Solution to the functional equation $f(z)=(-1)^zf(1-z)$???

Question

I was playing around with various functional equations and I created this particular one: $$f(z)=(-1)^zf(1-z)$$

Where $z$ is a complex number. I was wondering how I would go about finding a solution to this equation if one even exists.

Answer 1

Since $$f(z)=(-1)^zf(1-z)$$

If we replace $z$ with $1-z$ we get $$f(1-z)=(-1)^{1-z}f(z)$$ and thus

$$f(z) = (-1)^z(-1)^{1-z}f(z) = (-1)^1f(z)$$

so $f(z)=0$ for all $z$.