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## Problem

Consider the complex number in polar form , and the complex function in polar form .

Show that the Cauchy-Riemann equations in polar form are

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## Solution

It is important to know the Cartesian representation of the complex number and the Cauchy-Riemann equations first.

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Now use the multivariate product rule to take the the derivatives of , , , and .

For the function :

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Since , , , and , we get

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We repeat the same calculations for .

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Now use the Cauchy-Riemann equations in Cartesian form and make the right substitutions.

Therefore,

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