2F1 sometimes errors due to Gamma function:

[dlfivefifty]

n = 5;
a ,b = 0.1, 0.2
 _₂F₁(-n,n+a+b+1,a+1,z+eps()*im)

[dlfivefifty]

[1] nan_dom_err at ./math.jl:300 [inlined]
 [2] gamma(::Float64) at ./special/gamma.jl:3
 [3] reconeβ₀(::Float64, ::Float64, ::Float64, ::Complex{Float64}, ::Int64, ::Float64) at /Users/solver/.julia/v0.6/SingularIntegralEquations/src/HypergeometricFunctions/specialfunctions.jl:215
 [4] Bone(::Float64, ::Float64, ::Float64, ::Complex{Float64}, ::Int64, ::Float64) at /Users/solver/.julia/v0.6/SingularIntegralEquations/src/HypergeometricFunctions/specialfunctions.jl:229
 [5] _₂F₁one(::Float64, ::Float64, ::Float64, ::Complex{Float64}) at /Users/solver/.julia/v0.6/SingularIntegralEquations/src/HypergeometricFunctions/specialfunctions.jl:245
 [6] _₂F₁general(::Float64, ::Float64, ::Float64, ::Complex{Float64}) at /Users/solver/.julia/v0.6/SingularIntegralEquations/src/HypergeometricFunctions/Gauss.jl:95
 [7] _₂F₁general(::Float64, ::Float64, ::Float64, ::Complex{Float64}) at /Users/solver/.julia/v0.6/SingularIntegralEquations/src/HypergeometricFunctions/Gauss.jl:81
 [8] _₂F₁general(::Int64, ::Float64, ::Float64, ::Complex{Float64}) at /Users/solver/.julia/v0.6/SingularIntegralEquations/src/HypergeometricFunctions/Gauss.jl:82
 [9] _₂F₁(::Int64, ::Float64, ::Float64, ::Complex{Float64}) at /Users/solver/.julia/v0.6/SingularIntegralEquations/src/HypergeometricFunctions/Gauss.jl:51

[dlfivefifty]

Also, shouldn't we calculate 2F1_regularized to avoid issues with c being a negative integer?

[MikaelSlevinsky]

I thought c wouldn't be a negative integer for Jacobi functions, but probably the regularized version is more generic and robust. Cheers, Mikael On Dec 5, 2017, at 4:37 AM, Sheehan Olver <[email protected]<mailto:[email protected]>> wrote: Also, shouldn't we calculate 2F1_regularized to avoid issues with c being a negative integer? — You are receiving this because you are subscribed to this thread. Reply to this email directly, view it on GitHub<#105 (comment)>, or mute the thread<https://github.com/notifications/unsubscribe-auth/AHzBpbyQfoT_IzT6Yw94qwW5ogkG2VHuks5s9Rz4gaJpZM4Q2ErZ>.

[dlfivefifty]

According to F.W.J. Olver, one should always calculate the regularized version:

instead of F(a,b;c;z), the function on the left-hand side of Equation (9.02) on page 159 of my book is a much better choice. This is because it is an entire function of each of its parameters a, b, c and hence avoids computational problems that stem from the fact that as a function of c, F(a,b;c;z) has an infinite set of poles on the real axis. Similarly for other hgf”s.