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Kelvin functions
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<h2 id="berx">ber(<i>x</i>)</h2> <p>For integers <i>n</i>, ber<i>n</i>(<i>x</i>) has the series expansion </p> b e r n ( x ) = ( x 2 ) n ∑ k ≥ 0 cos [ ( 3 n 4 + k 2 ) π ] k ! Γ ( n + k + 1 ) ( x 2 4 ) k , {\displaystyle \mathrm {ber} _{n}(x)=\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}{\frac {\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]}{k!\Gamma (n+k+1)}}\left({\frac {x^{2}}{4}}\right)^{k},} <p>where Γ(<i>z</i>) is the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>. The special case ber0(<i>x</i>), commonly denoted as just ber(<i>x</i>), has the series expansion </p> b e r ( x ) = 1 + ∑ k ≥ 1 ( − 1 ) k [ ( 2 k ) ! ] 2 ( x 2 ) 4 k {\displaystyle \mathrm {ber} (x)=1+\sum _{k\geq 1}{\frac {(-1)^{k}}{[(2k)!]^{2}}}\left({\frac {x}{2}}\right)^{4k}} <p>and <a href="/facts/Asymptotic_series/4W79BNve">asymptotic series</a> </p> b e r ( x ) ∼ e x 2 2 π x ( f 1 ( x ) cos α + g 1 ( x ) sin α ) − k e i ( x ) π {\displaystyle \mathrm {ber} (x)\sim {\frac {e^{\frac {x}{\sqrt {2}}}}{\sqrt {2\pi x}}}\left(f_{1}(x)\cos \alpha +g_{1}(x)\sin \alpha \right)-{\frac {\mathrm {kei} (x)}{\pi }}} , <p>where </p> α = x 2 − π 8 , {\displaystyle \alpha ={\frac {x}{\sqrt {2}}}-{\frac {\pi }{8}},} f 1 ( x ) = 1 + ∑ k ≥ 1 cos ( k π / 4 ) k ! ( 8 x ) k ∏ l = 1 k ( 2 l − 1 ) 2 {\displaystyle f_{1}(x)=1+\sum _{k\geq 1}{\frac {\cos(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}} g 1 ( x ) = ∑ k ≥ 1 sin ( k π / 4 ) k ! ( 8 x ) k ∏ l = 1 k ( 2 l − 1 ) 2 . {\displaystyle g_{1}(x)=\sum _{k\geq 1}{\frac {\sin(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}.} <h2 id="beix">bei(<i>x</i>)</h2> <p>For integers <i>n</i>, bei<i>n</i>(<i>x</i>) has the series expansion </p> b e i n ( x ) = ( x 2 ) n ∑ k ≥ 0 sin [ ( 3 n 4 + k 2 ) π ] k ! Γ ( n + k + 1 ) ( x 2 4 ) k . {\displaystyle \mathrm {bei} _{n}(x)=\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}{\frac {\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]}{k!\Gamma (n+k+1)}}\left({\frac {x^{2}}{4}}\right)^{k}.} <p>The special case bei0(<i>x</i>), commonly denoted as just bei(<i>x</i>), has the series expansion </p> b e i ( x ) = ∑ k ≥ 0 ( − 1 ) k [ ( 2 k + 1 ) ! ] 2 ( x 2 ) 4 k + 2 {\displaystyle \mathrm {bei} (x)=\sum _{k\geq 0}{\frac {(-1)^{k}}{[(2k+1)!]^{2}}}\left({\frac {x}{2}}\right)^{4k+2}} <p>and asymptotic series </p> b e i ( x ) ∼ e x 2 2 π x [ f 1 ( x ) sin α − g 1 ( x ) cos α ] − k e r ( x ) π , {\displaystyle \mathrm {bei} (x)\sim {\frac {e^{\frac {x}{\sqrt {2}}}}{\sqrt {2\pi x}}}[f_{1}(x)\sin \alpha -g_{1}(x)\cos \alpha ]-{\frac {\mathrm {ker} (x)}{\pi }},} <p>where α, f 1 ( x ) {\displaystyle f_{1}(x)} , and g 1 ( x ) {\displaystyle g_{1}(x)} are defined as for ber(<i>x</i>). </p> <h2 id="kerx">ker(<i>x</i>)</h2> <p>For integers <i>n</i>, ker<i>n</i>(<i>x</i>) has the (complicated) series expansion </p> k e r n ( x ) = − ln ( x 2 ) b e r n ( x ) + π 4 b e i n ( x ) + 1 2 ( x 2 ) − n ∑ k = 0 n − 1 cos [ ( 3 n 4 + k 2 ) π ] ( n − k − 1 ) ! k ! ( x 2 4 ) k + 1 2 ( x 2 ) n ∑ k ≥ 0 cos [ ( 3 n 4 + k 2 ) π ] ψ ( k + 1 ) + ψ ( n + k + 1 ) k ! ( n + k ) ! ( x 2 4 ) k . {\displaystyle {\begin{aligned}&\mathrm {ker} _{n}(x)=-\ln \left({\frac {x}{2}}\right)\mathrm {ber} _{n}(x)+{\frac {\pi }{4}}\mathrm {bei} _{n}(x)\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\cos \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}.