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 machinereadable 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 Mathematics 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 hypergeometric 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
272 Chapter 6. Special Functions #include #include "complex.h" #include "nrutil.h" #define EPS 1.0e-6 Accuracy parameter fcomplex aa,bb,cc,z0,dz; Communicates with hypdrv int kmax,kount; Used by odeint float *xp,**yp,dxsav; fcomplex hypgeo(fcomplex a,fcomplex b,fcomplex c,fcomplex z) http://www Complex hypergeometric function 2F for complex a,b,c,and z,by direct integration of the hypergeometric equation in the complex plane.The branch cut is taken to lie along the real axis,Re z 1. 83g void bsstep(float y[],float dydx[],int nv,float *xx,float htry, nted for 19881992 float eps,float yscal[],float *hdid,float *hnext, void(*derivs,)(f1oat,f1oat，f1oat)); 1-600 void hypdrv(float s,float yy[],float dyyds[); void hypser(fcomplex a,fcomplex b,fcomplex c,fcomplex z, fcomplex *series,fcomplex *deriv); to any Cambridge from NUMERICAL RECIPES I void odeint(float ystart[],int nvar,float x1,float x2 float eps,float hi,float hmin,int *nok,int *nbad, void(*derivs.)(f1oat,f1oat[☐，f1oat[☐)， void(*rkqs)(f1oat☐，float☐，int,f1oat*,f1oat,f1oat, server computer, (Nor f1oat,f1oat*,f1oat*,void(*)(f1oat,f1oat门，f1oat口))： THE int nbad,nok; America fcomplex ans,y[3]; one paper UnN电.t ART float *yyi 是 kmax=0; Programs 1f(z.r*z.r+z.1*z.1=0.0?0.5:-0.5); aa=a; Load the global variables to pass pa- OF SCIENTIFIC COMPUTING(ISBN bb=b; rameters "over the head"of odeint 18881920 cc=c; to hypdrv. dz=Csub(z,z0); hypser(aa,bb,cc,z0,&y [1],&y [2]); Get starting function and derivative. yy=vector(1,4): yy[1]y[1].r; yy[2]=y[1].1; Numerical Recipes 10-621 -43108 yy[3]=y[2].r; yy[4]=y[2].1: odeint(yy,4,0.0,1.0,EPS,0.1,0.0001,&nok,&nbad,hypdrv,bsstep); (outside The arguments to odeint are the vector of independent variables,its length,the starting and ending values of the dependent variable,the accuracy parameter,an initial guess for North Software. stepsize,a minimum stepsize,the (returned)number of good and bad steps taken,and the names of the derivative routine and the(here Bulirsch-Stoer)stepping routine. Ame y[1]=Complex(yy[1],yy [2]); free_vector(yy,1,4)i return y[1];
272 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 machinereadable 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). #include #include "complex.h" #include "nrutil.h" #define EPS 1.0e-6 Accuracy parameter. fcomplex aa,bb,cc,z0,dz; Communicates with hypdrv. int kmax,kount; Used by odeint. float *xp,**yp,dxsav; fcomplex hypgeo(fcomplex a, fcomplex b, fcomplex c, fcomplex z) Complex hypergeometric function 2F1 for complex a, b, c, and z, by direct integration of the hypergeometric equation in the complex plane. The branch cut is taken to lie along the real axis, Re z > 1. { void bsstep(float y[], float dydx[], int nv, float *xx, float htry, float eps, float yscal[], float *hdid, float *hnext, void (*derivs)(float, float [], float [])); void hypdrv(float s, float yy[], float dyyds[]); void hypser(fcomplex a, fcomplex b, fcomplex c, fcomplex z, fcomplex *series, fcomplex *deriv); void odeint(float ystart[], int nvar, float x1, float x2, float eps, float h1, float hmin, int *nok, int *nbad, void (*derivs)(float, float [], float []), void (*rkqs)(float [], float [], int, float *, float, float, float [], float *, float *, void (*)(float, float [], float []))); int nbad,nok; fcomplex ans,y[3]; float *yy; kmax=0; if (z.r*z.r+z.i*z.i = 0.0 ? 