Special functions needed

[DavidSagan]

Some special functions that will be needed:

• cos_one = cos(x) - 1
• cosc = (1 - cos(x)) / x^2
• sqrt_one = sqrt(1+x) - 1 # Helpful identity: = x / (sqrt(1+x) + 1)
• complex error function (Faddeeva function)
• bessel functions of all sorts.
• elliptic integrals # sim_utils/special_functions/elliptic_integral_mod.f90 has code for this.
• generalized complete elliptic integral # See sim_utils/special_functions/gen_complete_elliptic.f90

PTC might have code for some of this.

[mattsignorelli]

We have sinc, though I had to add a constant factor of pi (sinc(x) = sin(x*pi)/x)) because that is what Julia base calculates. We also have complex erfc and erf so for the Fadeeva function I just need to add the scaling factor

[DavidSagan]

Julia defines sinc as the normalized version sing(x*pi)/(x*pi). I prefer the unnormalized version but oh well.

[mattsignorelli]

We could have the unnormalized version as "sa" for now https://en.wikipedia.org/wiki/Sinc_function I found on some forums / github issues that people want to change this but unfortunately it being a breaking change would have to wait until Julia 2.0

[DavidSagan]

How about sinc_u or sincu?