Liouvillian function

<h2 id="examples">Examples</h2>
<p>All <a href="/facts/Elementary_function/wVFyDwu8">elementary functions</a> are Liouvillian.
</p><p>Examples of well-known functions which are Liouvillian but not elementary are the <a href="/facts/Nonelementary_integral/IVp2vzq7">nonelementary antiderivatives</a>, for example:
</p>
<ul><li>The <a href="/facts/Error_function/Ca2H7pGr">error function</a>, 
  
    
      
        
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    {\displaystyle \mathrm {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt,}
  
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<li>The <a href="/facts/Exponential_integral/JF6aMcmL">exponential</a> (<i>Ei</i>), <a href="/facts/Logarithmic_integral/P3xwqRlA">logarithmic</a> (<i>Li</i> or <i>li</i>) and <a href="/facts/Fresnel_integral/5W7z760j">Fresnel</a> (<i>S</i> and <i>C</i>) integrals.</li></ul>
<p>All Liouvillian functions are solutions of <a href="/facts/Algebraic_differential_equation/g4efkb2F">algebraic differential equations</a>, but not conversely. Examples of functions which are solutions of algebraic differential equations but not Liouvillian include:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<ul><li>the <a href="/facts/Bessel_function/hSyvFPqC">Bessel functions</a> (except special cases);</li>
<li>the <a href="/facts/Hypergeometric_function/yTEC6NBJ">hypergeometric functions</a> (except special cases).</li></ul>
<p>Examples of functions which are <i>not</i> solutions of algebraic differential equations and thus not Liouvillian include all <a href="/facts/Transcendentally_transcendental_function/rP65a5NW">transcendentally transcendental functions</a>, such as:
</p>
<ul><li>the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>;</li>
<li>the <a href="/facts/Zeta_function/lf43angF">zeta function</a>.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Closed-form_expression/y0xCXlSk">Closed-form expression</a> – Mathematical formula involving a given set of operations</li>
<li><a href="/facts/Differential_Galois_theory/neKl6CLC">Differential Galois theory</a> – Study of Galois symmetry groups of differential fields</li>
<li><a href="/facts/Liouville%27s_theorem_(differential_algebra)/w9tx70G7">Liouville's theorem (differential algebra)</a> – Says when antiderivatives of elementary functions can be expressed as elementary functions</li>
<li><a href="/facts/Nonelementary_integral/IVp2vzq7">Nonelementary integral</a> – Integrals not expressible in closed-form from elementary functions</li>
<li><a href="/facts/Picard%E2%80%93Vessiot_theory/p62QH8wV">Picard–Vessiot theory</a> – Study of differential field extensions induced by linear differential equations</li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Davenport, J. H. (2007). "What Might 'Understand a Function' Mean". In Kauers, M.; Kerber, M.; Miner, R.; Windsteiger, W. (eds.). <a href="https://archive.org/details/towardsmechanize00kaue"><i>Towards Mechanized Mathematical Assistants</i></a>. Berlin/Heidelberg: Springer. pp. <a href="https://archive.org/details/towardsmechanize00kaue/page/n65">55</a>–65. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-73083-5.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>L. Chan, E.S. Cheb-Terrab, "Non-liouvillian solutions for second order Linear ODEs", Proceedings of the 2004 international symposium on Symbolic and algebraic computation (ISSAC '04), 2004, pp. 80–86 doi:10.1145/1005285.1005299 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Liouvillian function open-in-new

Liouvillian function