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Logarithmic derivative

Given a function typeset structure, the ratio

f^'/f

of its first derivative and itself is called the logarithmic derivative of typeset structure.

When typeset structure is a real function of real variable and takes strictly positive values then the chain rules gives

d/d x ln f(x) = f^'/f,

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 typeset structure be a meromorphic function in the region (i.e. open and connected set) typeset structureand typeset structure be a region such that its closure typeset structure and which boundary typeset structure is a continuous curve not containing a zero o pole of typeset structure. If��typeset structure and typeset structure denotes the number of zeros and poles of typeset structure lying inside typeset structure, respectively, then

N - P = 1/(2 π i) Underoverscript[∫, ∂ G, arg3] f^'(z)/f(z) d z ,

where typeset structure 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

Page created� Tuesday, 25th September 2007  (17 years ago).