0.5 : -0.5); aa=a; Load the global variables to pass parameters “over the head” of odeint to hypdrv. bb=b; cc=c; dz=Csub(z,z0); hypser(aa,bb,cc,z0,&y[1],&y[2]); Get starting function and derivative. yy=vector(1,4); yy[1]=y[1].r; yy[2]=y[1].i; yy[3]=y[2].r; yy[4]=y[2].i; odeint(yy,4,0.0,1.0,EPS,0.1,0.0001,&nok,&nbad,hypdrv,bsstep); The arguments to odeint are the vector of independent variables, its length, the starting and ending values of the dependent variable, the accuracy parameter, an initial guess for stepsize, a minimum stepsize, the (returned) number of good and bad steps taken, and the names of the derivative routine and the (here Bulirsch-Stoer) stepping routine. y[1]=Complex(yy[1],yy[2]); free_vector(yy,1,4); return y[1]; }
6.12 Hypergeometric Functions 273 #include "complex.h" #define ONE Complex(1.0,0.0) void hypser(fcomplex a,fcomplex b,fcomplex c,fcomplex z,fcomplex *series, fcomplex *deriv) Returns the hypergeometric series 2F1 and its derivative,iterating to machine accuracy.For zr=0.0; deriv->1=0.0; fac=Complex(1.0,0.0) temp=fac; .com or call 1-800-872- (including this one) granted for i aa-a; bb=b cc=c for(n=1;nr+=fac.r; deriv->i+fac.i; fac=Cmul(fac,RCmul(1.0/n,z)); *series=Cadd(temp,fac); if (series->r =temp.r&&series->i =temp.i)return 7423 (North America 电r:1s t tusers to make one paper 1988-1992 by Cambridge University Press.Programs from NUMERICAL RECIPES IN C: THE temp=*series; aa=Cadd(aa,ONE); bb=Cadd(bb,ONE); cc=Cadd(cc,ONE); only), copy for their nrerror("convergence failure in hypser"); #include "complex.h" #define ONE Complex(1.0,0.0) email to directcustsen extern fcomplex aa,bb,cc,z0,dz; Defined in hypgeo ART OF SCIENTIFIC COMPUTING(ISBN 0-521- void hypdrv(f1oats,float yy☐，float dyyds[☐) Computes derivatives for the hypergeometric equation,see text equation (5.14.4). fcomplex z,y[3],dyds [3]; @cambridge.org To order Numerical Recipes books or personal use.Further reproduction,or 1988-1992 by Numerical Recipes y[1]=Cop1ex(yy[1],yy[2]); -431085 y[2]=Complex(yy [3],yy [4]); z=Cadd(z0,RCmul(s,dz)); dyds [1]=Cmul(y [2],dz); dyds [2]=Cmul(Csub(Cmul(Cmul(aa,bb),y[1]),Cmul(Csub(cc. Software. Cmul(Cadd(Cadd(aa,bb),ONE),z)),y[2])), Cdiv(dz,Cmul(z,Csub(ONE,z)))); dyyds[1]-dyds[1].r; (outside North America) dyyds [2]=dyds[1].i; dyyds[3]=dyds[2].r; machine dyyds [4]=dyds [2].i; 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).[1]
6.12 Hypergeometric Functions 273 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 machinereadable 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). #include "complex.h" #define ONE Complex(1.0,0.0) void hypser(fcomplex a, fcomplex b, fcomplex c, fcomplex z, fcomplex *series, fcomplex *deriv) Returns the hypergeometric series 2F1 and its derivative, iterating to machine accuracy. For |z| ≤ 1/2 convergence is quite rapid. { void nrerror(char error_text[]); int n; fcomplex aa,bb,cc,fac,temp; deriv->r=0.0; deriv->i=0.0; fac=Complex(1.0,0.0); temp=fac; aa=a; bb=b; cc=c; for (n=1;nr+=fac.r; deriv->i+=fac.i; fac=Cmul(fac,RCmul(1.0/n,z)); *series=Cadd(temp,fac); if (series->r == temp.r && series->i == temp.i) return; temp= *series; aa=Cadd(aa,ONE); bb=Cadd(bb,ONE); cc=Cadd(cc,ONE); } nrerror("convergence failure in hypser"); } #include "complex.h" #define ONE Complex(1.0,0.0) extern fcomplex aa,bb,cc,z0,dz; Defined in hypgeo. void hypdrv(float s, float yy[], float dyyds[]) Computes derivatives for the hypergeometric equation, see text equation (5.14.4). { fcomplex z,y[3],dyds[3]; y[1]=Complex(yy[1],yy[2]); y[2]=Complex(yy[3],yy[4]); z=Cadd(z0,RCmul(s,dz)); dyds[1]=Cmul(y[2],dz); dyds[2]=Cmul(Csub(Cmul(Cmul(aa,bb),y[1]),Cmul(Csub(cc, Cmul(Cadd(Cadd(aa,bb),ONE),z)),y[2])), Cdiv(dz,Cmul(z,Csub(ONE,z)))); dyyds[1]=dyds[1].r; dyyds[2]=dyds[1].i; dyyds[3]=dyds[2].r; dyyds[4]=dyds[2].i; } CITED 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 by Dover Publications, New York). [1]
