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

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 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    Mehler, F. G., Math. Ann., Vol. 18 (1881).Google Scholar
  2. 2.
    Fok, V. A., Dokl. Akad. Nauk SSSR, Vol. 39, No. 7 (1943).Google Scholar
  3. 3.
    Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge Univ. Press, Cambridge (1931)Google Scholar
  4. 4.
    Weyl, H., J. rein. angew. Math., Vol. 141 (1912).Google Scholar
  5. 5.
    Titchmarsh, E. C., Eigenfunction Expansions Associated with Second-Order Differential Equations, Clarendon Press, Oxford (1950).Google Scholar
  6. 6.
    Kodaira, K., Am. J. Math., Vol. 72 (1950).Google Scholar
  7. 7.
    Lebedev, N. N., Certain Integral Transforms in Mathematical Physics, Dissertation [in Russian], (1951).Google Scholar

Copyright information

© Consultants Bureau, New York 1970

Authors and Affiliations

There are no affiliations available

Personalised recommendations