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Suppose I have three functions $f(z):\mathbb{C}\rightarrow\mathbb{C}$, $g(z):\mathbb{C}\rightarrow\mathbb{C}$, and $h(z):\mathbb{C}\rightarrow\mathbb{C}$.

What methods work for finding all $z$ such that $f(z)=g(z)=h(z)$? Variants of Newton-Raphson are acceptable, but I'm having difficulty encoding it.

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See Computing the Zeros of Analytic Functions for theory and code.

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If $f$ and $g$ are both holomorphic and are not the same function, then the set $\{z\mid f(z)=g(z)\}$ consists of at most countably many isolated points. Find those points by solving $f(z)-g(z)=0$ in whichever way works for those two functions, and then check each of them to see whether $f(z)=h(z)$.