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...[[special function]] that is closely related to the [[gamma function]] and to [[binomial coefficient]]s. It is defined by the [[integral]] A key property of the beta function is its close relationship to the [[gamma function]]: one has that<ref name=Davis622/> ...

...= https://books.google.com/books?id=2CAqsF-RebgC&pg=PA110 |title = Special functions of mathematics for engineers|page = 110|publisher = SPIE Press |date= 1998| ...entary]]) and [[sigmoid function|sigmoid]] function that occurs often in [[probability]], [[statistics]], and [[partial differential equation]]s. In many of these ...

* [[probability]], such as the [[Edgeworth series]]; * in [[physics]], where they give rise to the [[eigenstate]]s of the [[quantum harmonic oscillator]]; ...

{{Short description|Families of solutions to related differential equations}} [[File:Vibrating drum Bessel function.gif|thumb|right|Bessel functions are the radial part of the modes of vibration of a circular drum.]] ...

{{short description|Inverse of the exponential function, which maps products to sums}} [[File:Logarithm plots.png|right|thumb|upright=1.35|Plots of logarithm functions, with three commonly used bases. The special points {{math|log<sub>''b''</s ...