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14 Legendre and Related FunctionsReal Arguments

§14.19 Toroidal (or Ring) Functions

  1. §14.19(i) Introduction
  2. §14.19(ii) Hypergeometric Representations
  3. §14.19(iii) Integral Representations
  4. §14.19(iv) Sums
  5. §14.19(v) Whipple’s Formula for Toroidal Functions

§14.19(i) Introduction

When ν=n12, 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 =csinhηcosϕcoshηcosθ,
y =csinhηsinϕcoshηcosθ,
z =csinθcoshηcosθ,

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 Pν12μ(coshξ)=Γ(12μ)π1/2(1e2ξ)μe(ν+(1/2))ξ𝐅(12μ,12+νμ;12μ;1e2ξ),
14.19.3 𝑸ν12μ(coshξ)=π1/2(1e2ξ)μe(ν+(1/2))ξ𝐅(μ+12,ν+μ+12;ν+1;e2ξ).

§14.19(iii) Integral Representations

With ξ>0,

14.19.4 Pn12m(coshξ) =Γ(n+m+12)(sinhξ)m2mπ1/2Γ(nm+12)Γ(m+12)0π(sinϕ)2m(coshξ+cosϕsinhξ)n+m+(1/2)dϕ,
14.19.5 𝑸n12m(coshξ) =Γ(n+12)Γ(n+m+12)Γ(nm+12)0cosh(mt)(coshξ+coshtsinhξ)n+(1/2)dt,

§14.19(iv) Sums

With ξ>0,

14.19.6 𝑸12μ(coshξ)+2n=1Γ(μ+n+12)Γ(μ+12)𝑸n12μ(coshξ)cos(nϕ)=(12π)1/2(sinhξ)μ(coshξcosϕ)μ+(1/2),

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