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14 Legendre and Related FunctionsReal Arguments

§14.19 Toroidal (or Ring) Functions

Contents

§14.19(i) Introduction

When ν=n-12, n=0,1,2,, μ, and x(1,) solutions of (14.2.2) are known as toroidal or ring functions. This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates (η,θ,ϕ), which are related to Cartesian coordinates (x,y,z) by

14.19.1 x =csinhηcosϕcoshη-cosθ,
y =csinhηsinϕcoshη-cosθ,
z =csinθcoshη-cosθ,

where the constant c is a scaling factor. Most required properties of toroidal functions come directly from the results for Pνμ(x) and Qνμ(x). In particular, for μ=0 and ν=±12 see §14.5(v).

§14.19(ii) Hypergeometric Representations

With F as in §14.3 and ξ>0,

14.19.2 Pν-12μ(coshξ)=Γ(12-μ)π1/2(1-e-2ξ)μe(ν+(1/2))ξF(12-μ,12+ν-μ;1-2μ;1-e-2ξ),
μ12,32,52,.
14.19.3 Qν-12μ(coshξ)=π1/2(1-e-2ξ)μe(ν+(1/2))ξF(μ+12,ν+μ+12;ν+1;e-2ξ).

§14.19(iii) Integral Representations

With ξ>0,

14.19.4 Pn-12m(coshξ) =Γ(n+m+12)(sinhξ)m2mπ1/2Γ(n-m+12)Γ(m+12)0π(sinϕ)2m(coshξ+cosϕsinhξ)n+m+(1/2)dϕ,
14.19.5 Qn-12m(coshξ) =Γ(n+12)Γ(n+m+12)Γ(n-m+12)0cosh(mt)(coshξ+coshtsinhξ)n+(1/2)dt,
m<n+12.

§14.19(iv) Sums

With ξ>0,

14.19.6 Q-12μ(coshξ)+2n=1Γ(μ+n+12)Γ(μ+12)Qn-12μ(coshξ)cos(nϕ)=(12π)1/2(sinhξ)μ(coshξ-cosϕ)μ+(1/2),
μ>-12.

§14.19(v) Whipple’s Formula for Toroidal Functions