The generalized integro-exponential function

Author:
M. S. Milgram

Journal:
Math. Comp. **44** (1985), 443-458

MSC:
Primary 33A70; Secondary 65D15

DOI:
https://doi.org/10.1090/S0025-5718-1985-0777276-4

MathSciNet review:
777276

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Abstract | References | Similar Articles | Additional Information

Abstract: The generalized integro-exponential 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 first-order functions are reported in another section.

**[1]**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****[2]**I. K. Abu-Shumays,*Transcendental Functions Generalizing the Exponential Integrals*, Northwestern University (unpublished) report COO-2280-6, 1973.**[3]**Donald E. Amos,*Computation of exponential integrals*, ACM Trans. Math. Software**6**(1980), no. 3, 365–377. MR**585343**, https://doi.org/10.1145/355900.355908**[4]**L. Berg, "On the estimation of the remainder term in the asymptotic expansion of the exponential integral,"*Computing*, v. 18, 1977, pp. 361-363.**[5]**B. S. Berger,*Tables of Zeros and Weights for Gauss-Laguerre Quadrature to*24S*for n*= 400, 500, 600, Dept. of Mechanical Engineering, Univ. of Maryland, College Park, MD. (Unpublished report.)**[6]**W. F. Breig and A. L. Crosbie,*Numerical computation of a generalized exponential integral function*, Math. Comp.**28**(1974), 575–579; addendum, ibid. 28 (1974), no. 126, loose microfiche suppl. C9–C14. MR**0341811**, https://doi.org/10.1090/S0025-5718-1974-0341811-3**[7]**R. P. Brent, "A FORTRAN multiple-precision arithmetic package,"*ACM Trans. Math. Software*, v. 4, 1978, pp. 57-70;*ibid.*pp. 71-81.**[8]**S. Chandrasekhar,*Radiative transfer*, Dover Publications, Inc., New York, 1960. MR**0111583****[9]**W. J. Cody and Henry C. Thacher Jr.,*Rational Chebyshev approximations for the exponential integral 𝐸₁(𝑥)*, Math. Comp.**22**(1968), 641–649. MR**0226823**, https://doi.org/10.1090/S0025-5718-1968-0226823-X**[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. 456-497.**[12]**H. C. van de Hulst,*Scattering in a planetary atmosphere*, Astrophys. J.**107**(1948), 220–246. MR**0026409**, https://doi.org/10.1086/145005**[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. 72-75.**[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 AECL-4210, 1973.**[16]**C. Kaplan,*On a Generalization of the Exponential Integral*, Aerospace Research Lab. Report ARL-69-0120, 1969;*On Some Functions Related to the Exponential Integrals*, Aerospace Research Lab. Report ARL-70-0097, 1970;*Asymptotic and Series Expansion of the Generalized Exponential Integrals*, Air Force Office of Scientific Research Interim Report AFOSR-TR-72-2147, 1972.**[17]**J. Le Caine,*A Table of Integrals Involving the functions*, National Research Council of Canada Report NRC-1553, 1945, Section 1.6.**[18]**Y. L. Luke,*The Special Functions and Their Approximations*, Academic Press, New York, 1969.**[19]**A. S. Meligy and E. M. El Gazzy,*On the function ∫_{𝑧}^{∞}𝑒^{-𝑡}𝑡⁻ⁿ𝑑𝑡*, Proc. Cambridge Philos. Soc.**59**(1963), 735–737. MR**0153882****[20]**M. S. Milgram, "Approximate solutions to the half-space integral transport equation near a plane boundary,"*Canad. J. Phys.*, v. 58, 1980, pp. 1291-1310.**[21]**M. S. Milgram, "Some properties of the solution to the integral transport equation in semi-infinite plane geometry,"*Atomkernenergie*, v. 38, 1981, pp. 99-106.**[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. 207-217.**[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. 41-52.**[24]**Mohamed Adel Sharaf,*On the Λ-transform of the exponential integrals*, Astrophys. and Space Sci.**60**(1979), no. 1, 199–212. MR**523233**, https://doi.org/10.1007/BF00648252**[25]**R. R. Sharma and Bahman Zohuri,*A general method for an accurate evaluation of exponential integrals 𝐸₁(𝑥),𝑥>0*, J. Computational Phys.**25**(1977), no. 2, 199–204. MR**0474705****[26]**A. Stankiewicz,*The generalized integro-exponential functions*, Acta Univ. Wratislav.**188**(1973), 11–42. MR**0321265****[27]**Irene A. Stegun and Ruth Zucker,*Automatic computing methods for special functions. II. The exponential integral 𝐸_{𝑛}(𝑥)*, J. Res. Nat. Bur. Standards Sect. B**78B**(1974), 199–216. MR**0362844****[28]**H. Strubbe, "Development of the SCHOONSCHIP program,"*Comput. Phys. Comm.*, v. 18, 1979, pp. 1-5.**[29]**Riho Terras,*The determination of incomplete gamma functions through analytic integration*, J. Comput. Phys.**31**(1979), no. 1, 146–151. MR**531128**, https://doi.org/10.1016/0021-9991(79)90066-4

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DOI:
https://doi.org/10.1090/S0025-5718-1985-0777276-4

Article copyright:
© Copyright 1985
American Mathematical Society