Difference between revisions of "Gamma function"
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(Created page with "We define $$p_f(t,s)=e_{\frac{f}{\mathrm{id}}}(t,s),$$ where $\mathrm{id}$ denotes the identity map $\mathrm{id} \colon \mathbb{T} \rightarrow \mathbb{T}$ and $e_{\cdot}$ den...") |
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With these definitions, we have the gamma operator | With these definitions, we have the gamma operator | ||
$$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta.$$ | $$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta.$$ | ||
| + | |||
| + | =Properties of gamma functions= | ||
| + | |||
| + | ==Convergence== | ||
| + | |||
| + | =Examples of gamma functions= | ||
| + | |||
| + | =References= | ||
| + | <div id="tgfots"></div><bibtex> | ||
| + | @inproceedings{ | ||
| + | title="The Gamma Function on Time Scales", | ||
| + | author="Bohner, Martin and Karpuz, Başak", | ||
| + | booktitle="Dynamics of Continuous, Discrete \& Impulsive Systems. Series A. Mathematical Analysis", | ||
| + | volume="20", | ||
| + | year="2013", | ||
| + | pages="pp.507--522", | ||
| + | url="http://online.watsci.org/abstract_pdf/2013v20/v20n4a-pdf/7.pdf" | ||
| + | } | ||
| + | </bibtex> | ||
Revision as of 02:43, 24 July 2014
We define $$p_f(t,s)=e_{\frac{f}{\mathrm{id}}}(t,s),$$ where $\mathrm{id}$ denotes the identity map $\mathrm{id} \colon \mathbb{T} \rightarrow \mathbb{T}$ and $e_{\cdot}$ denotes the time scale exponential. Define the operations $$f \boxplus_{\mu} g := f+g+\dfrac{1}{\mathrm{id}}fg\mu$$ and $$f \boxminus_{\mu} g := \dfrac{(f-g)\mathrm{id}}{\mathrm{id} + g \mu}.$$ With these definitions, we have the gamma operator $$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta.$$
Properties of gamma functions
Convergence
Examples of gamma functions
References
<bibtex>@inproceedings{
title="The Gamma Function on Time Scales", author="Bohner, Martin and Karpuz, Başak", booktitle="Dynamics of Continuous, Discrete \& Impulsive Systems. Series A. Mathematical Analysis", volume="20", year="2013", pages="pp.507--522", url="http://online.watsci.org/abstract_pdf/2013v20/v20n4a-pdf/7.pdf"
} </bibtex>
