91Ó°ÊÓ

Prove that the logarithmic derivative of the product of two holomorphic functions equals the sum of their logarithmic derivatives.

Prove that a branch of the inverse of a holomorphic function is always univalent.

Describe the curves \(|f|=\) constant and arg \(f=\) constant for the function $$ f(z)=\exp \left(z^{2}\right) $$

What is the most general holomorphic function \(f\) in \(\mathbf{C}\) that satisfies \(f^{\prime}=c f\), where \(c\) is a constant?

Let \(a\) be a complex number of unit modulus and \(c\) an irrational real number. Prove that the values of \(a^{c}\) form a dense subset of the unit circle.

Suppose the holomorphic function \(f\) in \(\mathbf{C}\) has a holomorphic derivative and satisfies the differential equation \(f^{\prime \prime}=f .\) Prove that \(f\) has the form \(f(z)=a \cosh z+b \sinh z\), where \(a\) and \(b\) are constants.

In what sense is it true that \(\log a_{1} a_{2}=\log a_{1}+\log a_{2}\) for complex numbers \(a_{1}\) and \(a_{2} ?\)

Describe the images of the lines \(\operatorname{Re} z=\) constant and \(\operatorname{Im} z=\) constant under the \(\operatorname{map} z \mapsto \cos z\)

Prove that if \(f\) is a branch of \(z^{c}\) in an open set not containing 0 , then \(f\) is holomorphic and \(f^{\prime}\) is a branch of \(c z^{c-1}\).

Let \(G\) be the open set one obtains by removing from C the interval \([-1,1]\) on the real axis. Prove that there is a branch of the function \(\sqrt{\frac{z+1}{z-1}}\) in \(G\). (Suggestion: What is the image of \(G\) under the map \(\left.z \mapsto \frac{z+1}{z-1} ?\right)\)