Knowlpedia

Definition

Logarithmic derivative

The meromorphic function f prime over f, whose residues record zeros and poles.

Definition

Let ff be a nonzero . Its logarithmic derivative is

f′f.\frac{f'}{f}.

Where a local branch of log⁡f\log f exists, this is literally (log⁡f)′(\log f)', but the quotient is globally defined even when no single-valued logarithm exists.

Zeros, poles, and residues

If

f(z)=(z−a)mu(z),u(a)≠0,f(z)=(z-a)^m u(z),\qquad u(a)\ne0,

then

f′(z)f(z)=mz−a+u′(z)u(z).\frac{f'(z)}{f(z)}=\frac{m}{z-a}+\frac{u'(z)}{u(z)}.

Thus f′/ff'/f has a simple pole at each zero or pole of ff, with m=ord⁡a(f)m=\operatorname{ord}_a(f).

Argument principle

Integrating around a contour gives

12πi∫γf′f dz=∑aInd⁡(γ,a)ord⁡a(f).\frac{1}{2\pi i}\int_\gamma\frac{f'}f\,dz =\sum_a\operatorname{Ind}(\gamma,a)\operatorname{ord}_a(f).

This is the : the logarithmic derivative converts multiplicative zero-and-pole data into an additive .

Product behavior

The familiar rules

(fg)′fg=f′f+g′g,(f/g)′f/g=f′f−g′g\frac{(fg)'}{fg}=\frac{f'}f+\frac{g'}g, \qquad \frac{(f/g)'}{f/g}=\frac{f'}f-\frac{g'}g

mirror additivity of orders under products and quotients.

References
  1. Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979. Relevant: Chapter 5, §2.