The definition of _digamma(z::ComplexOrReal{Float64}) contains the formula
in which
_cotpi is defined as
|
_cotpi(x) = @static VERSION >= v"1.10.0-DEV.525" ? inv(tanpi(x)) : cospi(x) / sinpi(x) |
For large complex arguments,
_cotpi produces
NaN since
cospi and
sinpi produce
Inf. For example,
julia> cospi(500 * exp(im * 3pi/4))
Inf - Inf*im
julia> sinpi(500 * exp(im * 3pi/4))
Inf + Inf*im
julia> cospi(500 * exp(im * 3pi/4)) / sinpi(500 * exp(im * 3pi/4))
NaN + NaN*im
Nevertheless, the ordinary cot gives the correct answer
julia> cot(pi * 500 * exp(im * 3pi/4))
0.0 - 1.0im
A way to avoid NaN is to define cotpi as in mpmath
https://github.com/mpmath/mpmath/blob/b600dbcabf4b7406a61e82b9e607f754a9f12ff9/mpmath/libfp.py#L153-L161
The definition of
_digamma(z::ComplexOrReal{Float64})contains the formulaSpecialFunctions.jl/src/gamma.jl
Line 29 in cf35c58
in which
_cotpiis defined asSpecialFunctions.jl/src/gamma.jl
Line 63 in cf35c58
For large complex arguments,
_cotpiproducesNaNsincecospiandsinpiproduceInf. For example,Nevertheless, the ordinary
cotgives the correct answerA way to avoid
NaNis to definecotpias inmpmathhttps://github.com/mpmath/mpmath/blob/b600dbcabf4b7406a61e82b9e607f754a9f12ff9/mpmath/libfp.py#L153-L161