Logarithmic derivative

In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a [[function] f is defined by the formula

f′/f

where f′ is the derivative of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is indeed the formula for (log f)′, that is, the derivative of the logarithm of f, as follows from the chain rule.

The formula as given can be applied more widely; for example if f(z) is a meromorphic function, it makes sense at all complex values of z at which f has neither a zero nor a pole. Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily analysed in terms of the particular case

zⁿ

with n an integer, n≠0. The logarithmic derivative is then

n/z;

and one can draw the general conclusion that for f meromorphic, the singularities of the logarithmic derivative of f are all simple poles, with residue n from a zero of order n, residue −n from a pole of order n. This information is often exploited in contour integration.

Behind the use of the logarithmic derivative lie two basic facts about GL₁, that is, the multiplicative group of real numbers or other field. The differential operator

X^-1d/dX

is invariant under 'translation' (replacing X by aX for a constant). And the diffential form

dX/X

is likewise invariant. For functions F into GL₁, the formula

dF/F

is therefore a pullback of the invariant form.