Rearranging Cauchy Riemann equations

I've made use of the Cauchy - Riemann formulas to locate the analytic function:


But I'm having a mild rearranging trouble and also require to write it in regards to $z$, where $z=x+iy$.

Any kind of pointers would certainly be much valued

2022-06-07 14:34:13
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Answers: 4

If you recognize beforehand that the function is analytic (which you do if you've examined that it pleases the Cauchy-- Riemann formulas), you can get its expression in regards to $z$ merely by establishing $x=z$ and also $y=0$. This functions as a result of the individuality theory: 2 analytic features which settle on the actual axis has to concur almost everywhere.

2022-06-07 15:06:26

Hint : It aids seeing that $z^2=x^2-y^2+2ixy$.

2022-06-07 14:56:50

Given a function in regards to $x$ and also $y$ you can present officially new variables $z:=x+iy$ and also $\bar z:=x-iy$, i.e., $x=(z+\bar z)/2$, $y=(z-\bar z)/(2i)$. If the resulting expression in $z$ and also $\bar z$ does not have the variable $\bar z$ your function is in fact an analytic function of $z$.

2022-06-07 14:56:24

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

2022-06-07 14:56:19