Calculate imaginary part if real part is following

[Chromosom]

Homework Statement

$$f:\ v(x,y)=4xy+2x$$

The task is to calculate the imaginary part.

Homework Equations

The Attempt at a Solution

I have no idea what to do because in my opinion u(x,y) can be anything. For example: $$f(x,y)=4xy+2x+(3x-4y)\text i$$. But I must be wrong. I would appreciate if you just tell me what to do or where is my mistake, because I want to solve it alone :)

 

[jimmycricket]

Read some information on the Cauchy Riemann equations

 

[vanhees71]

Well, usually one writes
$$f(z)=f(x+\mathrm{i} y)=u(x,y)+\mathrm{i} v(x,y).$$
If you now assume that $f$ is a analytic function, you have the Cauchy-Riemann differential equations, relating the real and imaginary parts
$$\partial_x u=\partial_y v, \quad \partial_y u=-\partial_x v.$$
So if you have given $u$ (real part) you can determine the imaginary part (up to a constant) and vice versa.

Read the question carefully again, because it seems as if the imaginary part is given and you look for the real part and not the other way around.