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Given a function , the ratio
of its first derivative and itself is called the logarithmic derivative of .
When is a real function of real variable and takes strictly positive values then the chain rules gives
The observation is the motivation for the name.
The logarithmic derivative has many useful properties, For instance
In the theory of complex functions we have:
Theorem. Let be a meromorphic function in the region (i.e. open and connected set) and be a region such that its closure and which boundary is a continuous curve not containing a zero o pole of . If and denotes the number of zeros and poles of lying inside , respectively, then
where is an oriented boundary.
Cite this web-page as:
Štefan Porubský: Logarithmic derivative.