Special Functions part 13

For integer order, however, the routines in this section (and §) are simpler and faster. Their only drawback is that they are limited by the precision of the underlying rational approximations. For full double precision, it is best to work with the routines for fractional order in §. For any real ν, the Bessel function Jν (x) can be defined by the series representation Jν (x) = 1 x 2 ν ∞ k=0 | Hypergeometric Functions 271 CITED REFERENCES AND FURTHER READING Erdelyi A. Magnus W. Oberhettinger F. and Tricomi . 1953 Higher Transcendental Functions Vol. II New York McGraw-Hill . 1 Gradshteyn . and Ryzhik . 1980 Table of Integrals Series and Products New York Academic Press . 2 Carlson . 1977 SIAM Journal on Mathematical Analysis vol. 8 pp. 231-242. 3 Carlson . 1987 Mathematics ofComputation 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 . 1979 Numerische Mathematik vol. 33 pp. 1-16. 9 Carlson . and Notis . 1981 ACM Transactions on Mathematical Software vol. 7 pp. 398-403. 10 Carlson . 1978 SIAMJournal on Mathematical Analysis vol. 9 p. 524-528. 11 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 17. 12 Mathews J. and Walker . 1970 Mathematical Methods of Physics 2nd ed. Reading MA . Benjamin Addison-Wesley pp. 78-79. Hypergeometric Functions As was discussed in 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 . ab z a a 1 b b 1 z2 1 - 1 c c 1 2 a a 1 . a j 1 b b 1 . b j 1 zj c c 1 . c j 1 j 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 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 .