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33 Coulomb FunctionsVariables ρ,η

§33.7 Integral Representations

33.7.1 F(η,ρ)=ρ+12eiρ-(πη/2)|Γ(+1+iη)|01e-2iρtt+iη(1-t)-iηdt,
33.7.2 H-(η,ρ)=e-iρρ-(2+1)!C(η)0e-tt-iη(t+2iρ)+iηdt,
33.7.3 H-(η,ρ)=-ie-πηρ+1(2+1)!C(η)0(exp(-i(ρtanht-2ηt))(cosht)2+2+i(1+t2)exp(-ρt+2ηarctant))dt,
33.7.4 H+(η,ρ)=ie-πηρ+1(2+1)!C(η)-1-ie-iρt(1-t)-iη(1+t)+iηdt.

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)).