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The iterated equation of generalized axially symmetric potential theory, VI General solutions

Published online by Cambridge University Press:  09 April 2009

J. C. Burns
J. C. Burns
Affiliation:
Faculty of Military Studies, University of New South Wales, Royal Military College Duntroon, A.C.T. 2600, Australia
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The iterated equation of generalized axially symmetric potential theory [1] is the equation where, in its simplest form, the operator Lk is defined by the function f f(x, y) being assumed to belong to the class of C2n functions and the parameter l to take any real value. In appropriate circumstances, which will be indicated later, the operator can be generalized but as this can be done without altering the methods used, the operator will be taken in the form where r, θ are polar coordinates such that x = r cos θ, y = r sin μ = cosθ.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 3 , November 1974 , pp. 318 - 327
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Weinstein, A., ‘Generalized axially symmetric potential theory’, Bull. Amer. Math. Soc. 59 (1953), 20–28.CrossRefGoogle Scholar
[2]Burns, J. C., ‘The iterated equation of generalized axially symmetric potential theory, I. Particular solutions’, J. Austral. Math. Soc. 7 (1967), 263–276.CrossRefGoogle Scholar
[3]Weinstein, A., ‘On a class of partial differential equations of even order’, Ann. Mat. Pura Appl. 39 (1955), 245–254.CrossRefGoogle Scholar
[4]Payne, L. E., ‘Representation formulas for solutions of a class of partial differential equation’, J. Math. and Phys. 38 (1959), 145–149.CrossRefGoogle Scholar
[5]Burns, J. C., ‘The iterated equation of generalized axially symmetric potential theory, II. General solutions of Weinstein' type’, J. Austral. Math. Soc. 7 (1967), 277–289.CrossRefGoogle Scholar