Tài liệu Special Functions part 1 pptx

... IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-5 21- 4 310 8-5) Copyright (C) 19 88 -19 92 by Cambridge University Press.Programs Copyright (C) 19 88 -19 92 by Numerical Recipes Software. Permission is ... or call 1- 800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). Chapter 6. Special Functions 6.0 Introduction There is nothing particular...

Ngày tải lên: 24/12/2013, 12:16

... 19 76, Journal of Research of the National Bureau of Standards , vol. 80B, pp. 2 91 311 ; 19 81, op. cit. , vol. 86, pp. 6 61 686. Abramowitz, M., and Stegun, I.A. 19 64, Handbook of Mathematical Functions , ... vol. 28, pp. 811 – 816 . [3] 6 .11 Elliptic Integrals and Jacobian Elliptic Functions Elliptic integrals occur in many applications, because any integral of the form  R(t, s)...

Ngày tải lên: 21/01/2014, 18:20

... 1 Y 00 =  1 4π P 1 1 (x)= − (1 −x 2 ) 1/ 2 Y 11 = −  3 8π sin θe iφ P 0 1 (x)= xY 10 =  3 4π cos θ P 2 2 (x)= 3 (1 x 2 ) Y 22 = 1 4  15 2π sin 2 θe 2iφ P 1 2 (x)=−3 (1 x 2 ) 1/ 2 xY 21 = −  15 8π sin ... expressions, such as P m l (x)= ( 1) m (l + m)! 2 m m!(l −m)! (1 − x 2 ) m/2  1 − (l − m)(m + l +1) 1! (m +1)  1 x 2  + (l−m)(l − m − 1) (m + l +1) (...

Ngày tải lên: 26/01/2014, 15:20

... z)=  1 0 t z 1 (1 −t) w 1 dt (6 .1. 8) 216 Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-5 21- 4 310 8-5) Copyright (C) 19 88 -19 92 by ... recurrence relations, for example  n +1 k  = n +1 n−k +1  n k  =  n k  +  n k 1   n k +1  = n−k k +1  n k  (6 .1. 7) Finally, turning away from the combinatoria...

Ngày tải lên: 21/01/2014, 18:20

... than 1. 2 × 10 −7 . { float t,z,ans; z=fabs(x); t =1. 0/ (1. 0+0.5*z); ans=t*exp(-z*z -1. 265 512 23+t* (1. 00002368+t*(0.3740 919 6+t*(0.09678 418 + t*(-0 .18 628806+t*(0.27886807+t*( -1. 13520398+t* (1. 488 515 87+ t*(-0.82 215 223+t*0 .17 087277))))))))); return ... x)=e −x x a  1 x+ 1 a 1+ 1 x+ 2−a 1+ 2 x+ ···  (x>0) (6.2.6) It is computationally better to use the even pa...

Ngày tải lên: 21/01/2014, 18:20

... powers of x: E n (x)=−  1 (1 − n) − x (2 − n) · 1 + x 2 (3 − n) (1 · 2) −···+ (−x) n−2 ( 1) (n − 2)!  + (−x) n 1 (n − 1) ! [− ln x + ψ(n)] −  (−x) n 1 · n! + (−x) n +1 2 · (n +1) ! +···  (6.3.8) The ... ≡ B x (a, b) B(a, b) ≡ 1 B(a, b)  x 0 t a 1 (1 − t) b 1 dt (a, b > 0) (6.4 .1) It has the limiting values I 0 (a, b)=0 I 1 (a, b) =1 (6.4.2) and the symmetry relati...

Ngày tải lên: 21/01/2014, 18:20

... the hypergeometric series, 2 F 1 (a, b, c; z) =1+ ab c z 1! + a(a +1) b(b +1) c(c +1) z 2 2! + ··· + a(a +1) (a+j 1) b(b +1) (b+j 1) c(c +1) (c+j 1) z j j! + ··· (6 .12 .1) This series converges only ... ≡  φ 0  1 − k 2 sin 2 θdθ =sinφR F (cos 2 φ, 1 − k 2 sin 2 φ, 1) − 1 3 k 2 sin 3 φR D (cos 2 φ, 1 − k 2 sin 2 φ, 1) E(k) ≡ E(π/2,k)=R F (0, 1 − k 2 , 1) − 1 3 k 2...

Ngày tải lên: 21/01/2014, 18:20

... the hypergeometric series, 2 F 1 (a, b, c; z) =1+ ab c z 1! + a(a +1) b(b +1) c(c +1) z 2 2! + ··· + a(a +1) (a+j 1) b(b +1) (b+j 1) c(c +1) (c+j 1) z j j! + ··· (6 .12 .1) This series converges only ... hypdrv. bb=b; cc=c; dz=Csub(z,z0); hypser(aa,bb,cc,z0,&y [1] ,&y[2]); Get starting function and derivative. yy=vector (1, 4); yy [1] =y [1] .r; yy[2]=y [1] .i; yy[3]=y[2].r;...

Ngày tải lên: 21/01/2014, 18:20

... ≡ B x (a, b) B(a, b) ≡ 1 B(a, b)  x 0 t a 1 (1 − t) b 1 dt (a, b > 0) (6.4 .1) It has the limiting values I 0 (a, b)=0 I 1 (a, b) =1 (6.4.2) and the symmetry relation I x (a, b) =1 I 1 x (b, a)(6.4.3) If ... equations 6 .1. 9 and 6 .1. 3.) The continued fraction representation proves to be much more useful, I x (a, b)= x a (1 − x) b aB(a, b)  1 1+ d 1 1+ d 2 1+ ···  (6...

Ngày tải lên: 21/01/2014, 18:20

... (6.5.9). z=8.0/ax; y=z*z; xx=ax-0.78539 816 4; ans1 =1. 0+y*(-0 .10 98628627e-2+y*(0.2734 510 407e-4 +y*(-0.2073370639e-5+y*0.2093887 211 e-6))); ans2 = -0 .15 62499995e -1+ y*(0 .14 30488765e-3 +y*(-0.6 911 147651e-5+y*(0.76 210 9 516 1e-6 -y*0.93493 515 2e-7))); ans=sqrt(0.636 619 772/ax)*(cos(xx)*ans1-z*sin(xx)*ans2); } return ... America). z=8.0/x; y=z*z; xx=x-0.78539 816 4; ans1 =1. 0+y*...

Ngày tải lên: 21/01/2014, 18:20