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An Integral Formula For Qn (cos θ)

Published online by Cambridge University Press:  20 January 2009

E. T. Copson
Affiliation:
University College, Dundee.
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The function

is, as is well-known, a general solution of Laplace's equation of degree −1 in (x, y, z). In 1926* I proved that the particular solution r−1Q0 (z/r) cannot be represented in this form whereas the solution r−1Q0 (y/r) can. In the present note I find a very simple expression for the latter solution in the form (1.1), and I deduce from it an apparently new integral formula for Qn (cos θ).

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1945

References

page 81 note * Proc. Edin. Math. Soc. (1), 44 (1926), 2225. A slight correction is needed in that paper. I quoted an asymptotic formula for , due to Watson ; but as in that paper m is a positive integer, the expression given is the actual value of without the order term.Google Scholar

page 82 note * Hobson, , Spherical and Ellipsoidal Harmonics (Cambridge, 1931), 70.Google Scholar