On Spheroidal Functions

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概要

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• In this paper are investigated some mathematical properties of spheroidal functions satisfying the differential equation of the form: d/(dz){(1-z^2)(dZ)/(dz)}+(λ+κ^2z^2)Z=0. One solution pe_n(z) regular at |z|=1, which has already been known as special case of the generalized spheroidal function pe^m_n(z), is developed into a Legendre expansion and its coefficients are obtained explicitly for several cases. The solution of the first kind Re_n(z) and that of the second kind Se_n(z), which are valid especially when z>>1, are defined by the definite integrals and are also expressed in series forms in terms of the modified Bessel functions. An alternative expression for Se_n(z), which is conveniently used even when z is not so large, is also defined in like manner as in the case of the derivation of the modified Mathieu function FEK_n(z) or GEK_n(z). Further, the asymptotic behaviours of these functions Re_n(z) and Se_n(z) are obtained. Detailed calculations are developed for the case of a prolate spheroid. For the case of an oblate spheroid some essential parts only are given In Appendix.

• 社団法人日本物理学会の論文
• 1955-02-05

著者

• Aoi Tyusei

  Osaka Institute Of Technology

関連論文

• Numerical Analysis for Flows Around an Infinitely Long Square Cylinder Between Two Parallel Walls
• On Spheroidal Functions
• The Steady Flow of Viscous Fluid past a Fixed Spheroidal Obstacle at Small Reynolds Numbers

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