Digital Library of Mathematical Functions
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33 Coulomb FunctionsVariables ρ,η

§33.7 Integral Representations

33.7.1 F(η,ρ)=ρ+12ρ-(πη/2)|Γ(+1+η)|01-2ρtt+η(1-t)-ηt,
33.7.2 H-(η,ρ)=-ρρ-(2+1)!C(η)0-tt-η(t+2ρ)+ηt,
33.7.3 H-(η,ρ)=--πηρ+1(2+1)!C(η)0(exp(-(ρtanht-2ηt))(cosht)2+2+(1+t2)exp(-ρt+2ηarctant))t,
33.7.4 H+(η,ρ)=-πηρ+1(2+1)!C(η)-1--ρt(1-t)-η(1+t)+ηt.

Noninteger powers in (33.7.1)–(33.7.4) and the arctangent assume their principal values (§§4.2(i), 4.2(iv), 4.23(ii)).