Find v such that f(z) = u+iv is analytic

[Vitani11]

Homework Statement

Find v such that f(z) = u+iv is analytic.

Homework Equations

du/dx = dv/dy and dv/dx=-du/dy

The Attempt at a Solution

I'm not sure what I'm supposed to do. I think I need to find U in order to find V because if a function is analytic it satisfies the Cauchy Riemann equations. I tried to play around with cauchy riemann equations to get dv in terms of everything else but that's not helping. Can I also use the Laplace equation to solve this? This is so general that I know it's probably simple, but at the same time it makes it hard to understand what my answer is supposed to look like. What should my approach be?

 

[mfb]

You can find v as function of u.
It is a bit confusing to use the same symbols for the complex components of the argument (first equation) and the complex components of the function ("relevant equations").

 

[nuuskur]

Suppose you are given $u:=u(x,y)$ and you know $f$ is differentiable so it satisfies C-R, in particular,$\frac{du}{dx} = \frac{dv}{dy}$.
Therefore $v = \int u_x dy + g(x)$, where $u_x$ is the partial derivative w.r.t $x$. You add a function of $x$ because when you differentiate w.r.t $y$, the $g(x)$ vanishes.

 

[mfb]

That's not how I would interpret the u and v given in the first post.

@Vitani11: Is that the full and exact problem statement? "Find v such that f(z) = u+iv is analytic." Nothing else given?

 

[nuuskur]

The problem has to start from somewhere. We must either have $u$ or $v$. I agree that the problem is ambiguous.

 

[Vitani11]

That is 100% all that was given, promise. Anyway thank you I now think I can solve this.