Roots of and the evaluation of integrals with cylindrical function kernels

Authors:
Srinivas Tadepalli and Costas Emmanuel Synolakis

Journal:
Quart. Appl. Math. **52** (1994), 103-112

MSC:
Primary 33C10

DOI:
https://doi.org/10.1090/qam/1262322

MathSciNet review:
MR1262322

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Abstract: An elementary proof is presented showing that the function , where is a natural number, has no zeroes in the lower and upper half-planes respectively. The roots of are given for certain values of and their locations are plotted. Cartesian maps (mappings of constant coordinate lines) of are obtained, and special features of these maps are discussed. Some integrals with cylindrical kernels involving are obtained in terms of the zeroes of .

**[1]**C. E. Synolakis,*On the roots of*, Quart. Appl. Math.**XLVI**, 105-107 (1988)**[2]**A. D. Rawlins,*Note on the roots of*, Quart. Appl. Math.**XLVII**, 323-324 (1989)**[3]**D. A. MacDonald,*The roots of*, Quart. Appl. Math.**XLVII**, 375-378 (1989)**[4]**H. Bateman,*Higher Transcendental Functions*, McGraw-Hill, New York, 1953**[5]**G. N. Watson,*A Treatise on the Theory of Bessel Functions*, Cambridge University Press, London, 1980, pp. 484-485, 502**[6]**Abramowitz and Stegun,*Handbook of Mathematical Functions*. New York: Dover Publications, New York, 1970**[7]**Stephen Wolfram,*Mathematica--A System for Doing Mathematics by Computer*, Addison-Wesley, New York, 1988

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DOI:
https://doi.org/10.1090/qam/1262322

Article copyright:
© Copyright 1994
American Mathematical Society