2., duplicatio formula. ( /2 s c :, 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. (Legedre s duplicatio formula 72. Z ( i 73 : g 2 (z dz = π dz z 2 ( < z <., /2. z = si θ, dθ ( π/2 θ π/2 (. π /2 + θ/π = /2 + (arcsi z/π ( z. 74. g 3 (z dz = dz ( z., / g 4 (z dz = 2 π z2 dz ( z., /4. 72 Legedre s duplicatio formula Gauss s multiplicatio theorem : Γ(s = s /2 Γ(sΓ(s + /Γ(s + 2/ Γ(s + ( /. (2π( /2 Γ(3s = 3 3s /2 Γ(sΓ(s + /3Γ(s + 2/3/(2π ,..,.,..,,.

67 9.8. Maxwell-Boltzma ( z = cos θ, si 2 2 π si2 θ dθ ( θ π g (z. /. Z ( i. c /c +2 = ( / = /., Z ( i / : c z 2 ( z 2 ( 3/2 dz = c (c c +2 = c = c +. z 2 ( z 2. Y ( i = Z ( i,, ( y dy g = ( ( 3/2 y2 dy c., ν = ( /2, ( 3/2 3/2 ( /2 ( y2 = ( y2 y2 /2 e y2 /2 /2 c = 2ν + 2 2ν B(ν, ν 2ν 2 2ν 2 πν = 2π ν 2 2ν 75 Wiger. N M i e M 2 ii /2 dm ii i<j e M 2 ij /2 dm ij, M., /4, N /4 Wiger..,,, (, Tate. Tate p p + 2 p si (28. si 2,, Dedekid si2-, 6, (25.. si 2. SU(2 (Haar SU(2 si 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 si 2. Tate p SU(2 3 S 3 = SU(2.

68 68 9. : 77. Wallis 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 ( i : lim ( y g = lim ( y 2 / ( 3/2 2 2 B(, = e y2/ π, y g(y, /2 g(y i dω g(y e y2 dy 2π C S R (., S = { (y,..., y R y y 2 = }, C, dω. Maxwell-Boltzma. 9.9 Maxwell-Boltzma (2 x i., m m... m S m = { (x m+,..., x x 2 m+ + + x 2 = } dω, r = x 2 m+ + + x 2, x m+,..., x r m dx dx = r m dx dx m dr dω., r r = x x 2, r = r 2 x 2 x 2 m r / r = r/r, dx dx = r(r 2 x 2 ( m 2/2 dx dx m dr dω. x i (i =,..., m z i = x /r (i =,..., m, dx dx = r ( z 2 z 2 m ( m 2/2 dz dr dω., ρ(r, g(z,..., z m ρ(r dx dx R = c ( m g(z,..., z m ( z 2 zm 2 ( m 2/2 dz dz m. z 2 + +z2 m< ( 77 c = ( y2 / ( 3/2 dy, lim ( y 2 / ( 3/2 = e y2 /2 lim c = 2π.,. 78, Wallis.

69 9.9. Maxwell-Boltzma (2 69 c ( m = r ρ(r dr dω S m. m = c ( =., ρ(r = e r2 /2 /(2π /2, ( dω = (2π /2 S r e r2 /2 dr = 2/2 π /2 2 /2 Γ(/2 = 2π/2 Γ(/2 = (dω S. : r s e r2 /2 dr = e t (2t (s 2/2 dt = 2 s/2 Γ(s/2. π/2 Γ(/2 + r 2 /2 = t, r dr = dt, r s dr = r s 2 r dr., r ρ(r dr., c ( dω S ( m = = dω S ( m. m c ( m. c ( m 79 : c ( m = ( z 2 zm 2 ( m 2/2 dz dz m. z 2+ +z2 m< = t /2 t /2 t i >, m i= t i< = Γ(/2m Γ(( m/2. Γ(/2 m ( t t m ( m 2/2 dt dt 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< 79 = m + 2 c (m+2 m dt dt m du t p t p m m ( t t m p m+p m+ u p m ( u p m+. = π m/2 /Γ(m/2 + m.

70 7 9. : (B. (, (Z (,..., Z ( m g (z,..., z m dz dz m = c ( m ( z 2 zm 2 ( m 2/2 dz dz m., σ >, (Y (,..., Y m ( = σ (Z (,..., Z m ( ( ( m 2/2 m y 2 σ 2 i dy dy m 8., ( lim σ 2 i= m i= y 2 i m 2 2 = exp ( 2σ 2 (Y (,..., Y m ( m 8., C ( g(y σ,..., y m dω σ S ( g(y (2σ 2 m/2,..., y m exp m y 2 R 2σ m 2 i dy dy m., σ S = { (y,..., y R y y 2 = σ 2 } σ, C ( σ, dω. Maxwell-Boltzma, σ 2 Boltzma kt. i= m i= y 2 i 9. < p <.. B p, p ( P (B p, = k = p k ( p k (k =,, 2,..., k. p p( p, E[e itb p, ] = (pe it + q.., p, (B p, p/ p( p.. Γ(s + = s!, ( s t = s!/(t!(s t! ( α + β 2 8 Y ( i 8 Y ( i B(α, β = (α + β! (α!(β! = (α + β α.,.

