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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Bateman function</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In mathematics, the <b>Bateman function</b> (or <i>k</i>-function) is a special case of the <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">confluent hypergeometric function</a> studied by <a href="Harry_Bateman" title="Harry Bateman">Harry Bateman</a>(1931).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Bateman defined it by
</p>
<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 \displaystyle k_{\nu }(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\cos(x\tan \theta -\nu \theta )\,d\theta .}">
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<annotation encoding="application/x-tex">{\displaystyle \displaystyle k_{\nu }(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\cos(x\tan \theta -\nu \theta )\,d\theta .}</annotation>
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</math></span><img src="./6d9507d662ca3b5bb270b4065cb24bc70b5101d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.754ex; height:6.343ex;" alt="{\displaystyle \displaystyle k_{\nu }(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\cos(x\tan \theta -\nu \theta )\,d\theta .}" loading="lazy"></span></dd></dl>
<p><a href="Harry_Bateman" title="Harry Bateman">Bateman</a> discovered this function, when <a href="Theodore_von_K%C3%A1rm%C3%A1n" title="Theodore von Kármán">Theodore von Kármán</a> asked for the solution of the following differential equation which appeared in the theory of <a href="Turbulence" title="Turbulence">turbulence</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
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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 x{\frac {d^{2}u}{dx^{2}}}=(x-\nu )u}">
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<annotation encoding="application/x-tex">{\displaystyle x{\frac {d^{2}u}{dx^{2}}}=(x-\nu )u}</annotation>
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</math></span><img src="./e76597130b57f03e6ac78c7b06502eecd9dd8310.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.407ex; height:6.009ex;" alt="{\displaystyle x{\frac {d^{2}u}{dx^{2}}}=(x-\nu )u}" loading="lazy"></span></dd></dl>
<p>and Bateman found this function as one of the solutions. Bateman denoted this function as "k" function in honor of <a href="Theodore_von_K%C3%A1rm%C3%A1n" title="Theodore von Kármán">Theodore von Kármán</a>.
</p><p>The Bateman function for <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 x>0}">
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<annotation encoding="application/x-tex">{\displaystyle x>0}</annotation>
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</math></span><img src="./80d24be5f0eb4a9173da6038badc8659546021d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}" loading="lazy"></span> is the related to the <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">Confluent hypergeometric function</a> of the second kind as follows
</p>
<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 k_{\nu }(x)={\frac {e^{-x}}{\Gamma \left(1+{\frac {1}{2}}\nu \right)}}U\left(-{\frac {1}{2}}\nu ,0,2x\right),\quad x>0.}">
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<annotation encoding="application/x-tex">{\displaystyle k_{\nu }(x)={\frac {e^{-x}}{\Gamma \left(1+{\frac {1}{2}}\nu \right)}}U\left(-{\frac {1}{2}}\nu ,0,2x\right),\quad x>0.}</annotation>
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</math></span><img src="./39dcfa313c79e124b57f159545027eaa9afd9845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.584ex; height:7.009ex;" alt="{\displaystyle k_{\nu }(x)={\frac {e^{-x}}{\Gamma \left(1+{\frac {1}{2}}\nu \right)}}U\left(-{\frac {1}{2}}\nu ,0,2x\right),\quad x>0.}" loading="lazy"></span></dd></dl>
<p>This is not to be confused with another function of the same name which is used in Pharmacokinetics.
