which is related to the gamma function by B(z, w) = hence #include float beta(float z, float w) Returns the value of the beta function B(z, w). { float gammln(float xx); return exp(gammln(z)+gammln(w)-gammln(z+w)); } | 216 Chapter 6. Special Functions which is related to the gamma function by B z w r z w hence include float beta float z float w Returns the value of the beta function B z w . float gammln float xx return exp gammln z gammln w -gammln z w CITED REFERENCES AND FURTHER READING Abramowitz M. and Stegun . 1964 Handbook of Mathematical Functions Applied Mathematics Series Volume 55 Washington National Bureau of Standards reprinted 1968 by Dover Publications New York Chapter 6. Lanczos C. 1964 SIAM Journal on Numerical Analysis ser. B vol. 1 pp. 86-96. 1 Incomplete Gamma Function Error Function Chi-Square Probability Function Cumulative Poisson Function The incomplete gamma function is defined by P a x Y a xX - e-tta-1dt a 0 v 7 r a V a J0 till It has the limiting values P a 0 0 and P a 1 1 The incomplete gamma function P a x is monotonic and for a greater than one or so rises from near-zero to near-unity in a range of x centered on about a 1 and of width about Pa see Figure . The complement of P a x is also confusingly called an incomplete gamma function Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 Q a x 1 P a x Í e tta 1dt a 0 r a r a Jx Incomplete Gamma Function 217 It has the limiting values Q a 0 1 and Q a 1 0 The notations P a x y a x and T a x are standard the notation Q a x is specific to this book. There is a series development for y a x as follows y a X e-xxa X-----r a -----xn 6 2 5 Y a x e X r a 1 n X n n v 7 One does not actually need to compute a new T a 1 n for each n one rather uses equation and the previous coefficient. A continued fraction development for T a x is r a x e-xxa a 2- -2-------------------- x 0 7 7 x 1 x 1 x 7 77 7 It is computationally better to use the even part of which converges twice as fast see z z T 1 1 1 - a 2 2 - a T a x e-xxM--------------------------------- x 0 x 1 a x 3 a x 5 a j Sample page from .
