安格尔函数

安格尔函数是英国数学家安格尔在1855年首先研究的积分函数

${\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta }$

Anger function 2D animiation

AngerJ(v,x) Maple animation for v=5

Anger function 3D plot

微分方程

安格尔函数满足下列微分方程

${\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y=(z-\nu )\sin(\pi z)/\pi }$

参考文献

• C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29
• Paris, R. B., Anger-Weber Functions, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
• Prudnikov, A.P., Anger function, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
• Prudnikov, A.P., Weber function, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
• G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952)
• H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76