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<title>Bateman transform</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Bateman transform</span></span>
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<p>In the <a href="Mathematics" title="Mathematics">mathematical</a> study of <a href="Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>, the <b>Bateman transform</b> is a method for solving the <a href="Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a> in four dimensions and <a href="Wave_equation" title="Wave equation">wave equation</a> in three by using a <a href="Line_integral" title="Line integral">line integral</a> of a <a href="Holomorphic_function" title="Holomorphic function">holomorphic function</a> in three <a href="Complex_number" title="Complex number">complex variables</a>. It is named after the mathematician <a href="Harry_Bateman" title="Harry Bateman">Harry Bateman</a>, who first published the result in (<a href="#CITEREFBateman1904">Bateman 1904</a>).
</p><p>The formula asserts that if <i>ƒ</i> is a holomorphic function of three complex variables, then
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (w,x,y,z)=\oint _{\gamma }f{\big (}(w+ix)+(iy+z)\zeta ,(iy-z)+(w-ix)\zeta ,\zeta {\big )}\,d\zeta }">
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<annotation encoding="application/x-tex">{\displaystyle \phi (w,x,y,z)=\oint _{\gamma }f{\big (}(w+ix)+(iy+z)\zeta ,(iy-z)+(w-ix)\zeta ,\zeta {\big )}\,d\zeta }</annotation>
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</math></span><img src="./4951de5a9c83834bfcc767647a32bc6b69d00461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:66.861ex; height:6.009ex;" alt="{\displaystyle \phi (w,x,y,z)=\oint _{\gamma }f{\big (}(w+ix)+(iy+z)\zeta ,(iy-z)+(w-ix)\zeta ,\zeta {\big )}\,d\zeta }" loading="lazy"></span></dd></dl>
<p>is a solution of the Laplace equation, which follows by differentiation under the integral. Furthermore, Bateman asserted that the most general solution of the Laplace equation arises in this way.
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFBateman1904" class="citation cs2"><a href="Harry_Bateman" title="Harry Bateman">Bateman, Harry</a> (1904), <a rel="nofollow" class="external text" href="https://archive.today/20130415131037/http://plms.oxfordjournals.org/cgi/reprint/s2-1/1/451">"The solution of partial differential equations by means of definite integrals"</a>, <i><a href="Proceedings_of_the_London_Mathematical_Society" class="mw-redirect" title="Proceedings of the London Mathematical Society">Proceedings of the London Mathematical Society</a></i>, <b>1</b> (1): <span class="nowrap">451–</span>458, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-1.1.451">10.1112/plms/s2-1.1.451</a>, archived from <a rel="nofollow" class="external text" href="http://plms.oxfordjournals.org/cgi/reprint/s2-1/1/451">the original</a> on 2013-04-15</cite>.</li>
<li><cite id="CITEREFEastwood2002" class="citation cs2"><a href="Michael_Eastwood" title="Michael Eastwood">Eastwood, Michael</a> (2002), <a rel="nofollow" class="external text" href="http://www.msri.org/ext/concepts/eastwood3.pdf"><i>Bateman's formula</i></a> <span class="cs1-format">(PDF)</span>, <a href="Mathematical_Sciences_Research_Institute" class="mw-redirect" title="Mathematical Sciences Research Institute">MSRI</a></cite>.</li></ul>
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