Expression for function f:(a+bi) -> (a^2+b)+ai

Question

Let $f:(a+bi) \to (a^2+b)+ai$

Is there an expression for f such that does not use functions Re and Im?

You can try substituting in $a = \frac{1}{2}(z + \bar z)$ and $b = \frac{1}{2i}(z - \bar z)$ and seeing if your expression simplifies.
Following Rahul's suggestion, $f(z) = \frac14 z^2 + \frac12 |z|^2 + \frac14 \overline{z}^2 + i\overline{z}$.
Seems like using $\overline{z}$ is cheating if we are not allowed Im.
What is an "expression"? If you allow $\overline{z}$, then you can express Re and Im in terms of it as Rahul writes. If you allow $|z|$ then $|z|^2/z=\overline{z}$ for all nonzero $z$. If you limit yourself to analytic functions, then the answer is "no" as lhf explains.

lhf · Answer 1 · 2010-12-21 17:41:17Z

up vote 8 down vote

No expression as an analytic function because it does not satisfy the Cauchy-Riemann equations.

answered Dec 21 '10 at 17:41

lhf
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