71 9.. Poisso 7, 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 =, α = k, ( f k+, k+ (p dp = ( + p k ( p k dp ( < p < k, p = (k + /( + 2, ((k + ( k + /(( ( + 2, p = k/ 82., A B k, A B α = k +, β = k Poisso Poisso. < µ < Z, N, p = µ/ : P (N = k = = ( µ k ( µ ( k k ( µ µ k ( µ ( k k! ( k, ( P (N = k = µ µ k ( µ ( k ( k µ µk e k! k!. N k =,, 2,... µ µk P (N = k = e k!, N µ Poisso. N Poisso.. µ. N µ/n. N. µ N, N Poisso. Poisso. 82 k p (, p, p,. 83.

72 72 9. : µ Poisso µ : µ µk (N = E[N] = ke k! = µe µ E[N(N ] = µ 2 e µ k= k=2 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. Poisso : E[e itn ] = e µ k= µ k (k! = µ, itk µk e k! = exp(µ(eit = ( exp(e it µ., Poisso µ.,, µ,, X = N µ µ Y = (N µ2 µ 84. shape α = k + >, scale τ = : f k+, (µ dµ = e µ µ k dµ (µ >. k! µ = k +, k +, µ = k., k, µ shape α = k +, scale τ = 85. µ, t > Poisso, T N t µt Poisso : µt (µtk P (N t = k = e k! (k =,, 2, 3,.... α = k + >, τ = /t : µt (µtk f k+,/t (µ dµ = e t dµ. k! µ = (k + /t, (k + /t 2, µ = k/t. k µ t (µ >, t µ,. 84, N (observed, µ (expectated, O, E, Y = (O E 2 /E. 85.

73 ( E[ ] 86 : E[αX + βy ] = αe[x] + βe[y ] (. f E[f(X] (. E[] = (.. X ( E [ X ] <. µ X = E[X] X. X µ X, (X µ X 2 X, σx 2, σ X.. E [ X r] < X r, E[X r ] X r. X X µ X = E[X], 2 E[X 2 ] = σx 2 + µ2 X σ2 X = E[X2 ] E[X] 2. X φ X (t = E[e itx ] X. t. 87. X, Y, X Y. X r, φ X (t t = r, φ (k X ( = E[Xk ] (k =,,..., r. X Y. Cauchy-Schwarz, E[ (X µ X (Y µ Y ] σ X σ Y, σ XY = E[(X µ X (Y µ Y ] well-defied, E[(X µx (Y µ Y ] σx σ Y. σ XY X Y. ρ XY = σ XY /(σ X σ Y X Y.., θ cos θ. X X µ X, E(X µ X (Y µ Y ],. X i, i,..., i r, E[f (X i f r (X ir ] = E[f (X i ] E[f r (X ir ] (f k. X Y X Y,. 86 (Ω, F, µ X. X X(x µ(dx Ω E[ ]. 87 (Ω, F, µ X : Ω R. R Borel A µ X (A = µ(x (A, R µ X. µ X X. µ X Lebesgue f(x, f(x X. R g(x X g g(x. g(x. g(x, g(x E[g(X] = R g(x µ X(dx. X f(x E[g(X] = g(xf(x dx. R

74 74. : Tauber D α α >, X D α, Y D β, X, Y., X + Y D α+β, D α. X,..., X r, φ X + +X r = r i=k φ X k., φ Dα = ϕ α, D α. : Tauber. Tauber.. f(t t > 88, α, a >. 89, x f(t dt ax α (x f(x aαx α (x. ( x.., f. f, c >, cx f(t dt x cx x ax α, f(t dt f(x x f(t dt c x f(t dt. ( x c x cx x f(t dt f(t dt ax α ax α c x 9, f(x ax α x f(t dt ax α c x f(t dt ax α. (2 c c α c lim if x f(x lim sup axα x f(x ax c α. (3 α c c. lim x α lim if x f(x lim sup axα x f(x α. axα f(x ax α = α, f(x aαxα (x. 88 x x f(x f(x.,.. 89 F (x G(x (x lim x (F (x/g(x = 9 cx f(t dt ac α x α (x.

