Common integrals in quantum field theory: Difference between revisions

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See [[Static forces and virtual-particle exchange#Selected examples|Static forces and virtual-particle exchange]] for an application of this integral.

In the small m limit the integral reduces to {{math|{{sfrac|1|4''πr''}}}}.

To derive this result note:

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====Angular integration in cylindrical coordinates====

There are two important integrals. The angular integration of an exponential in cylindrical coordinates can be written in terms of Bessel functions of the first kind<ref name="Zwillinger_2014">{{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 |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[[Academic Press, Inc.]] |date=2015 |orig-year=October 2014 |edition=8 |language=English |isbn=0-12-384933-0 |id={{ISBN |978-0-12-384933-5}} |lccn=2014010276 <!-- |url=http://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21 -->|title-link=Gradshteyn and Ryzhik <!-- |pages= was on page 402 and 679 in 1965 edition, but page numbers probably changed meanwhile -->}}</ref><ref>{{cite book|author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)| publisher=Wiley| year=1998| isbn=0-471-30932-X}} p. 113</ref>

:<math>\int_0^{2 \pi} {d\varphi \over 2 \pi} \exp\left( i p \cos( \varphi) \right)=J_0 (p)</math>