I am new to complex analysis and I would be grateful to be guided in the following problem. We know that if $f$ is a function from $\Bbb C \to \Bbb R$, then $|f|$ is a function from from $\Bbb R^2 \to\Bbb R$. In complex analysis, the maximum modulus principle tells us that an analytic function over a compact domain takes its maximum value somewhere on the boundary. I want to know if there are methods (analytic or numerical) suited to finding the maximum or minimum of the modulus of an analytic function on a curve. For instance, how can we find the maximum and minimum of a complex valued function, say for example $$f(z) = e^z+\sin z$$ on the unit circle? In general, how do we optimise functions from $\Bbb R^n \to\Bbb R$? I would be highly obliged if somebody could kindly provide links to relevant literature/references.
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1$\begingroup$ You just set the gradient (w.r.t. the two real variables) to zero. That's standard multivariable (real) calculus. Questions like this would be better suited at math.stackexchange. $\endgroup$– DirkOct 13, 2022 at 13:30
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$\begingroup$ Parametrize the unit circle, and find the maximum of the function of one variable. $\endgroup$– Alexandre EremenkoOct 13, 2022 at 13:43
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1$\begingroup$ If your function is a polynomial (or close enough to one) soliton.vm.bytemark.co.uk/pub/jjg/en/code/steckin $\endgroup$– J.J. GreenOct 13, 2022 at 14:10
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$\begingroup$ For circles Jack's lemma is useful too (maximum modulus on circle $|z|=r$ at some $w$ implies $wf'(w)/f(w) \ge 0$ - so both real and nonnegative - if $f$ analytic inside the circle, while minimum modulus and $f(w) \ne 0$ implies $wf'(w)/f(w)$ real) $\endgroup$– ConradOct 13, 2022 at 17:26
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$\begingroup$ @Conrad,Thank you for for letting me know about Jack's lemma $\endgroup$– AgnostMysticOct 15, 2022 at 2:00
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