Prove analytic function is constant, if argument is constant Show that analytic function with constant argument is constant.. Need help with this one, unable to prove.. 
This is true by the definition of a function. 
Agreed, but still need a formal proof. Can you please help? 
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In this context, I think that argument refers to the value of when expressing a complex number in polar form: . 
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From Cauchy Riemann eqs with theta constant, $\displaystyle \partial u/\partial r=\frac{1}{r}\partial v/\partial \theta=0\\ \partial v/\partial r=\frac{1}{r}\partial u/\partial \theta=0 $ So u and v are constant. So f(z)=u+iv=constant. 
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