The generalized integroexponential function
Author:
M. S. Milgram
Journal:
Math. Comp. 44 (1985), 443458
MSC:
Primary 33A70; Secondary 65D15
MathSciNet review:
777276
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Abstract: The generalized integroexponential function is defined in terms of the exponential integral (incomplete gamma function) and its derivatives with respect to order. A compendium of analytic results is given in one section. Rational minimax approximations sufficient to permit the computation of the first six firstorder functions are reported in another section.
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Computational Phys. 25 (1977), no. 2, 199–204.
MR
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A.
Stankiewicz, The generalized integroexponential functions,
Acta Univ. Wratislav. 188 (1973), 11–42. MR 0321265
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Irene
A. Stegun and Ruth
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Bur. Standards Sect. B 78B (1974), 199–216. MR 0362844
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(81d:65010), http://dx.doi.org/10.1016/00219991(79)900664
 [1]
 M. Abramowitz & I. Stegun, Handbook of Mathematical Functions, Chapter 5, U. S. National Bureau of Standards, Washington, DC, 1964. MR 0167642 (29:4914)
 [2]
 I. K. AbuShumays, Transcendental Functions Generalizing the Exponential Integrals, Northwestern University (unpublished) report COO22806, 1973.
 [3]
 D. E. Amos, "Computation of exponential integrals," ACM Trans. Math. Software, v. 6, 1980, pp. 365377. MR 585343 (82b:65011)
 [4]
 L. Berg, "On the estimation of the remainder term in the asymptotic expansion of the exponential integral," Computing, v. 18, 1977, pp. 361363.
 [5]
 B. S. Berger, Tables of Zeros and Weights for GaussLaguerre Quadrature to 24S for n = 400, 500, 600, Dept. of Mechanical Engineering, Univ. of Maryland, College Park, MD. (Unpublished report.)
 [6]
 W. F. Breig & A. L. Crosbie, "Numerical computation of a generalized exponential integral function," Math. Comp., v. 28, 1974, pp. 575579. MR 0341811 (49:6557)
 [7]
 R. P. Brent, "A FORTRAN multipleprecision arithmetic package," ACM Trans. Math. Software, v. 4, 1978, pp. 5770; ibid. pp. 7181.
 [8]
 S. Chandrasekhar, Radiative Transfer, Dover, New York, 1960. MR 0111583 (22:2446)
 [9]
 W. J. Cody & H. C. Thacher, Jr., "Rational Chebyshev approximations for the exponential integral ." Math. Comp., v. 22, 1968, pp. 641649. MR 0226823 (37:2410)
 [10]
 DCADRE, IMSL Library, 6th Floor, GNB Bldg., 7500 Bellaire Blvd., Houston, TX.
 [11]
 E. A. Gussman, "Modification to the weighting function method for the calculation of Fraunhofer lines in solar and stellar spectra," Z. Astrophys., v. 65, 1967, pp. 456497.
 [12]
 H. C. Van de Hulst, "Scattering in a planetary atmosphere," Astrophys. J., v. 107, 1948, pp. 220246. MR 0026409 (10:151a)
 [13]
 H. C. Van de Hulst, Multiple Light Scattering, Vol. 1, Academic Press, New York, 1980.
 [14]
 D. R. Jeng, E. J. Lee & K. J. de Witt, "Exponential integral kernels appearing in the radiative heat flux," Indian J. Tech., v. 13, 1975, pp. 7275.
 [15]
 J. H. Johnson & J. M. Blair, REMES2: A FORTRAN Programme to Calculate Rational Minimax Approximations to a Given Function, Atomic Energy of Canada Ltd., Report AECL4210, 1973.
 [16]
 C. Kaplan, On a Generalization of the Exponential Integral, Aerospace Research Lab. Report ARL690120, 1969; On Some Functions Related to the Exponential Integrals, Aerospace Research Lab. Report ARL700097, 1970; Asymptotic and Series Expansion of the Generalized Exponential Integrals, Air Force Office of Scientific Research Interim Report AFOSRTR722147, 1972.
 [17]
 J. Le Caine, A Table of Integrals Involving the functions , National Research Council of Canada Report NRC1553, 1945, Section 1.6.
 [18]
 Y. L. Luke, The Special Functions and Their Approximations, Academic Press, New York, 1969.
 [19]
 A. S. Meligy & E. M. El Gazzy, "On the function ," Proc. Cambridge Philos. Soc., v. 59, 1963, pp. 735737. MR 0153882 (27:3843)
 [20]
 M. S. Milgram, "Approximate solutions to the halfspace integral transport equation near a plane boundary," Canad. J. Phys., v. 58, 1980, pp. 12911310.
 [21]
 M. S. Milgram, "Some properties of the solution to the integral transport equation in semiinfinite plane geometry," Atomkernenergie, v. 38, 1981, pp. 99106.
 [22]
 M. S. Milgram, "Solution of the integral transport equation across a place boundary," Proc. ANS/ENS International Topical Meeting on Advances in Mathematical Methods for the Solution of Nuclear Engineering Problems, Munich, FDR, (1981), pp. 207217.
 [23]
 W. Neuhaus & S. Schottlander, "The development of Aireys converging factors of the exponential integral to a representation with remainder term," Computing, v. 15, 1975, pp. 4152.
 [24]
 M. A. Sharaf, "On the transform of the exponential integrals," Astrophys. and Space Sci., v. 60, 1979, pp. 199212. MR 523233 (81g:65186)
 [25]
 R. R. Sharma & B. Zohuri, "A general method for an accurate evaluation of exponential integrals , ," J. Comput. Phys., v. 25, 1977, pp. 199204. MR 0474705 (57:14339)
 [26]
 A. Stankiewicz, "The generalized integroexponential functions," Acta Univ. Wratislav., No. 188, 1973, pp. 1142. MR 0321265 (47:9798)
 [27]
 I. A. Stegun & R. Zucker, "Automatic computing methods for special functions. Part II," J. Res. Nat. Bur. Standards, v. 78B, 1974, pp. 199218. MR 0362844 (50:15282)
 [28]
 H. Strubbe, "Development of the SCHOONSCHIP program," Comput. Phys. Comm., v. 18, 1979, pp. 15.
 [29]
 R. Terras, "The determination of incomplete gamma functions through analytic integration," J. Comput. Phys., v. 31, 1979, pp. 146151. MR 531128 (81d:65010)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507772764
PII:
S 00255718(1985)07772764
Article copyright:
© Copyright 1985
American Mathematical Society
