c6-2 (779501), страница 2
Текст из файла (страница 2)
It is defined as the probability that the number of Poissonrandom events occurring will be between 0 and k − 1 inclusive, if the expected meannumber is x. It has the limiting valuesPx (< 1) = e−xPx (< ∞) = 1(6.2.14)Its relation to the incomplete gamma function is simplyPx (< k) = Q(k, x) = gammq (k, x)(6.2.15)Chi-Square Probability FunctionP (χ2 |ν) is defined as the probability that the observed chi-square for a correctmodel should be less than a value χ2 . (We will discuss the use of this function inChapter 15.) Its complement Q(χ2 |ν) is the probability that the observed chi-squarewill exceed the value χ2 by chance even for a correct model. In both cases ν is aninteger, the number of degrees of freedom.
The functions have the limiting valuesP (0|ν) = 0Q(0|ν) = 1P (∞|ν) = 1Q(∞|ν) = 0(6.2.16)(6.2.17)and the following relation to the incomplete gamma functions,ν,P (χ |ν) = P2ν,Q(χ2 |ν) = Q22χ2ν χ2= gammp,22 2χ2ν χ2= gammq,22 2(6.2.18)(6.2.19)Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).z=fabs(x);t=1.0/(1.0+0.5*z);ans=t*exp(-z*z-1.26551223+t*(1.00002368+t*(0.37409196+t*(0.09678418+t*(-0.18628806+t*(0.27886807+t*(-1.13520398+t*(1.48851587+t*(-0.82215223+t*0.17087277)))))))));return x >= 0.0 ? ans : 2.0-ans;222Chapter 6.Special FunctionsCITED REFERENCES AND FURTHER READING:Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 byDover Publications, New York), Chapters 6, 7, and 26.Pearson, K.
(ed.) 1951, Tables of the Incomplete Gamma Function (Cambridge: CambridgeUniversity Press).The standard definition of the exponential integral isZ∞En (x) =1e−xtdt,tnx > 0,n = 0, 1, . . .(6.3.1)The function defined by the principal value of the integralZEi(x) = −∞−xe−tdt =tZx−∞etdt,tx>0(6.3.2)is also called an exponential integral.
Note that Ei(−x) is related to −E1 (x) byanalytic continuation.The function En (x) is a special case of the incomplete gamma functionEn (x) = xn−1 Γ(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:n1 n+1 21···(6.3.4)En (x) = e−xx+ 1+ x+ 1+ x+We use it in its more rapidly converging even form,1·n2(n + 1)1−x···En (x) = ex+n− x+n+2− 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 representationEn (x) =(−x)n−1[− ln x + ψ(n)] −(n − 1)!∞Xm=0m6=n−1(−x)m(m − n + 1)m!(6.3.6)The quantity ψ(n) here is the digamma function, given for integer arguments byψ(1) = −γ,ψ(n) = −γ +n−1Xm=11m(6.3.7)Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).6.3 Exponential Integrals.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
