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Wright omega function
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<h2 id="uses">Uses</h2> <p>One of the main applications of this function is in the resolution of the equation <i>z</i> = ln(<i>z</i>), as the only solution is given by <i>z</i> = <i>e</i>−ω(<i>π</i> <i>i</i>). </p><p><i>y</i> = ω(<i>z</i>) is the unique solution, when z ≠ x ± i π {\displaystyle z\neq x\pm i\pi } for <i>x</i> ≤ −1, of the equation <i>y</i> + ln(<i>y</i>) = <i>z</i>. Except for those two values, the Wright omega function is <a href="/facts/Continuous_function/sKbl02pB">continuous</a>, even <a href="/facts/Analytic_function/JhL3z8FP">analytic</a>. </p> <h2 id="properties">Properties</h2> <p>The Wright omega function satisfies the relation W k ( z ) = ω ( ln ( z ) + 2 π i k ) {\displaystyle W_{k}(z)=\omega (\ln(z)+2\pi ik)} . </p><p>It also satisfies the <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> </p> d ω d z = ω 1 + ω {\displaystyle {\frac {d\omega }{dz}}={\frac {\omega }{1+\omega }}} <p>wherever ω is analytic (as can be seen by performing <a href="/facts/Separation_of_variables/oJdkw5xq">separation of variables</a> and recovering the equation ln ( ω ) + ω = z {\displaystyle \ln(\omega )+\omega =z} ), and as a consequence its <a href="/facts/Integral/V7lyV997">integral</a> can be expressed as: </p> ∫ ω n d z = { ω n + 1 − 1 n + 1 + ω n n if n ≠ − 1 , ln ( ω ) − 1 ω if n = − 1. {\displaystyle \int \omega ^{n}\,dz={\begin{cases}{\frac {\omega ^{n+1}-1}{n+1}}+{\frac {\omega ^{n}}{n}}&{\mbox{if }}n\neq -1,\\\ln(\omega )-{\frac {1}{\omega }}&{\mbox{if }}n=-1.\end{cases}}} <p>Its <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> around the point a = ω a + ln ( ω a ) {\displaystyle a=\omega _{a}+\ln(\omega _{a})} takes the form : </p> ω ( z ) = ∑ n = 0 + ∞ q n ( ω a ) ( 1 + ω a ) 2 n − 1 ( z − a ) n n ! {\displaystyle \omega (z)=\sum _{n=0}^{+\infty }{\frac {q_{n}(\omega _{a})}{(1+\omega _{a})^{2n-1}}}{\frac {(z-a)^{n}}{n!}}} <p>where </p> q n ( w ) = ∑ k = 0 n − 1 ⟨ ⟨ n + 1 k ⟩ ⟩ ( − 1 ) k w k + 1 {\displaystyle q_{n}(w)=\sum _{k=0}^{n-1}{\bigg \langle }\!\!{\bigg \langle }{\begin{matrix}n+1\\k\end{matrix}}{\bigg \rangle }\!\!{\bigg \rangle }(-1)^{k}w^{k+1}} <p>in which </p> ⟨ ⟨ n k ⟩ ⟩ {\displaystyle {\bigg \langle }\!\!{\bigg \langle }{\begin{matrix}n\\k\end{matrix}}{\bigg \rangle }\!\!{\bigg \rangle }} <p>is a <a href="/facts/Eulerian_number/lMWud3wa">second-order Eulerian number</a>. </p> <h2 id="values">Values</h2> ω ( 0 ) = W 0 ( 1 ) ≈ 0.56714 ω ( 1 ) = 1 ω ( − 1 ± i π ) = − 1 ω ( − 1 3 + ln ( 1 3 ) + i π ) = − 1 3 ω ( − 1 3 + ln ( 1 3 ) − i π ) = W − 1 ( − 1 3 e − 1 3 ) ≈ − 2.237147028 {\displaystyle {\begin{array}{lll}\omega (0)&=W_{0}(1)&\approx 0.56714\\\omega (1)&=1&\\\omega (-1\pm i\pi )&=-1&\\\omega (-{\frac {1}{3}}+\ln \left({\frac {1}{3}}\right)+i\pi )&=-{\frac {1}{3}}&\\\omega (-{\frac {1}{3}}+\ln \left({\frac {1}{3}}\right)-i\pi )&=W_{-1}\left(-{\frac {1}{3}}e^{-{\frac {1}{3}}}\right)&\approx -2.237147028\\\end{array}}} <h2 id="plots">Plots</h2> <h2 id="notes">Notes</h2> <ul><li><a href="https://www.uwo.ca/apmaths/faculty/jeffrey/pdfs/wrightomega.pdf">"The Wright ω function", Robert Corless and David Jeffrey</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Not to be confused with the Fox–Wright function, also known as Wright function. <a href="/wiki/Fox%E2%80%93Wright_function" target="_blank">/wiki/Fox%E2%80%93Wright_function</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>