Are there any Haskell libraries for integrating complex functions?

Question

1. How to numerically integrate complex, complex-valued functions in Haskell?
2. Are there any existing libraries for it? numeric-tools operates only on reals.

I am aware that on complex plane there's only line integrals, so the interface I am interested in is something like this:

i = integrate f x a b precision

to calculate integral along straight line from a to b of function f on point x. i, x, a, b are all of Complex Double or better Num a => Complex a type.

Please... :)

Answer 1

You can make something like this yourself. Suppose you have a function realIntegrate of type (Double -> Double) -> (Double,Double) -> Double, taking a function and a tuple containing the lower and upper bounds, returning the result to some fixed precision. You could define realIntegrate f (lo,hi) = quadRomberg defQuad (lo,hi) f using numeric-tools, for example.

Then we can make your desired function as follows - I'm ignoring the precision for now (and I don't understand what your x parameter is for!):

integrate :: (Complex Double -> Complex Double) -> Complex Double -> Complex Double -> Complex Double
integrate f a b = r :+ i where
    r = realIntegrate realF (0,1)
    i = realIntegrate imagF (0,1)
    realF t = realPart (f (interpolate t)) -- or realF = realPart . f . interpolate
    imagF t = imagPart (f (interpolate t))
    interpolate t = a + (t :+ 0) * (b - a)

So we express the path from a to b as a function on the real interval from 0 to 1 by linear interpolation, take the value of f along that path, integrate the real and imaginary parts separately (I don't know if this can give numerically badly behaving results, though) and reassemble them into the final answer.

I haven't tested this code as I don't have numeric-tools installed, but at least it typechecks :-)