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Anger function
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<h2 id="relation-between-weber-and-anger-functions">Relation between Weber and Anger functions</h2> <p>The Anger and Weber functions are related by </p> sin ( π ν ) J ν ( z ) = cos ( π ν ) E ν ( z ) − E − ν ( z ) , − sin ( π ν ) E ν ( z ) = cos ( π ν ) J ν ( z ) − J − ν ( z ) , {\displaystyle {\begin{aligned}\sin(\pi \nu )\mathbf {J} _{\nu }(z)&=\cos(\pi \nu )\mathbf {E} _{\nu }(z)-\mathbf {E} _{-\nu }(z),\\-\sin(\pi \nu )\mathbf {E} _{\nu }(z)&=\cos(\pi \nu )\mathbf {J} _{\nu }(z)-\mathbf {J} _{-\nu }(z),\end{aligned}}} <p>so in particular if ν is not an <a href="/facts/Integer/PUwwrawx">integer</a> they can be expressed as linear combinations of each other. If ν is an integer then Anger functions Jν are the same as Bessel functions <i>J</i>ν, and Weber functions can be expressed as finite linear combinations of <a href="/facts/Struve_function/cU5tVgjR">Struve functions</a>. </p> <h2 id="power-series-expansion">Power series expansion</h2> <p>The Anger function has the <a href="/facts/Power_series/vYh6sTps">power series</a> expansion<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> J ν ( z ) = cos π ν 2 ∑ k = 0 ∞ ( − 1 ) k z 2 k 4 k Γ ( k + ν 2 + 1 ) Γ ( k − ν 2 + 1 ) + sin π ν 2 ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 2 2 k + 1 Γ ( k + ν 2 + 3 2 ) Γ ( k − ν 2 + 3 2 ) . {\displaystyle \mathbf {J} _{\nu }(z)=\cos {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{4^{k}\Gamma \left(k+{\frac {\nu }{2}}+1\right)\Gamma \left(k-{\frac {\nu }{2}}+1\right)}}+\sin {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2^{2k+1}\Gamma \left(k+{\frac {\nu }{2}}+{\frac {3}{2}}\right)\Gamma \left(k-{\frac {\nu }{2}}+{\frac {3}{2}}\right)}}.} <p>While the Weber function has the power series expansion<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> E ν ( z ) = sin π ν 2 ∑ k = 0 ∞ ( − 1 ) k z 2 k 4 k Γ ( k + ν 2 + 1 ) Γ ( k − ν 2 + 1 ) − cos π ν 2 ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 2 2 k + 1 Γ ( k + ν 2 + 3 2 ) Γ ( k − ν 2 + 3 2 ) . {\displaystyle \mathbf {E} _{\nu }(z)=\sin {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{4^{k}\Gamma \left(k+{\frac {\nu }{2}}+1\right)\Gamma \left(k-{\frac {\nu }{2}}+1\right)}}-\cos {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2^{2k+1}\Gamma \left(k+{\frac {\nu }{2}}+{\frac {3}{2}}\right)\Gamma \left(k-{\frac {\nu }{2}}+{\frac {3}{2}}\right)}}.} <h2 id="differential-equations">Differential equations</h2> <p>The Anger and Weber functions are solutions of inhomogeneous forms of Bessel's equation </p> z 2 y ′ ′ + z y ′ + ( z 2 − ν 2 ) y = 0. {\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y=0.} <p>More precisely, the Anger functions satisfy the equation<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> z 2 y ′ ′ + z y ′ + ( z 2 − ν 2 ) y = ( z − ν ) sin ( π ν ) π , {\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y={\frac {(z-\nu )\sin(\pi \nu )}{\pi }},} <p>and the Weber functions satisfy the equation<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> z 2 y ′ ′ + z y ′ + ( z 2 − ν 2 ) y = − z + ν + ( z − ν ) cos ( π ν ) π . {\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y=-{\frac {z+\nu +(z-\nu )\cos(\pi \nu )}{\pi }}.} <h2 id="recurrence-relations">Recurrence relations</h2> <p>The Anger function satisfies this inhomogeneous form of <a href="/facts/Recurrence_relation/JolXC71M">recurrence relation</a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> z J ν − 1 ( z ) + z J ν + 1 ( z ) = 2 ν J ν ( z ) − 2 sin π ν π . {\displaystyle z\mathbf {J} _{\nu -1}(z)+z\mathbf {J} _{\nu +1}(z)=2\nu \mathbf {J} _{\nu }(z)-{\frac {2\sin \pi \nu }{\pi }}.