How to numerically calculate value of complex function given derivative of this function in Haskell?

Question

Given:

1. Haskell
2. Complex-valued function df/dz defined on complex plane U (let's say z is a Complex Double).
3. Point z1 from the U on which df/dz is defined.

Question:

How to get value of function f(z) for which df/dz is a derivative, in point z1? I. e. how to restore value of original function given only it's derivative, assuming complex plane?

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This question is somewhat related to my previous question about calculating integrals of complex functions, but they are about different things. Here I am interested not in calculating some scalar value, but in finding the origin function given it's derivative. It's essentially calculating the indefinite integral of this derivative.

Answer 1

(Runge–Kutta in Haskell)

You can use some numeric solver like Runge-Kutta

-- define 4th order Runge-Kutta map (RK4) 
rk4 :: Floating a => (a -> a) -> a -> a -> a
rk4 f h x = x + (1/6) * (k1 + 2*k2 + 2*k3 + k4)
            where k1 = h * f (x)
                  k2 = h * f (x + 0.5*k1)
                  k3 = h * f (x + 0.5*k2)
                  k4 = h * f (x + k3)

in that case function signature is Floating but you can use RealFloat instead (you can use runge-kutta in complex).

Complete example:

Prelude> import Data.Complex
Prelude Data.Complex> let rk4 f h x = x + (1/6) * (k1 + 2*k2 + 2*k3 + k4) where {k1 = h * f(x);k2 = h * f (x + 0.5*k1);k3 = h * f (x + 0.5*k2);k4 = h * f (x + k3)}
Prelude Data.Complex> let f z = 2 * z
Prelude Data.Complex> rk4 f (0.1 :+ 0.2) (0.3 :+ 1.2)
(-0.2334199999999999) :+ 1.4925599999999999
Prelude Data.Complex>

On the other hand, @leftaroundabout suggest extend that behavior to VectorSpace (great! of course! :D )