</p>
<div class="mw-heading mw-heading2"><h2 id="Havelock_function">Havelock function</h2></div>
<p>Complementary to the Bateman function, one may also define the Havelock function, named after <a href="Thomas_Henry_Havelock" title="Thomas Henry Havelock">Thomas Henry Havelock</a>. In fact, both the Bateman and the Havelock functions were first introduced by Havelock in 1927,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> while investigating the surface elevation of the uniform stream past an immersed circular cylinder. The Havelock function is defined by
</p>
<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 \displaystyle h_{\nu }(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\sin(x\tan \theta -\nu \theta )\,d\theta .}">
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<annotation encoding="application/x-tex">{\displaystyle \displaystyle h_{\nu }(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\sin(x\tan \theta -\nu \theta )\,d\theta .}</annotation>
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</math></span><img src="./22cb6309c53f80e04524ca8046e9b17e8bb181e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.627ex; height:6.343ex;" alt="{\displaystyle \displaystyle h_{\nu }(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\sin(x\tan \theta -\nu \theta )\,d\theta .}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<ul><li><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 k_{0}(x)=e^{-|x|}}">
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<mi>k</mi>
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<annotation encoding="application/x-tex">{\displaystyle k_{0}(x)=e^{-|x|}}</annotation>
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</math></span><img src="./c27a1973b3279a550085c59df59ed726868e4c87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.952ex; height:3.343ex;" alt="{\displaystyle k_{0}(x)=e^{-|x|}}" loading="lazy"></span></li>
<li><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 k_{-n}(x)=k_{n}(-x)}">
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<mi>k</mi>
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<annotation encoding="application/x-tex">{\displaystyle k_{-n}(x)=k_{n}(-x)}</annotation>
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</math></span><img src="./2719d81158d26f465f92875df0fd3abdcbadc55b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.323ex; height:2.843ex;" alt="{\displaystyle k_{-n}(x)=k_{n}(-x)}" loading="lazy"></span></li>
<li><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 k_{n}(0)={\frac {2}{n\pi }}\sin {\frac {n\pi }{2}}}">
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle k_{n}(0)={\frac {2}{n\pi }}\sin {\frac {n\pi }{2}}}</annotation>
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</math></span><img src="./0f7310ddc975296f2736211869237bee35d0e864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.255ex; height:5.176ex;" alt="{\displaystyle k_{n}(0)={\frac {2}{n\pi }}\sin {\frac {n\pi }{2}}}" loading="lazy"></span></li>
<li><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 k_{2}(x)=(x+|x|)e^{-|x|}}">
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<mi>k</mi>
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<annotation encoding="application/x-tex">{\displaystyle k_{2}(x)=(x+|x|)e^{-|x|}}</annotation>
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</math></span><img src="./8e921ae4635d279b2ec07f254f35325da0d0fe3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.555ex; height:3.343ex;" alt="{\displaystyle k_{2}(x)=(x+|x|)e^{-|x|}}" loading="lazy"></span></li>
<li><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 |k_{n}(x)|\leq 1}">
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<annotation encoding="application/x-tex">{\displaystyle |k_{n}(x)|\leq 1}</annotation>
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</math></span><img src="./29f2413816d7b8af59ab9fb8d7774daae14f2abc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.123ex; height:2.843ex;" alt="{\displaystyle |k_{n}(x)|\leq 1}" loading="lazy"></span> for real values of <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 n}">
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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> and <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 x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span></li>
<li><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 k_{2n}(x)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k_{2n}(x)=0}</annotation>
</semantics>
</math></span><img src="./995d550db78e399dc9392ce5c2284ca5be434ffc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.652ex; height:2.843ex;" alt="{\displaystyle k_{2n}(x)=0}" loading="lazy"></span> for <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 x<0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo><</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x<0}</annotation>
</semantics>
</math></span><img src="./1a4dbbf970b2d2863dcab589eafe006f08e727d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}" loading="lazy"></span> if <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 n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> is a positive integer</li>
<li><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 k_{1}(x)={\frac {2x}{\pi }}[K_{1}(x)+K_{0}(x)],\ x<0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
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<mi>π<!-- π --></mi>
</mfrac>
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<mo stretchy="false">[</mo>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo>,</mo>
<mtext> </mtext>
<mi>x</mi>
<mo><</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k_{1}(x)={\frac {2x}{\pi }}[K_{1}(x)+K_{0}(x)],\ x<0}</annotation>
</semantics>
</math></span><img src="./e8e069c788758c30a7518601f006744ddbb66eed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.503ex; height:5.176ex;" alt="{\displaystyle k_{1}(x)={\frac {2x}{\pi }}[K_{1}(x)+K_{0}(x)],\ x<0}" loading="lazy"></span>, where <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 K_{n}(-x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K_{n}(-x)}</annotation>
</semantics>
</math></span><img src="./96ca6ca34133d54404315ebaa8a646e18fbd4611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.139ex; height:2.843ex;" alt="{\displaystyle K_{n}(-x)}" loading="lazy"></span> is the <a href="Modified_Bessel_function_of_the_second_kind" class="mw-redirect" title="Modified Bessel function of the second kind">Modified Bessel function of the second kind</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFBateman1931" class="citation cs2"><a href="Harry_Bateman" title="Harry Bateman">Bateman, H.</a> (1931), "The k-function, a particular case of the confluent hypergeometric function", <i><a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>, <b>33</b> (4): <span class="nowrap">817–</span>831, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1989510">10.2307/1989510</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1989510">1989510</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1501618">1501618</a></cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Bateman_function">"Bateman function"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Martin, P. A., & Bateman, H. (2010). from Manchester to Manuscript Project. Mathematics Today, 46, 82-85. <a rel="nofollow" class="external free" href="http://www.math.ust.hk/~machiang/papers_folder/http___www.ima.org.uk_mathematics_mt_april10_harry_bateman_from_manchester_to_manuscript_project.pdf">http://www.math.ust.hk/~machiang/papers_folder/http___www.ima.org.uk_mathematics_mt_april10_harry_bateman_from_manchester_to_manuscript_project.pdf</a></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Havelock, T. H. (1927). The method of images in some problems of surface waves. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115(771), 268-280.</span>
</li>
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