# Bessel function of second with complex order

**URL:** <https://discourse.julialang.org/t/bessel-function-of-second-with-complex-order/20443>\
**Category:** General Usage\
**Created:** [February 4, 2019, 6:48pm UTC](https://discourse.julialang.org/t/bessel-function-of-second-with-complex-order/20443 "2019-02-04T18:48:36Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![Tanjona\_Radonirina\_R](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tanjona_radonirina_r/32/7010_2.png) [@Tanjona\_Radonirina\_R](https://discourse.julialang.org/u/Tanjona_Radonirina_R)\
**Post date:** [February 4, 2019, 6:48pm UTC](https://discourse.julialang.org/t/bessel-function-of-second-with-complex-order/20443/1 "2019-02-04T18:48:36Z")

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Is there a math library in Julia which can compute a modified Bessel function of the second kind with complex order and argument? I am only aware of `SpecialFunctions.jl` which contains a method to compute a modified Bessel function of the second kind with a complex argument of real order.

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**Author:** ![simonbyrne](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/simonbyrne/32/19_2.png) [@simonbyrne](https://discourse.julialang.org/u/simonbyrne)\
**Post date:** [February 4, 2019, 7:22pm UTC](https://discourse.julialang.org/t/bessel-function-of-second-with-complex-order/20443/2 "2019-02-04T19:22:00Z")

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Your best bet might be [Nemo.jl](http://nemocas.github.io/Nemo.jl/latest/).

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**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [February 4, 2019, 8:58pm UTC](https://discourse.julialang.org/t/bessel-function-of-second-with-complex-order/20443/3 "2019-02-04T20:58:40Z")

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I had a student working on a related problem at one point — my recollection is that we couldn’t find any free code for this, and very little literature. He developed some prototypes, but as I recall he ran into tricky accuracy issues with the second-kind Bessel functions for complex arguments that were close to integers. We also wanted the derivative with respect to the order, however, so maybe the problem was easier if you don’t need that. I’ll bug him to see whether he has any notes he can post.

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**Author:** ![JeffreySarnoff](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jeffreysarnoff/32/1980_2.png) [@JeffreySarnoff](https://discourse.julialang.org/u/JeffreySarnoff)\
**Post date:** [February 5, 2019, 12:08am UTC](https://discourse.julialang.org/t/bessel-function-of-second-with-complex-order/20443/4 "2019-02-05T00:08:12Z")

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If you want to use a package that is in development,  
`add SpecialFunctions` and  
`add ArbNumerics` (must use version 0.3.5+, just merged).

```julia
using ArbNumerics
import SpecialFunctions: besselj, bessely

for F in (:besselj, :bessely)
   @eval function $F(order::Complex{T}, argument::Complex{T}) where {T}
       result = $F(ArbComplex(order), ArbComplex(argument))
       return Complex{T}(result)
   end
end

```

then

```julia
order = 0.0 + 0.5im; argument = 1.0 + 1.0im;

besselj(order, argument)
# 0.6415000256096106 - 0.2518459064184279im

bessely(order, argument)
# 0.12578491928293956 + 0.17534238733104368im

```

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<div class="post-metadata">

**Author:** ![Tanjona\_Radonirina\_R](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tanjona_radonirina_r/32/7010_2.png) [@Tanjona\_Radonirina\_R](https://discourse.julialang.org/u/Tanjona_Radonirina_R)\
**Post date:** [February 7, 2019, 9:01am UTC](https://discourse.julialang.org/t/bessel-function-of-second-with-complex-order/20443/5 "2019-02-07T09:01:37Z")

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Thank you all for your response. I finally ended up constructing my own function using `SpecialFunctions.jl`. The lines of codes below seem to be performing very well.

```julia
using SpecialFunctions

# Define the modified Bessel Function of the second kind with complex order & argument
function CBesselK(nu, z)

	Knu = exp(lgamma(nu)) * (z/2)^(-nu)
	Kmu = exp(lgamma(-nu)) * (z/2)^nu

	bound = 20

	precision = 1e-10

	s = 0

	if (abs(z) < 10)
		for i = 0:bound
			var = i
			facto = factorial(i)

			r = 0.5 * Knu * ((z/2)^(2*var) * exp(-lgamma(1+var-nu)+lgamma(1-nu))/facto) + 0.5 * Kmu * ((z/2)^(2*var) * exp(-lgamma(1+var+nu)+lgamma(1+nu))/facto)
			s+=r

			if ((abs(real(r/s))<precision)&&(abs(imag(r/s))<precision))
				break
			end
		end
	else
		for i = 0:bound
			var = i
			facto = factorial(i)

			r = sqrt(pi/(2*z)) * exp(-z) * exp(lgamma(0.5+var+nu)+lgamma(0.5+var-nu)-lgamma(0.5+nu)-lgamma(0.5-nu)/facto) * (-1/2*z)^var
			s+=r

			if ((abs(real(r/s))<precision)&&(abs(imag(r/s))<precision))
				break
			end
		end
	end

	return s

end

```

Comparing the performance with **Mathematica** :

```julia
# CBesselK
order = exp(1-3im)
arg = (4+6im)*exp(2*im*pi)
CBesselK(order, arg)
0.011090297922744 - 0.005887726316203im

# Mathematica 
N[BesselK[Exp[1 - 3 I], (4 + 6 I)*Exp[2*I*Pi]], 15]
0.011090297922749 - 0.005887726316251 I

```

Cheers!