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The application of the Hankel transformation to spherically symmetric coverage problems

Published online by Cambridge University Press:  14 July 2016

Gerhard Schroeter
Gerhard Schroeter*
Affiliation:
Industrieanlagen-Betriebsgesellschaft mbH
*
∗Postal address: Industrieanlagen-Betriebsgesellschaft mbH, Einsteinstrasse 20, 8012 Ottobrunn, W. Germany.
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Abstract

We derive an integral expression for the expected coverage of an n-dimensional target when the value density function of the target, the damage function, and the distribution of the damage function center around the target center are spherically symmetric. This integral expression is transformed by means of Parseval's theorem for Hankel transforms to an integral of the product of the Hankel transforms of order ½n − 1 of all underlying functions.

Keywords

HANKEL TRANSFORMEXPECTED COVERAGERANDOM WALKCHARACTERISTIC FUNCTIONSYSTEM BIASWEAPON SYSTEMS ANALYSIS
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 4 , December 1980 , pp. 1121 - 1126
Copyright
Copyright © Applied Probability Trust 

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References

Davies, B. (1978) Integral Transforms and their Applications. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Erdelyi, A. (1954) Tables of Integral Transforms, Vol. II. McGraw Hill, New York.Google Scholar
Guenther, W. C. (1961) On the probability of capturing a randomly selected point in three dimensions. SIAM Rev. 3, 247–251.CrossRefGoogle Scholar
Guenther, W. C. (1964) A generalization of the integral of the circular coverage function. Amer. Math. Monthly 71, 278–283.Google Scholar
Guenther, W. C. and Terragno, P. J. (1964) A review of the literature on a class of coverage problems. Ann. Math. Statist. 35, 232–260.Google Scholar
Kluyver, J. C. (1906) A local probability problem. Proc. Sect. Sci., K. Akad. van Wet. Amsterdam 8, 341–350.Google Scholar
Lord, R. D. (1954a) The use of the Hankel transform in statistics, I. Biometrika 41, 44–55.CrossRefGoogle Scholar
Lord, R. D. (1954b) The use of the Hankel transform in statistics, II. Biometrika 41, 344–350.Google Scholar
Rayleigh, Lord (1919) On the problem of random vibrations, and of random flights in one, two, or three dimensions. Phil. Mag. (6) 37, 321–347.Google Scholar
Schroeter, G. (1976) Probability of kill and expected destroyed value if the underlying distributions are rotationally symmetric. Operat. Res. 24, 586–591.CrossRefGoogle Scholar
Schroeter, G. (1977) The application of a Hankel transformation to a class of circularly symmetric coverage problems. Operat. Res. 25, 675–687.CrossRefGoogle Scholar
Schroeter, G. (1980) Expected coverage of a randomly located target by multiple independent salvos. Operat. Res. 28.Google Scholar
Smith, E. H. and Stone, D. E. (1961) A note on the expected coverage of one circle by another, the case of offset aim. SIAM Rev. 3, 51–53.Google Scholar
Sneddon, I. N. (1951) Fourier Transforms. McGraw Hill, New York.Google Scholar
Sneddon, I. N. (1972) The Use of Integral Transforms. McGraw Hill, New York.Google Scholar
Watson, G. N. (1958) A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.Google Scholar