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

Abstract

The solution of a number of problems in mathematical physics reduces to finding a function φ_μ(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 ²_p (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.