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)


The solution of a number of problems in mathematical physics reduces to finding a function φμ(ν)) from the condition
$$\eqalign{ & {\psi _\mu }\left( x \right) = \int\limits_0^\infty {P_{i\upsilon - {1 \over 2}}^{ - \;\mu }} \left( x \right){\varphi _\mu }\left( \upsilon \right)d\upsilon , \cr & \left( {R{e_\mu } >- {1 \over 2},,\;\,x \ge 1} \right), \cr} $$
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.


Mathematical Physic Arbitrary Function Inversion Formula Consultant Bureau Legendre Function 


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Copyright information

© Consultants Bureau, New York 1970

Authors and Affiliations

  • B. G. Nikolaev

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