# Substitution rule in complex analysis

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.