Editing Logarithmic derivative (section)

==Complex analysis==
{{See also|Argument principle}}
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 [[zeros and poles|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

{{block indent | em = 1.5 | text = {{math|''z<sup>n</sup>''}}}}

with ''n'' an integer, {{math|''n'' ≠ 0}}. The logarithmic derivative is then
<math display="block">n/z</math>
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 (complex analysis)|residue]] ''n'' from a zero of order ''n'', residue &minus;''n'' from a pole of order ''n''. See [[argument principle]]. This information is often exploited in [[contour integration]].<ref>{{Cite book |last=Gonzalez|first=Mario|url=https://books.google.com/books?id=ncxL7EFr7GsC&dq=%22logarithmic+derivative%22+AND+%22complex+analysis%22&pg=PA740 | title=Classical Complex Analysis |date=1991-09-24 |publisher=CRC Press|isbn=978-0-8247-8415-7|language=en}}</ref><ref>{{Cite web|date=7 June 2020|title=Logarithmic residue - Encyclopedia of Mathematics|url=http://encyclopediaofmath.org/index.php?title=Logarithmic_residue&oldid=47703|access-date=2021-08-12|website=encyclopediaofmath.org}}</ref>{{Verify source|date=August 2021}}

In the field of [[Nevanlinna theory]], an important lemma states that the proximity function of a logarithmic derivative is small with respect to the Nevanlinna characteristic of the original function, for instance <math>m(r,h'/h) = S(r,h) = o(T(r,h))</math>.<ref>{{Cite book|last=Zhang|first=Guan-hou|url=https://books.google.com/books?id=Ne7OpHc3lOQC&dq=%22nevanlinna+theory%22+AND+%22second+fundamental+theorem%22+AND+%22logarithmic+derivative%22&pg=PP9 | title=Theory of Entire and Meromorphic Functions: Deficient and Asymptotic Values and Singular Directions| date=1993-01-01| publisher=American Mathematical Soc.|isbn=978-0-8218-8764-6|pages=18|language=en|access-date=12 August 2021}}</ref>{{Verify source|date=August 2021}}