75 .2. Laplace Tauber 75 f. f. f (,(2., (3 lim if lim sup,., f. a f( = a f(t..2. a, a 2, a 3,..., a, α >. a k a α (. k= a aα α (.2 Laplace Tauber Stoe-Weierstrass ϕ(y [, ], g(y [, ] c (,, g(c ± = lim ε g(c ± ε., ε >, P (y, Q(y P (y g(y Q(y ( y,. g(yϕ(y dy ε g(yϕ(y dy P (yϕ(y dy Q(yϕ(y dy g(yϕ(y dy, g(yϕ(y dy + ε. Q(y. (g(y g(y P (y. g(c g(c+. (g(c g(c + g(y g( y. ϕ(y N = ϕ(y dy = ϕ(y dy, N <. g(y [, ], M > g(y M ( y. ε >. 9.

76 76. : Tauber c δ >, g(y g δ (y : g(y ( y c δ, g δ (y = max{a(y c + g(c +, g(y} (c δ y c, g(y (c y. a = (g(c + g(c δ/δ, a(y c + g(c + = a(y (c δ + g(c δ. M g(y g δ (y M ( y. g δ (yϕ(y M ϕ(y ( y lim δ g δ (yϕ(y = g(yϕ(y (y c Lebesgue, lim δ g δ (yϕ(y dy =, δ > g(yϕ(y dy g δ (yϕ(y dy g(yϕ(y dy. g(yϕ(y dy + ε 3. Stoe-Weierstrass, Q(y Q(y g δ (y ε ε ( y 3N 3N. g(y g δ (y Q(y ( y, Q(yϕ(y dy Q(y g δ (y ε ϕ(y dy + 3N ε ϕ(y dy + 3N = ε 3N N + = g(yϕ(y dy + ε.. g(yϕ(y dy + ε 3 + g(yϕ(y dy + ε 3 + ε 3N N g δ (yϕ(y dy + ε 3N ϕ(y dy ε ϕ(y dy 3N.4. f(t t >, a, α >. /x e xt f(t dt a x α (x f(t dt a x α Γ(α + (x.. (.

77 .2. Laplace Tauber 77. F (x = e xt f(t dt, F (x a/x α (x, k =,, 2,..., F ((k + x = : e xt ( e xt k f(t dt a (k + α x = a α x α Γ(α c = α Γ(α p(y e xt p(e xt f(t dt a x α Γ(α [, ] ϕ(y e t ( e t k t α dt (x. e ct t α dt (c >. e t p(e t t α dt (x. ϕ(y = ( log y α ( < y, ϕ( = 92, y = e [, ] g(y g(y = { ( y < e y (e y..3, ε >, P (y, Q(y P (y g(y Q(y ( y, g(yϕ(y dy ε g(yϕ(y dy P (yϕ(y dy Q(yϕ(y dy. y = e t, e t g(e t t α dy ε e t g(e t t α dy, f(t 93, e xt P (e xt f(t dt e t P (e t t α dy e t Q(e t t α dy e xt g(e xt f(t dt g(yϕ(y dy, g(yϕ(y dy + ε e t g(e t t α dy, e t g(e t t α dy + ε. e xt Q(e xt f(t dt 92 ϕ(y dy = ( log yα dy y = e t, ϕ(y dy = e t t α dt = Γ(α. ϕ(y = ( log y α α > [, ]. 93 f(t.

78 78. : Tauber, a/(x α Γ(α +, x, e t P (e t t α x α Γ(α + dt lim if e xt g(e xt f(t dt x a x α Γ(α + lim sup e xt g(e xt f(t dt e t Q(e t t α dt x a 2, ε > : x α Γ(α + lim x a 94 : e xt g(e xt f(t dt = e xt g(e xt f(t dt a x α Γ(α e t g(e t t α dt. e t g(e t t α dt (x. e e xt t /x, t /x e xt g(e xt =, t > /x g(e xt =, : /x f(t dt a x α Γ(α. t α dt = a x α αγ(α = a x α Γ(α + (x...5. f(t t >, α, a >.,. t f(t dt e xt f(t dt a x α (x atα Γ(α +, atα f(t Γ(α (x a f( = a f(t, Stieltjes, y = e x x y,. ( e x x (x..6. a, a, a 2,..., α, a >. lim( yα a y = a y a k k= = aα Γ(α + (. (. 94 Stoe-Weierstrass Jova Karamata.

79 .2. Laplace Tauber 79.. x >, y = e x, y x (x, F (x := e x a a (x. x α = k =,, 2,..., F ((k + x = e x (e x k a = : a (k + α x = a α x α Γ(α c = α Γ(α, p(y = e x p(e x a a x α Γ(α g(y = 95 : x= e x g(e x a a x α Γ(α e t ( e t k t α dt (x. e ct t α dt (c >. e t p(e t t α dt (x. { ( y < e, y (e y e t g(e t t α dt (x. e e x /x, /x e x g(e xx =, : /x a a x α Γ(α t α dt = a x α αγ(α = a x α Γ(α + x = /, k =,,...,, (x. a k k= aα Γ(α + (...7. a, a, a 2,..., α, a >. lim( yα a y = a y = 95.3., a,..4. Jova Karamata.

