222 Chapter 6.Special Functions CITED REFERENCES AND FURTHER READING: Abramowitz,M.,and Stegun,I.A.1964,Handbook of Mathematical Functions,Applied Mathe- matics Series,Volume 55(Washington:National Bureau of Standards;reprinted 1968 by Dover Publications,New York),Chapters 6,7,and 26. Pearson,K.(ed.)1951,Tables of the Incomplete Gamma Function (Cambridge:Cambridge University Press). 6.3 Exponential Integrals The standard definition of the exponential integral is 鱼 En()= d tn x>0,n=0,1,. (6.3.1) The function defined by the principal value of the integral ⊙ 令 =-= x>0 (6.3.2) Press. is also called an exponential integral.Note that Ei(-)is related to -E1(x)by analytic continuation. 9 Program The function En(x)is a special case of the incomplete gamma function En(x)=r"-1I(1-n,x) (6.3.3) OF SCIENTIFIC( We can therefore use a similar strategy for evaluating it.The continued fraction- just equation(6.2.6)rewritten-converges for all z >0: En(x)=e-x 1n1n+12 COMPUTING (ISBN x+1+x+1++… (6.3.4) 1920 Numerical 10-521 We use it in its more rapidly converging even form, 43108 1 1.n 2(n+1) 、 En(x）=e- (6.3.5) (outside Recipes x+n-x+n+2-x+n+4- The continued fraction only really converges fast enough to be useful for z1. Software. For 0<z1,we can use the series representation E.(回= a2-lnx+(n]、 (-x)m (6.3.6) m-n+1)m! 020 The quantity (n)here is the digamma function,given for integer arguments by -1 (1)=-7, =+ (6.3.7)222 Chapter 6. Special Functions Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). CITED REFERENCES AND FURTHER READING: Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathe￾matics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications, New York), Chapters 6, 7, and 26. Pearson, K. (ed.) 1951, Tables of the Incomplete Gamma Function (Cambridge: Cambridge University Press). 6.3 Exponential Integrals The standard definition of the exponential integral is En(x) =
∞ 1 e−xt tn dt, x > 0, n = 0, 1,... (6.3.1) The function defined by the principal value of the integral Ei(x) = −
∞ −x e−t t dt =
x −∞ et t dt, x > 0 (6.3.2) is also called an exponential integral. Note that Ei(−x) is related to −E1(x) by analytic continuation. The function En(x) is a special case of the incomplete gamma function En(x) = xn−1Γ(1 − n, x) (6.3.3) We can therefore use a similar strategy for evaluating it. The continued fraction — just equation (6.2.6) rewritten — converges for all x > 0: En(x) = e−x 1 x + n 1 + 1 x + n + 1 1 + 2 x + ··· (6.3.4) We use it in its more rapidly converging even form, En(x) = e−x 1 x + n − 1 · n x + n + 2 − 2(n + 1) x + n + 4 − ··· (6.3.5) The continued fraction only really converges fast enough to be useful for x >∼ 1. For 0 < x <∼ 1, we can use the series representation En(x) = (−x)n−1 (n − 1)! [− ln x + ψ(n)] − ∞ m=0 m=n−1 (−x)m (m − n + 1)m! (6.3.6) The quantity ψ(n) here is the digamma function, given for integer arguments by ψ(1) = −γ, ψ(n) = −γ + n −1 m=1 1 m (6.3.7)