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I've used the Cauchy-Riemann equations to find the analytic function:


But I'm having a slight rearranging problem and need to write it in terms of $z$, where $z=x+iy$.

Any suggestions would be much appreciated

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up vote 5 down vote accepted

If you know in advance that the function is analytic (which you do if you've checked that it satisfies the Cauchy–Riemann equations), you can obtain its expression in terms of $z$ simply by setting $x=z$ and $y=0$. This works because of the uniqueness theorem: two analytic functions which agree on the real axis must agree everywhere.

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Given a function in terms of $x$ and $y$ you can introduce formally new variables $z:=x+iy$ and $\bar z:=x-iy$, i.e., $x=(z+\bar z)/2$, $y=(z-\bar z)/(2i)$. If the resulting expression in $z$ and $\bar z$ does not contain the variable $\bar z$ your function is actually an analytic function of $z$.

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Hint:It helps noticing that $z^2=x^2-y^2+2ixy$.

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In addition to the formula for $z^2$, note that $iz = ix - y$.

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