Conformal mapping between disk and the complement of a spiral

Question

The Riemann mapping theorem guarantees the existence of a biholomorphic mapping between the unit disk and the complement in the complex plane of an (archimedean or logarithmic) spiral ... is it known an explicit formula for such a map? Anyone knows something about such functions and can provide me any reference?

Thanks in advance

Answer 1

A hint:

Given a $\lambda>0$ the function $$f(z):=e^{(\lambda+i)z}$$ maps the strip $0< {\rm Im}(z)<{2\pi\over\lambda}$ conformally onto the complement of the logarithmic spiral with polar representation $r(\phi)=e^{\lambda\phi}$ $\ (-\infty<\phi<\infty$).