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I am trying to revise for an exam and I cannot get my head around what this question is asking me:

Characterise those holomorphic functions $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $\hat f$ is holomorphic, where $\hat f$ is the function sending $x+iy$ to $v(x,y)+iu(x,y)$ for $x,y$ are real numbers and $u$ is the real part and $v$ is he imaginary part.

The question is very vague and i'm not really sure what its asking me to do and has a few different acceptable answers according to the feedback.

Any help would be great

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    "For which holomorphic $f \colon \mathbb{C} \to \mathbb{C}$, where $f(x+iy) = u(x,y) + iv(x,y)$ with $u,v$ real valued functions, is the function $\hat{f}\colon (x+iy) \mapsto v(x,y) + iu(x,y)$ also holomorphic?" Note that $\hat{f}(z) = i\cdot \overline{f(z)}$. – 2017-02-02
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    And recall that functions with constant real part or imaginary part are constant. – 2017-02-02
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    You can apply the CR equations: We get both $u_x=v_y, u_y=-v_x$ and $v_x=u_y, v_y=-u_x.$ – 2017-02-02
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    thanks for the responses, but what does the question mean 'characterise'? Could I use the CR equations to show that $\hat f$ is differentiable everywhere (as shown above as they are satisfied) and these alone will answer the question? – 2017-02-02
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    "characterize" means find an equivalent description, for example consider this: Characterize the continuous functions on $[a,b]$ such that $\int_0^1|f| = 0.$ Answer: $f\equiv 0.$ – 2017-02-02

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Assume such $\hat{f}$ is holomorphic. Consider the entire function $f\hat{f}=i\bar{f}f=i|f|^2=i(u^2+v^2)$. We know that the real part is 0 and since $\mathbb{C}$ is a domain, that $f \hat{f}$ is constant. In other words, we have $u^2+v^2=C$. But then there also holds that $|f|^2=C$ and therefore $f$ has to be constant according to Liouvilles theorem. So the only function statisfying this relation are the constant ones.