6.12 Hypergeometric Functions 271 CITED REFERENCES AND FURTHER READING: Erdelyi,A.,Magnus,W.,Oberhettinger,F.,and Tricomi,F.G.1953,Higher Transcendental Functions,Vol.Il,(New York:McGraw-Hill).[1] Gradshteyn,I.S.,and Ryzhik,I.W.1980,Table of Integrals,Series,and Products (New York: Academic Press).[2] Carlson,B.C.1977.SIAM Journal on Mathematical Analysis,vol.8,pp.231-242.[3] Carlson,B.C.1987.Mathematics of Computation,vol.49,pp.595-606 [4];1988,op.cit.,vol.51 pp.267-280[51989,op.ct,vol.53,pp.327-3336l1991,op.cit,vol.56,pp.267-280. 7] Bulirsch,R.1965,Numerische Mathematik,vol.7,pp.78-90;1965,op.cit.,vol.7,pp.353-354; 1969,op.ct.,vol.13,pp.305-315.[8] Carlson,B.C.1979,Numerische Mathematik,vol.33,pp.1-16.[9] Carlson,B.C.,and Notis,E.M.1981,ACM Transactions on Mathematical Software,vol.7, 菲 Pp.398-403.[10] 世慧三 君 Carlson.B.C.1978.SIAM Journal on Mathematical Analysis,vol.9,p.524-528.[11] ICAL 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),Chapter 17.[12] RECIPES Mathews,J.,and Walker,R.L.1970,Mathematical Methods of Physics,2nd ed.(Reading,MA: W.A.Benjamin/Addison-Wesley),pp.78-79. 立 Press. 6.12 Hypergeometric Functions 9 As was discussed in 85.14,a fast,general routine for thethe complex hyperge- IENTIFIC ometric function 2F(a,b,c;z),is difficult or impossible.The function is defined as 61 the analytic continuation of the hypergeometric series, 2(a,6,c)=1+地之 a(a+1)b(b+1)z2 c元+ cc+1)21+ (ISBN +aa+1..(a+j-10b6+1).6+j-1)z 22 十·… c(c+1)..(c+j-1) Numerica 10.621 (6.12.1) 43106 This series converges only within the unit circle z<1 (see [1]).but one's interest in the function is not confined to this region. Section 5.14 discussed the method of evaluating this function by direct path integration in the complex plane.We here merely list the routines that result. Implementation of the function hypgeo is straightforward,and is described by comments in the program.The machinery associated with Chapter 16's routine for integrating differential equations,odeint,is only minimally intrusive,and need not even be completely understood:use of odeint requires one zeroed global variable. one function call,and a prescribed format for the derivative routine hypdrv. The function hypgeo will fail,of course,for values of z too close to the singularity at 1.(If you need to approach this singularity,or the one at oo,use the "linear transformation formulas"in $15.3 of [11.)Away from z =1,and for moderate values of a,b,c,it is often remarkable how few steps are required to integrate the equations.A half-dozen is typical.6.12 Hypergeometric Functions 271 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: Erd´elyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. 1953, Higher Transcendental Functions, Vol. II, (New York: McGraw-Hill). [1] Gradshteyn, I.S., and Ryzhik, I.W. 1980, Table of Integrals, Series, and Products (New York: Academic Press). [2] Carlson, B.C. 1977, SIAM Journal on Mathematical Analysis, vol. 8, pp. 231–242. [3] Carlson, B.C. 1987, Mathematics of Computation, vol. 49, pp. 595–606 [4]; 1988, op. cit., vol. 51, pp. 267–280 [5]; 1989, op. cit., vol. 53, pp. 327–333 [6]; 1991, op. cit., vol. 56, pp. 267–280. [7] Bulirsch, R. 1965, Numerische Mathematik, vol. 7, pp. 78–90; 1965, op. cit., vol. 7, pp. 353–354; 1969, op. cit., vol. 13, pp. 305–315. [8] Carlson, B.C. 1979, Numerische Mathematik, vol. 33, pp. 1–16. [9] Carlson, B.C., and Notis, E.M. 1981, ACM Transactions on Mathematical 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 Mathe￾matics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications, New York), Chapter 17. [12] Mathews, J., and Walker, R.L. 1970, Mathematical Methods of Physics, 2nd ed. (Reading, MA: W.A. Benjamin/Addison-Wesley), pp. 78–79. 6.12 Hypergeometric Functions As was discussed in §5.14, a fast, general routine for the the complex hyperge￾ometric function 2F1(a, b, c; z), is difficult or impossible. The function is defined as the analytic continuation of the hypergeometric series, 2F1(a, b, c; z)=1+ ab c z 1! + a(a + 1)b(b + 1) c(c + 1) z2 2! + ··· + a(a + 1)...(a + j − 1)b(b + 1)...(b + j − 1) c(c + 1)...(c + j − 1) zj j! + ··· (6.12.1) This series converges only within the unit circle |z| < 1 (see [1]), but one’s interest in the function is not confined to this region. Section 5.14 discussed the method of evaluating this function by direct path integration in the complex plane. We here merely list the routines that result. Implementation of the function hypgeo is straightforward, and is described by comments in the program. The machinery associated with Chapter 16’s routine for integrating differential equations, odeint, is only minimally intrusive, and need not even be completely understood: use of odeint requires one zeroed global variable, one function call, and a prescribed format for the derivative routine hypdrv. The function hypgeo will fail, of course, for values of z too close to the singularity at 1. (If you need to approach this singularity, or the one at ∞, use the “linear transformation formulas” in §15.3 of [1].) Away from z = 1, and for moderate values of a, b, c, it is often remarkable how few steps are required to integrate the equations. A half-dozen is typical