On the integral $\smallint ^{\infty }_{0}e^{-\mu t}t^{\nu -1} \textrm {log}^{m} t dt$
HTML articles powered by AMS MathViewer
- by K. S. Kölbig PDF
- Math. Comp. 41 (1983), 171-182 Request permission
Abstract:
A recurrence relation is given for the integral in the title. Formulae which allow easy evaluation by formula manipulation are given for integer and half-integer values of $\nu$. Explicit expressions for these integrals for small values of m and $\nu = n$, $\nu = n + \frac {1}{2}$ are also presented.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 20402, 1966. Fifth printing, with corrections; National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents). MR 0208798 W. Börsch-Supan, "On the evaluation of the function $\phi (\lambda ) = \frac {1}{{2\pi i}}\;\smallint _{\sigma - i\infty }^{\sigma + i\infty }{e^{u\ln u + \lambda u}} du$ for real values of $\lambda$," J. Res. Nat. Bur. Standards, v. 65B, 1961, pp. 245-250.
- Jason A. Carlson Gallas, Table errata: Table of integrals, series and products [English translation of the fourth Russian edition, Academic Press, New York, 1965; MR 33 #5952] by I. S. Gradšteĭn and I. M. Ryžik, Math. Comp. 35 (1980), no. 152, 1444. MR 583522, DOI 10.1090/S0025-5718-1980-0583522-8
- H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947. MR 0022294
- Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128, DOI 10.1007/978-94-010-2196-8 I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980. G. H. Hardy, Ramanujan, Cambridge Univ. Press, Cambridge; Macmillan, New York, 1940, pp. 195-198. A. C. Hearn, REDUCE Users Manual, 2nd ed., Univ. of Utah, Salt Lake City, Utah, 1973.
- Peter Henrici, Applied and computational complex analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR 0372162 L. Landau, "On the energy loss of fast particles by ionization," J. Physics, v. 8, 1944, pp. 201-205.
- N. W. McLachlan, P. Humbert, and L. Poli, Supplément au formulaire pour le calcul symbolique, Mémor. Sci. Math., no. 113, Gauthier-Villars, Paris, 1950 (French). MR 0038478
- R. H. Ritchie and A. Y. Sakakura, Asymptotic expansions of solutions of the heat conduction equation in internally bounded cylindrical geometry, J. Appl. Phys. 27 (1956), 1453–1459. MR 83911, DOI 10.1063/1.1722288 V. E. Wood, "Some integrals of Ramanujan and related contour integrals," Math. Comp., v. 20, 1966, pp. 424-429.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Math. Comp. 41 (1983), 171-182
- MSC: Primary 33A70; Secondary 33A15, 44A10, 80A99
- DOI: https://doi.org/10.1090/S0025-5718-1983-0701632-1
- MathSciNet review: 701632