Derivative and Exp
SCIENTIA
Series A: Mathematical Sciences, Vol. 22 (2012), 129-151
Universidad T´cnica Federico Santa Mar´ e ıa
Valpara´
ıso, Chile
ISSN 0716-8446 c Universidad T´cnica Federico Santa Mar´ 2012 e ıa
The integrals in Gradshteyn and Ryzhik.
Part 22: Bessel-K functions
Larry Glasser, Karen T. Kohl, Christoph Koutschan, Victor H. Moll, and Armin Straub
Abstract. The table of Gradshteyn and Ryzhik contains many integrals that can be evaluated using the modified Bessel function. Some examples are discussed and typos in the table are corrected.
1. Introduction
This paper is part of the collection initiated in [12], aiming to evaluate the entries in [8] and to provide some context. This table contains a large variety of entries involving the Bessel functions. The goal of the current work is to evaluate some entries in [8] where the integrand is an elementary function and the result involves the so-called modified Bessel function of the second kind, denoted by Kν (x). Other types of integrals containing Bessel functions will appear in a future publication. This introduction contains a brief description of the Bessel functions. The reader is referred to [3, 13, 14, 15] for more information about this class of functions.
The Bessel differential equation du d2 u
+x
+ (x2 − ν 2 )u = 0 dx2 dx arises from the solution of Laplace’s equation
(1.1)
x2
∂2U
∂2U
∂2U
+
+
=0
∂x2
∂y 2
∂z 2 in spherical or cylindrical coordinates. The method of Frobenius shows that, for any ν ∈ R, the function
(1.2)
(1.3)
Jν (x) =
∞
k=0
(−1)k
Γ(ν + 1 + k) k!
x
2
ν+2k
2000 Mathematics Subject Classification. Primary 33.
Key words and phrases. Integrals, Bessel functions, automatic proofs, method of brackets.
The third author was partially supported by the DDMF project of the Microsoft Research-INRIA
Joint Centre. The fourth author wishes to acknowledge the partial support of NSF-DMS 0713836.
1
2
L. GLASSER, K. KOHL, C.
Series A: Mathematical Sciences, Vol. 22 (2012), 129-151
Universidad T´cnica Federico Santa Mar´ e ıa
Valpara´
ıso, Chile
ISSN 0716-8446 c Universidad T´cnica Federico Santa Mar´ 2012 e ıa
The integrals in Gradshteyn and Ryzhik.
Part 22: Bessel-K functions
Larry Glasser, Karen T. Kohl, Christoph Koutschan, Victor H. Moll, and Armin Straub
Abstract. The table of Gradshteyn and Ryzhik contains many integrals that can be evaluated using the modified Bessel function. Some examples are discussed and typos in the table are corrected.
1. Introduction
This paper is part of the collection initiated in [12], aiming to evaluate the entries in [8] and to provide some context. This table contains a large variety of entries involving the Bessel functions. The goal of the current work is to evaluate some entries in [8] where the integrand is an elementary function and the result involves the so-called modified Bessel function of the second kind, denoted by Kν (x). Other types of integrals containing Bessel functions will appear in a future publication. This introduction contains a brief description of the Bessel functions. The reader is referred to [3, 13, 14, 15] for more information about this class of functions.
The Bessel differential equation du d2 u
+x
+ (x2 − ν 2 )u = 0 dx2 dx arises from the solution of Laplace’s equation
(1.1)
x2
∂2U
∂2U
∂2U
+
+
=0
∂x2
∂y 2
∂z 2 in spherical or cylindrical coordinates. The method of Frobenius shows that, for any ν ∈ R, the function
(1.2)
(1.3)
Jν (x) =
∞
k=0
(−1)k
Γ(ν + 1 + k) k!
x
2
ν+2k
2000 Mathematics Subject Classification. Primary 33.
Key words and phrases. Integrals, Bessel functions, automatic proofs, method of brackets.
The third author was partially supported by the DDMF project of the Microsoft Research-INRIA
Joint Centre. The fourth author wishes to acknowledge the partial support of NSF-DMS 0713836.
1
2
L. GLASSER, K. KOHL, C.
References: Symb. Comp., 10:571–591, 1990. [4] F. Chyzak. An extension of Zeilberger’s fast algorithm to general holonomic functions. Discrete Mathematics, 217(1-3):115–134, 2000. [5] I. Gonzalez and V. Moll. Definite integrals by the method of brackets. Part 1. Adv. Appl. Math., 45:50–73, 2010. [9] K. Kohl. Algorithmic methods for definite integration. PhD thesis, Tulane University, 2011. [10] C. Koutschan. HolonomicFunctions (user’s guide). Technical Report 10-01, RISC Report Series, Johannes Kepler University Linz, 2010. http://www.risc.uni-linz.ac.at/research/combinat/ software/HolonomicFunctions/. Scientia, 14:1–6, 2007. [14] G. N. Watson. A treatise on the Theory of Bessel Functions. Cambridge University Press, 1966. [15] E. T. Whittaker and G. N. Watson. Modern Analysis. Cambridge University Press, 1962. Received May 15, 2012, revised October 20, 2012 ´