On the integral
Author:
K. S. Kölbig
Journal:
Math. Comp. 41 (1983), 171182
MSC:
Primary 33A70; Secondary 33A15, 44A10, 80A99
MathSciNet review:
701632
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 halfinteger values of . Explicit expressions for these integrals for small values of m and , are also presented.
 [1]
Handbook of mathematical functions, with formulas, graphs and
mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun.
Fifth printing, with corrections. National Bureau of Standards Applied
Mathematics Series, Vol. 55, National Bureau of Standards, Washington,
D.C., (for sale by the Superintendent of Documents, U.S. Government
Printing Office, Washington, D.C., 20402), 1966. MR 0208798
(34 #8607)
 [2]
W. BörschSupan, "On the evaluation of the function for real values of ," J. Res. Nat. Bur. Standards, v. 65B, 1961, pp. 245250.
 [3]
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
(82c:65003), http://dx.doi.org/10.1090/S00255718198005835228
 [4]
H.
S. Carslaw and J.
C. Jaeger, Conduction of Heat in Solids, Oxford, at the
Clarendon Press, 1947. MR 0022294
(9,188a)
 [5]
Louis
Comtet, Advanced combinatorics, Revised and enlarged edition,
D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite
expansions. MR
0460128 (57 #124)
 [6]
I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980.
 [7]
G. H. Hardy, Ramanujan, Cambridge Univ. Press, Cambridge; Macmillan, New York, 1940, pp. 195198.
 [8]
A. C. Hearn, REDUCE Users Manual, 2nd ed., Univ. of Utah, Salt Lake City, Utah, 1973.
 [9]
Peter
Henrici, Applied and computational complex analysis,
WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1974.
Volume 1: Power series—integration—conformal
mapping—location of zeros; Pure and Applied Mathematics. MR 0372162
(51 #8378)
 [10]
L. Landau, "On the energy loss of fast particles by ionization," J. Physics, v. 8, 1944, pp. 201205.
 [11]
N.
W. McLachlan, P.
Humbert, and L.
Poli, Supplément au formulaire pour le calcul
symbolique, Mémor. Sci. Math. no. 113, GauthierVillars, Paris,
1950 (French). MR 0038478
(12,408b)
 [12]
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 0083911
(18,780d)
 [13]
V. E. Wood, "Some integrals of Ramanujan and related contour integrals," Math. Comp., v. 20, 1966, pp. 424429.
 [1]
 M. Abramowitz & I. A. STEGUN (EDITORS), Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, 5th printing with corrections, Nat. Bur. Standards Appl. Math. Series, No. 55, U.S. Government Printing Office, Washington, D. C., 1966. MR 0208798 (34:8607)
 [2]
 W. BörschSupan, "On the evaluation of the function for real values of ," J. Res. Nat. Bur. Standards, v. 65B, 1961, pp. 245250.
 [3]
 J. A. Carlson Gallas, "Table erratum 572," Math. Comp., v. 35, 1980, p. 1444. MR 583522 (82c:65003)
 [4]
 H. S. Carslaw & J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford Univ. Press, New York, 1959, pp. 339341. MR 0022294 (9:188a)
 [5]
 L. Comtet, Advanced Combinatorics, Reidel, Dordrecht and Boston, 1974, pp. 216217. MR 0460128 (57:124)
 [6]
 I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980.
 [7]
 G. H. Hardy, Ramanujan, Cambridge Univ. Press, Cambridge; Macmillan, New York, 1940, pp. 195198.
 [8]
 A. C. Hearn, REDUCE Users Manual, 2nd ed., Univ. of Utah, Salt Lake City, Utah, 1973.
 [9]
 P. Henrici, Applied and Computational Complex Analysis, vol. 1, Wiley, New York, 1974. MR 0372162 (51:8378)
 [10]
 L. Landau, "On the energy loss of fast particles by ionization," J. Physics, v. 8, 1944, pp. 201205.
 [11]
 N. W. McLachlan, P. Humbert & L. Poli, "Supplément au formulaire pour le calcul symbolique," Mémor. Sci. Math., Fasc. 113, GauthierVillars, Paris, 1950. MR 0038478 (12:408b)
 [12]
 R. H. Ritchie & A. Y. Sakakura, "Asymptotic expansions of solutions of the heat conduction equation in internally bounded cylindrical geometry," J. Appl. Phys., v. 27, 1956, pp. 14531459. MR 0083911 (18:780d)
 [13]
 V. E. Wood, "Some integrals of Ramanujan and related contour integrals," Math. Comp., v. 20, 1966, pp. 424429.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
33A70,
33A15,
44A10,
80A99
Retrieve articles in all journals
with MSC:
33A70,
33A15,
44A10,
80A99
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198307016321
PII:
S 00255718(1983)07016321
Article copyright:
© Copyright 1983
American Mathematical Society
