# MacRobert E function > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/MacRobert_E_function > Markdown URL: https://mediated.wiki/source/MacRobert_E_function.md > Source: https://en.wikipedia.org/wiki/MacRobert_E_function > Source revision: 1301851611 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In mathematics, the '''E-function''' was introduced by {{harvs|txt|authorlink=Thomas Murray MacRobert|first=Thomas Murray|last=MacRobert|year=1937–1938}} to extend the [generalized hypergeometric series](/source/generalized_hypergeometric_function) ''p''''F''''q''(·) to the case ''p'' > ''q'' + 1. The underlying objective was to define a very general function that includes as particular cases the majority of the [special function](/source/special_function)s known until then. However, this function had no great impact on the literature as it can always be expressed in terms of the [Meijer G-function](/source/Meijer_G-function), while the opposite is not true, so that the G-function is of a still more general nature. It is defined as: \begin{align} E(p;\alpha_r;\rho_s;z) \equiv {} & \frac{\Gamma(\alpha_{q+1})}{\prod_{k=1}^q\Gamma(\rho_k-\alpha_k)} \prod_{\mu=1}^q \int_0^\infty\lambda_\mu^{\rho_\mu-\alpha_\mu-a}(\lambda_\mu+1)^{-\rho_\mu} \, d\lambda_\mu \\ & \times \prod_{\nu=2}^{p-q-1} \int_0^\infty \lambda_{q+\nu}^{\alpha_{q+\nu}-1}\exp(-\lambda_{q+\nu}) \, d\lambda_{q+\nu} \\ & \times \int_0^\infty\lambda_p^{\alpha_p-1}\exp(-\lambda_p) \left[\frac{\prod_{k=q+2}^p\lambda_k}{z\prod_{k=1}^q\lambda_k+1}+1\right] \, d\lambda_p \end{align} ==Definition== There are several ways to define the MacRobert E-function; the following definition is in terms of the [generalized hypergeometric function](/source/generalized_hypergeometric_function): * when ''p'' ≤ ''q'' and ''x'' ≠ 0, or ''p'' = ''q'' + 1 and |''x''| > 1: :: E \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, x \right) = \frac{\prod_{j=1}^p \Gamma (a_j)} {\prod_{j=1}^q \Gamma (b_j)} \;_p F_q \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, -x^{-1} \right) * when ''p'' ≥ ''q'' + 2, or ''p'' = ''q'' + 1 and |''x''| < 1: :: E \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, x \right) = \sum_{h=1}^p \frac{\prod_{j=1}^p \Gamma (a_j - a_h)^*} {\prod_{j=1}^q \Gamma (b_j - a_h)} \Gamma (a_h) \; x^{a_h} \;_{q+1}F_{p-1} \!\left( \left. \begin{matrix} a_h, 1 + a_h - b_1, \dots, 1 + a_h - b_q \\ 1 + a_h - a_1, \dots, *, \dots, 1 + a_h - a_p \end{matrix} \; \right| \, (-1)^{p-q} \;x \right). The asterisks here remind us to ignore the contribution with index ''j'' = ''h'' as follows: In the product this amounts to replacing Γ(0) with 1, and in the argument of the hypergeometric function this amounts to shortening the vector length from ''p'' to ''p'' − 1. Evidently, this definition covers all values of ''p'' and ''q''. ==Relationship with the Meijer G-function== The MacRobert E-function can always be expressed in terms of the [Meijer G-function](/source/Meijer_G-function): : E \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, x \right) = G_{q+1,\,p}^{\,p,\,1} \!\left( \left. \begin{matrix} 1, \mathbf{b_q} \\ \mathbf{a_p} \end{matrix} \; \right| \, x \right) where the parameter values are unrestricted, i.e. this relation holds without exception. ==References== * {{cite book | last= Andrews | first= L. C. | year= 1985 | title= Special Functions for Engineers and Applied Mathematicians | location= New York | publisher= MacMillan | isbn= 0-02-948650-5 }} * {{cite book | last1= Erdélyi | first1= A. | author1-link= Arthur Erdélyi | last2= Magnus | first2= W. | last3= Oberhettinger | first3= F. | name-list-style= amp | last4= Tricomi | first4= F. G. | year= 1953 | title= Higher Transcendental Functions | volume= 1 | url= http://apps.nrbook.com/bateman/Vol1.pdf | location= New York | publisher= McGraw–Hill | archive-date= 2011-08-11 | access-date= 2012-06-04 | archive-url= https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf | url-status= dead }} (see § 5.2, "Definition of the E-Function", p. 203) *{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[Academic Press, Inc.](/source/Academic_Press%2C_Inc.) |date=2015 |orig-year=October 2014 |edition=8 |language=English |isbn=978-0-12-384933-5 |lccn=2014010276 |title-link=Gradshteyn and Ryzhik |chapter=9.4. }} * {{cite journal | last= MacRobert | first= T. M. | authorlink= Thomas Murray MacRobert | title= Induction proofs of the relations between certain asymptotic expansions and corresponding generalised hypergeometric series | year= 1937–38 | journal= Proc. R. Soc. Edinburgh | volume= 58 | pages= 1–13 | jfm= 64.0337.01 }} * {{cite journal | last= MacRobert | first= T. M. | url= http://www.digizeitschriften.de/resolveppn/GDZPPN00229060X | title= Barnes integrals as a sum of E-functions | journal= Mathematische Annalen | volume= 147 | issue= 3 | year= 1962 | pages= 240–243 | zbl= 0100.28601 | doi=10.1007/bf01470741| s2cid= 121048026 | url-access= subscription }} ==External links== * {{mathworld | urlname= MacRobertsE-Function | title= MacRobert's E-Function}} {{DEFAULTSORT:MacRobert E function}} Category:Hypergeometric functions --- Adapted from the Wikipedia article [MacRobert E function](https://en.wikipedia.org/wiki/MacRobert_E_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/MacRobert_E_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.