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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. Retrieved 2024/12/29 from Interactive Information Portal for Algorithmic Mathematics, Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic, web-page http://www.cs.cas.cz/portal/AlgoMath/MathematicalAnalysis/ComplexAnalysis/LogarithmicDerivative.htm