\end{aligned}}} <p>The special case ker0(<i>x</i>), commonly denoted as just ker(<i>x</i>), has the series expansion </p> k e r ( x ) = − ln ( x 2 ) b e r ( x ) + π 4 b e i ( x ) + ∑ k ≥ 0 ( − 1 ) k ψ ( 2 k + 1 ) [ ( 2 k ) ! ] 2 ( x 2 4 ) 2 k {\displaystyle \mathrm {ker} (x)=-\ln \left({\frac {x}{2}}\right)\mathrm {ber} (x)+{\frac {\pi }{4}}\mathrm {bei} (x)+\sum _{k\geq 0}(-1)^{k}{\frac {\psi (2k+1)}{[(2k)!]^{2}}}\left({\frac {x^{2}}{4}}\right)^{2k}} <p>and the asymptotic series </p> k e r ( x ) ∼ π 2 x e − x 2 [ f 2 ( x ) cos β + g 2 ( x ) sin β ] , {\displaystyle \mathrm {ker} (x)\sim {\sqrt {\frac {\pi }{2x}}}e^{-{\frac {x}{\sqrt {2}}}}[f_{2}(x)\cos \beta +g_{2}(x)\sin \beta ],} <p>where </p> β = x 2 + π 8 , {\displaystyle \beta ={\frac {x}{\sqrt {2}}}+{\frac {\pi }{8}},} f 2 ( x ) = 1 + ∑ k ≥ 1 ( − 1 ) k cos ( k π / 4 ) k ! ( 8 x ) k ∏ l = 1 k ( 2 l − 1 ) 2 {\displaystyle f_{2}(x)=1+\sum _{k\geq 1}(-1)^{k}{\frac {\cos(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}} g 2 ( x ) = ∑ k ≥ 1 ( − 1 ) k sin ( k π / 4 ) k ! ( 8 x ) k ∏ l = 1 k ( 2 l − 1 ) 2 . {\displaystyle g_{2}(x)=\sum _{k\geq 1}(-1)^{k}{\frac {\sin(k\pi /4)}{k!(8x)^{k}}}\prod _{l=1}^{k}(2l-1)^{2}.} <h2 id="keix">kei(<i>x</i>)</h2> <p>For integer <i>n</i>, kei<i>n</i>(<i>x</i>) has the series expansion </p> k e i n ( x ) = − ln ( x 2 ) b e i n ( x ) − π 4 b e r n ( x ) − 1 2 ( x 2 ) − n ∑ k = 0 n − 1 sin [ ( 3 n 4 + k 2 ) π ] ( n − k − 1 ) ! k ! ( x 2 4 ) k + 1 2 ( x 2 ) n ∑ k ≥ 0 sin [ ( 3 n 4 + k 2 ) π ] ψ ( k + 1 ) + ψ ( n + k + 1 ) k ! ( n + k ) ! ( x 2 4 ) k . {\displaystyle {\begin{aligned}&\mathrm {kei} _{n}(x)=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} _{n}(x)-{\frac {\pi }{4}}\mathrm {ber} _{n}(x)\\&-{\frac {1}{2}}\left({\frac {x}{2}}\right)^{-n}\sum _{k=0}^{n-1}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {(n-k-1)!}{k!}}\left({\frac {x^{2}}{4}}\right)^{k}\\&+{\frac {1}{2}}\left({\frac {x}{2}}\right)^{n}\sum _{k\geq 0}\sin \left[\left({\frac {3n}{4}}+{\frac {k}{2}}\right)\pi \right]{\frac {\psi (k+1)+\psi (n+k+1)}{k!(n+k)!}}\left({\frac {x^{2}}{4}}\right)^{k}.\end{aligned}}} <p>The special case kei0(<i>x</i>), commonly denoted as just kei(<i>x</i>), has the series expansion </p> k e i ( x ) = − ln ( x 2 ) b e i ( x ) − π 4 b e r ( x ) + ∑ k ≥ 0 ( − 1 ) k ψ ( 2 k + 2 ) [ ( 2 k + 1 ) ! ] 2 ( x 2 4 ) 2 k + 1 {\displaystyle \mathrm {kei} (x)=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} (x)-{\frac {\pi }{4}}\mathrm {ber} (x)+\sum _{k\geq 0}(-1)^{k}{\frac {\psi (2k+2)}{[(2k+1)!]^{2}}}\left({\frac {x^{2}}{4}}\right)^{2k+1}} <p>and the asymptotic series </p> k e i ( x ) ∼ − π 2 x e − x 2 [ f 2 ( x ) sin β + g 2 ( x ) cos β ] , {\displaystyle \mathrm {kei} (x)\sim -{\sqrt {\frac {\pi }{2x}}}e^{-{\frac {x}{\sqrt {2}}}}[f_{2}(x)\sin \beta +g_{2}(x)\cos \beta ],} <p>where <i>β</i>, <i>f</i>2(<i>x</i>), and <i>g</i>2(<i>x</i>) are defined as for ker(<i>x</i>). </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bessel_function/hSyvFPqC">Bessel function</a></li></ul> <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_379.htm">"Chapter 9"</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. 379. <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>Olver, F. W. J.; Maximon, L. C. (2010), <a href="http://dlmf.nist.gov/10">"Bessel functions"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li>Weisstein, Eric W. "Kelvin Functions." From MathWorld—A Wolfram Web Resource. <a href="http://mathworld.wolfram.com/KelvinFunctions.html">[1]</a></li> <li>GPL-licensed C/C++ source code for calculating Kelvin functions at codecogs.com: <a href="https://web.archive.org/web/20070407195618/http://www.codecogs.com/d-ox/maths/special/bessel/kelvin.php">[2]</a></li></ul>