Knowlpedia

Definition

Holomorphic logarithm of a nonvanishing function

A branch L with exp(L)=f, available on a simply connected domain for a zero-free holomorphic function.

A holomorphic logarithm of a nonvanishing ff on a domain Ω\Omega is a holomorphic LL satisfying eL(z)=f(z)e^{L(z)}=f(z). If Ω\Omega is simply connected, such a logarithm exists. Choosing its value L(z0)L(z_0) at one point fixes the branch.

Construction

The function f′/ff'/f is holomorphic. On a simply connected domain it has a primitive, so define

L(z)=L(z0)+∫z0zf′(w)f(w) dw,eL(z0)=f(z0).L(z)=L(z_0)+\int_{z_0}^z\frac{f'(w)}{f(w)}\,dw, \qquad e^{L(z_0)}=f(z_0).

The derivative of fe−Lfe^{-L} is zero and its value at z0z_0 is one, proving the claim. Any two branches on a connected domain differ by a constant in 2πiZ2\pi i\mathbb Z. Nonvanishing alone does not suffice on arbitrary domains; f(z)=zf(z)=z on a punctured disc has no single-valued holomorphic logarithm there.

References