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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.
Some basic calculus applications:
The logarithmic derivative idea is closely connected to the integrating factor method, for first order differential equations . In operator terms, write
and let M denote the operator of multiplication by some given function G(x). Then
can be written (by the product rule) as
where M* now denotes the multiplication operator by the logarithmic derivative
In practice we are given an operator such as
and wish to solve equations
for the function h, given f. This then reduces to solving
which has as solution
with any indefinite integral of F.
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 poleIn complex analysis, a pole of a function is a certain type of simple singularity that behaves like the singularity of f ''z 1 z n at z 0; a pole of a function f is a point a such that f ''z approaches infinity as z approaches a''. Formally, suppose U is. Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily analysed in terms of the particular case
with n an integer, n≠0. The logarithmic derivative is then
and one can draw the general conclusion that for f meromorphic, the singularities of the logarithmic derivative of f are all simple poles, with residueThe term residue has several meanings in mathematics. Additionally, it has meaning in molecular biology. In complex analysis, the residue is a complex number which describes the behavior of path integrals of a meromorphic function around a singularity. n from a zero of order n, residue −n from a pole of order n. See argument principleIn complex analysis, the argument principle (or Cauchy argument principle establishes the value of integrals of the form : where N is the number of zeros, P the number of poles, and f has a finite number of zeros and poles inside a closed contour C''.. This information is often exploited in contour integration.