How would I go about using the Cauchy-Riemann equations to show that is an entire function for any n using polar coordinates?
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Well for starters, what is f(z) in polar form? What is the polar form of the Cauchy-Riemann equations?
Cauchy-Riemann is and I split f(z) up like so: I'm stuck on what to do next... am I allowed to differentiate as I would over R?
Originally Posted by jgv115 Cauchy-Riemann is and I split f(z) up like so: I'm stuck on what to do next... am I allowed to differentiate as I would over R? Just take your derivatives. $u_r(r,\theta)=n r^{n-1}\cos(n\theta)$ $u_\theta(r,\theta) = -n r^n \sin(n\theta)$ etc. and then check that the polar CR eqs are satisfied.
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