Tài liệu Special Functions part 10 pdf

... 258Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge ... -sign;sums=sum;sum=sumc;} else {sumc=sum;sum=sums;256Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge ... -n*(n+1);b=Cadd(b,Complex(4.0,0.0));d=Cdiv(ONE,Cadd(RCmul(a,d),b)); Denominators cannot be zero.6 .10 Dawson’s Integral259Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge...

... expint");else {if (n == 0) ans=exp(-x)/x; Special case.else {if (x == 0.0) ans=1.0/nm1; Another special case.else {226Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN ... functionfor several pairs (a, b).224Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge ... Wrench[2]can provide you with the first 328 digits ifnecessary!#include <math.h>#define MAXIT 100 Maximum allowed number of iterations.#define EULER 0.5772156649 Euler’s constant γ.#define...

... (6.5.8).y=x*x;ans1=x*(-0.4900604943e13+y*(0.1275274390e13+y*(-0.5153438139e11+y*(0.7349264551e9+y*(-0.4237922726e7+y*0.8511937935e4)))));ans2=0.2499580570e14+y*(0.4244419664e12+y*(0.3733650367e10+y*(0.2245904002e8+y*(0 .102 0426050e6+y*(0.3549632885e3+y)))));230Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright ... -2957821389.0+y*(7062834065.0+y*(-512359803.6+y* (108 79881.29+y*(-86327.92757+y*228.4622733))));ans2=40076544269.0+y*(745249964.8+y*(7189466.438+y*(47447.26470+y*(226 .103 0244+y*1.0))));ans=(ans1/ans2)+0.636619772*bessj0(x)*log(x);} else { Fitting function (6.5 .10) .234Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by ... America).z=8.0/x;y=z*z;xx=x-0.785398164;ans1=1.0+y*(-0 .109 8628627e-2+y*(0.2734 5104 07e-4+y*(-0.2073370639e-5+y*0.2093887211e-6)));ans2 = -0.1562499995e-1+y*(0.1430488765e-3+y*(-0.6911147651e-5+y*(0.76 2109 5161e-6+y*(-0.934945152e-7))));ans=sqrt(0.636619772/x)*(sin(xx)*ans1+z*cos(xx)*ans2);}return...

... and Pearson, C.E. 1966, Functions of a Complex Variable(New York:McGraw-Hill), pp. 220ff.6.7 Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions Many algorithms ... {y=2.0/x;ans=(exp(-x)/sqrt(x))*(1.25331414+y*(0.23498619+y*(-0.3655620e-1+y*(0.1504268e-1+y*(-0.780353e-2+y*(0.325614e-2+y*(-0.68245e-3)))))));}return ans;}236Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge ... Wiley), §6.8, p. 141. [1]6.6 Modified Bessel Functions of Integer OrderThe modified Bessel functions In(x) and Kn(x) are equivalent to the usualBessel functions Jnand Ynevaluated for purely...

... trade@cup.cam.ac.uk (outside North America).Chapter 6. Special Functions 6.0 IntroductionThere is nothing particularly special about a special function, except thatsome person in authority ... the moniker. Special functions are sometimes called higher transcendental functions( higher than what?) or functionsof mathematicalphysics (but they occur inother fields also) or functions that ... second-orderdifferential equations (but not all special functions do). One might simply call them“useful functions and let it go at that; it is surely only a matter of taste which functions we have chosen to include...

... from the combinatorial functions with integer valuedarguments, we come to the beta function,B(z, w)=B(w, z)=10tz−1(1 −t)w−1dt (6.1.8)216Chapter 6. Special Functions Sample page from ... term is parametrized by .For γ =5,N=6, and a certain set of c’s,theerrorissmallerthan||<2 10 10 .Impressed? If not, then perhaps you will be impressed by the fact that (with thesesame parameters) ... plane Re z>0.6.1 Gamma, Beta, and Related Functions 215Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge University...

... gcf");*gammcf=exp(-x+a*log(x)-(*gln))*h; Put factors in front.}216Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge ... about a+1. In these respective218Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge ... (χ2|ν)=Pν2,χ22=gammpν2,χ22(6.2.18)Q(χ2|ν)=Qν2,χ22=gammqν2,χ22(6.2.19)222Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge...

... tox/h,andx0≡n0h,x≡x−x0,andn≡n−n0,sothatF(x)≈1√πNn=−Nnodde−(x−nh)2n+ n0, (6 .10. 4)6.11 Elliptic Integrals and Jacobian Elliptic Functions 261Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) ... Mathematical Functions , Applied Mathe-matics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 byDover Publications, New York), Chapters 5 and 7.6 .10 Dawson’s IntegralDawson’s ... defined byF (x)=e−x2x0et2dt (6 .10. 1)The function can also be related to the complex error function byF (z)=i√π2e−z2[1 −erfc(−iz)] . (6 .10. 2)A remarkable approximation for F...

... significant figures near the zero.270Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge ... (0.5*C2)#define C8 (C3+C3)268Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge ... 1)K(k)=RF(0, 1 − k2, 1)(6.11.19)266Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge...

... equations. A half-dozen is typical.272Chapter 6. Special Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge ... Software,vol.7,pp. 398–403. [10] Carlson, B.C. 1978,SIAM Journal on Mathematical Analysis, vol. 9, p. 524–528. [11]Abramowitz, M., and Stegun, I.A. 1964,Handbook of Mathematical Functions , Applied ... 6.12 Hypergeometric Functions 271Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-4 3108 -5)Copyright (C) 1988-1992 by Cambridge University...