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Let $lambda_{mathbb{C}}$ be the Lebesgue measure in the complex plane.
Let $f$ be an entire function and $g$ a continuous function.
I ask:
When does the substitution rule hold
$$int_{mathbb{C}} g(f(z)) f'(z) dlambda_{mathbb{C}}(z)=int_{f(mathbb{C})} g(w) dlambda_{mathbb{C}}(w)?$$
I suppose that $f$ should be a diffeomorphism could be sufficient, is that correct?
Answer
This is hardly ever correct. For example, let $f(z) = iz,$ $g(z) = e^{-|z|}.$ Then the left side equals
$$iint_{mathbb C} e^{-|z|} , dlambda(z),$$
the right side equals
$$int_{mathbb C} e^{-|w|} , dlambda(w).$$
Note two things: i) The only entire diffeomorphisms of $mathbb C$ into $mathbb C$ are of the form $f(z) = az+b.$ ii) If $f$ is a holomorphic injective map from $U$ onto $V,$ then the correct change of variables is
$$ int_{U} g(f(z))|f'(z)|^2 , dlambda(z) = int_{V} g(w) , dlambda(w).$$
Just compute the real Jacobian of the map $zto f(z)$ to see this.