} <p>While the Weber function satisfies this inhomogeneous form of <a href="/facts/Recurrence_relation/JolXC71M">recurrence relation</a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> z E ν − 1 ( z ) + z E ν + 1 ( z ) = 2 ν E ν ( z ) − 2 ( 1 − cos π ν ) π . {\displaystyle z\mathbf {E} _{\nu -1}(z)+z\mathbf {E} _{\nu +1}(z)=2\nu \mathbf {E} _{\nu }(z)-{\frac {2(1-\cos \pi \nu )}{\pi }}.} <h2 id="delay-differential-equations">Delay differential equations</h2> <p>The Anger and Weber functions satisfy these homogeneous forms of <a href="/facts/Delay_differential_equation/fM5wzXKr">delay differential equations</a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> J ν − 1 ( z ) − J ν + 1 ( z ) = 2 ∂ ∂ z J ν ( z ) , {\displaystyle \mathbf {J} _{\nu -1}(z)-\mathbf {J} _{\nu +1}(z)=2{\dfrac {\partial }{\partial z}}\mathbf {J} _{\nu }(z),} E ν − 1 ( z ) − E ν + 1 ( z ) = 2 ∂ ∂ z E ν ( z ) . {\displaystyle \mathbf {E} _{\nu -1}(z)-\mathbf {E} _{\nu +1}(z)=2{\dfrac {\partial }{\partial z}}\mathbf {E} _{\nu }(z).} <p>The Anger and Weber functions also satisfy these inhomogeneous forms of <a href="/facts/Delay_differential_equation/fM5wzXKr">delay differential equations</a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> z ∂ ∂ z J ν ( z ) ± ν J ν ( z ) = ± z J ν ∓ 1 ( z ) ± sin π ν π , {\displaystyle z{\dfrac {\partial }{\partial z}}\mathbf {J} _{\nu }(z)\pm \nu \mathbf {J} _{\nu }(z)=\pm z\mathbf {J} _{\nu \mp 1}(z)\pm {\frac {\sin \pi \nu }{\pi }},} z ∂ ∂ z E ν ( z ) ± ν E ν ( z ) = ± z E ν ∓ 1 ( z ) ± 1 − cos π ν π . {\displaystyle z{\dfrac {\partial }{\partial z}}\mathbf {E} _{\nu }(z)\pm \nu \mathbf {E} _{\nu }(z)=\pm z\mathbf {E} _{\nu \mp 1}(z)\pm {\frac {1-\cos \pi \nu }{\pi }}.} <ul><li><a href="/facts/Milton_Abramowitz/6yRyez7P">Abramowitz, Milton</a>; <a href="/facts/Irene_Stegun/FjSS1LDR">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a href="http://www.math.ubc.ca/~cbm/aands/page_498.htm">"Chapter 12"</a>. <a href="/facts/Abramowitz_and_Stegun/CNGSV3Ed"><i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i></a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 498. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-61272-0. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://www.loc.gov/item/65012253">65-12253</a>.</li> <li>C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29</li> <li>Prudnikov, A.P. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Weber_function">"Weber function"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li> <li><a href="/facts/G.N._Watson/VLfFjgjh">G.N. Watson</a>, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952)</li> <li>H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Prudnikov, A.P. (2001) [1994], "Anger function", Encyclopedia of Mathematics, EMS Press <a href="/wiki/Anatolii_Platonovich_Prudnikov" target="_blank">/wiki/Anatolii_Platonovich_Prudnikov</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Paris, R. B. (2010), "Anger-Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Paris, R. B. (2010), "Anger-Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Paris, R. B. (2010), "Anger-Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Paris, R. B. (2010), "Anger-Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Paris, R. B. (2010), "Anger-Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Paris, R. B. (2010), "Anger-Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Paris, R. B. (2010), "Anger-Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Paris, R. B. (2010), "Anger-Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. <a href="978-0-521-19225-5" target="_blank">978-0-521-19225-5</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>