The Expansion of an Arbitrary Function in Terms of an Integral of Associated Legendre Functions of First Kind with Complex Index

  • B. G. Nikolaev
Part of the Seminars in Mathematics book series (SM, volume 9)

Abstract

The solution of a number of problems in mathematical physics reduces to finding a function φμ(ν)) from the condition
$$\eqalign{ & {\psi _\mu }\left( x \right) = \int\limits_0^\infty {P_{i\upsilon - {1 \over 2}}^{ - \;\mu }} \left( x \right){\varphi _\mu }\left( \upsilon \right)d\upsilon , \cr & \left( {R{e_\mu } >- {1 \over 2},,\;\,x \ge 1} \right), \cr} $$
(1)
where P p 2 (x) is an associated Legendre function of first kind and ψ μ (x) is a function defined on I ≤ x < ∞ . In other words, the problem reduces to inverting the integral (1) and expanding a given function ψ μ (x) in an integral of associated Legendre functions. The necessity of such transformations arises, for example, in problems related to the use of toroidal and ellipsoidal coordinates.

Keywords

Mathematical Physic Arbitrary Function Inversion Formula Consultant Bureau Legendre Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Consultants Bureau, New York 1970

Authors and Affiliations

  • B. G. Nikolaev

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