Roots of $J_\gamma (z)\pm iJ_{\gamma +1}(z)=0$ 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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: An elementary proof is presented showing that the function $f\left ( z \right ) = \\ {J_\gamma }\left ( z \right ) \pm i{J_{\gamma + 1}} \left ( z \right )$, where $\gamma$ is a natural number, has no zeroes in the lower and upper half-planes respectively. The roots of $f\left ( z \right )$ are given for certain values of $\gamma$ and their locations are plotted. Cartesian maps (mappings of constant coordinate lines) of $f\left ( z \right )$ are obtained, and special features of these maps are discussed. Some integrals with cylindrical kernels involving $f\left ( z \right )$ are obtained in terms of the zeroes of ${J_\gamma }\left ( z \right )$.
C. E. Synolakis, On the roots of $f\left ( z \right ) = {j_0}\left ( z \right ) - i{j_1}\left ( z \right )$, Quart. Appl. Math. XLVI, 105–107 (1988)
A. D. Rawlins, Note on the roots of $f\left ( z \right ) = {j_0}\left ( z \right ) - i{j_1}\left ( z \right )$, Quart. Appl. Math. XLVII, 323–324 (1989)
D. A. MacDonald, The roots of $f\left ( z \right ) = {j_0}\left ( z \right ) - i{j_1}\left ( z \right )$, Quart. Appl. Math. XLVII, 375–378 (1989)
H. Bateman, Higher Transcendental Functions, McGraw-Hill, New York, 1953
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, London, 1980, pp. 484–485, 502
Abramowitz and Stegun, Handbook of Mathematical Functions. New York: Dover Publications, New York, 1970
Stephen Wolfram, Mathematica—A System for Doing Mathematics by Computer, Addison-Wesley, New York, 1988
C. E. Synolakis, On the roots of $f\left ( z \right ) = {j_0}\left ( z \right ) - i{j_1}\left ( z \right )$, Quart. Appl. Math. XLVI, 105–107 (1988)
A. D. Rawlins, Note on the roots of $f\left ( z \right ) = {j_0}\left ( z \right ) - i{j_1}\left ( z \right )$, Quart. Appl. Math. XLVII, 323–324 (1989)
D. A. MacDonald, The roots of $f\left ( z \right ) = {j_0}\left ( z \right ) - i{j_1}\left ( z \right )$, Quart. Appl. Math. XLVII, 375–378 (1989)
H. Bateman, Higher Transcendental Functions, McGraw-Hill, New York, 1953
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, London, 1980, pp. 484–485, 502
Abramowitz and Stegun, Handbook of Mathematical Functions. New York: Dover Publications, New York, 1970
Stephen Wolfram, Mathematica—A System for Doing Mathematics by Computer, Addison-Wesley, New York, 1988
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
33C10
Retrieve articles in all journals
with MSC:
33C10
Additional Information
Article copyright:
© Copyright 1994
American Mathematical Society