80 8. : Tauber a k k= aα Γ(α +, a aα Γ(α (..4 Stieltjes φ(t φ(t = (t <, a, α >. F (x := e xt dφ(t a x α (x. φ(t atα Γ(α + (t, λ, a, α >. e λx a (x x α = #{ λ t } atα Γ(α +. t = λ. (t ( /α Γ(α + λ /α ( a.3 Wallis ( y /2 = = a y ( y < a ( /2 a = ( = ( + ( ! 3 (2 = = (2! 2! 2 2 (! = ( 2 > ,.6..

81 .3. Wallis (2 = 2!. a + = + /2 + a < a. a. ( y /2 = a y =.7 : a = 2 2 ( 2 /2 Γ(/2 = π (. Wallis 97.. a Wallis a / π (, p,k = a k a k., mi{k, k} p,k π, a < b, lim a k/ b p,k = lim k( k = π a k/ b π k k ( k ( k = π b a dx x( x. π (x( x /2 dx. : x π dt = 2 x t( t π dy y 2 = 2 π arcsi x. (x, y = (/2, /2 x. Wallis. Wallis Tauber (.7.., 98.. /2, Tauber Wallis. 98 Frak Spitzer, Priciples of Radom Walk, Spriger GTM 34 ( pp ,, (24.,.

82 82. : Tauber, : # { (x i i= {±} a < #{ k x + + x k > } < b} lim = b dx, 2 π a x( x lim 2 <x,...,x <, a<#{ k x + +x k > }<b dx dx = π b a dx x( x., x + + x k > (, #{ k x + + x k > }, a < #{ k x + + x k > } < b a b...4 x x 2 + x 4 x 8 + x 6 x 32 + x? F (x F (x = x x 2 + x 4 x 8 + x 6 x 32 + = ( k x 2k ( x < k=. x F (x??, /2., F (x 2 = x F (x., x F (x.5. F ( , F ( F (x x /2 a, s F (x = a x, s = a + a + + a. = a x = ( x s x = = s = a + a + + a Peter Dure, Sums for Diverget Series: A Tauberia Adveture, 23- (. G. H. Hardy 97. x F (x.5..

83 .5. Laplace-Stieltjes 83. F (x = x( x + x 4 ( x 4 + x 6 ( x 6 + = ( xf(x, f(x = x + x 4 ( + x + x 2 + x 3 + x 6 ( + x + + x 5 + = x + (x 4 + x 5 + x 6 + x 7 + (x 6 + x x 3 +., s.,.6 (α =,., s + s + + s lim 2 2k k 2 2k s + s + + s lim 2 2k k 2 2k lim F (x = lim ( x s x = a x x lim = s = a k= = lim k k 4 k k = lim k 2 4k = lim k = lim k 4 k 4 4 k = 3, 4 k 4 2 4k = 2 3, k= s. x F (x..5 Laplace-Stieltjes.6. F (x R 2., lim x F (x = lim x F (x =, F (x. F (x x. F (x. F (x, x R, F (x = lim ε F (x ε (. F (x F (x = F (x + = lim ε F (x + ε. F (x F (x > F x. F (x x F (x F (x >. F (x. F (x ( lim ε F (x + ε = F (x. 2 F (x x x F (x F (x.

84 84. : Tauber F (x Lebesgue-Stieltjes µ F ((a, b] = F (b F (a ( a < b Borel µ F (., Borel A f(x df (x = f(x µ F (dx A, Lebesgue-Stieltjes. µ F (dx df (x. F (x x = a µ F ({a} =, F (b F (a = df (x = df (x (a < b (a,b]. 2 a = b x = a F (x. R Borel µ x R µ((, x] < F (x = µ((, x] F (x lim x F (x =, µ = µ F. [a, b] (a < b Riema-Stieltjes [a, b], A [a,b] a = x < x < x 2 < < x = b, c i [x i, x i ] b a f(xdf (x = lim f(c i (F (x i F (x i i=.. f(x b f(x df (x = f(x µ F (dx. F ( =. µ (, Borel a M(s = µ Laplace. f(x µ(dx Laplace. [a,b] e sx µ(dx e sx f(x µ(dx.. µ (µ((, < M(s = e sx µ(dx < (s. µ, M(σ < e σx µ(dx, Laplace Laplace. e σx µ(dx/m(σ,.

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