Principal component regression

In mathematics, the Anger function, introduced by Template:Harvs, is a function defined as

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

and is closely related to Bessel functions.

The Weber function (also known as Lommel-Weber function), introduced by Template:Harvs, is a closely related function defined by

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

and is closely related to Bessel functions of the second kind.

Relation between Weber and Anger functions

The Anger and Weber functions are related by

$\sin(\pi \nu )\mathbf {J} _{\nu }(z)=\cos(\pi \nu )\mathbf {E} _{\nu }(z)-\mathbf {E} _{-\nu }(z)$

$-\sin(\pi \nu )\mathbf {E} _{\nu }(z)=\cos(\pi \nu )\mathbf {J} _{\nu }(z)-\mathbf {J} _{-\nu }(z)$

so in particular if ν is not an integer they can be expressed as linear combinations of each other. If ν is an integer then Anger functions J_ν are the same as Bessel functions J_ν, and Weber functions can be expressed as finite linear combinations of Struve functions.

Differential equations

The Anger and Weber functions are solutions of inhomogenous forms of Bessel's equation $z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y=0$. More precisely, the Anger functions satisfy the equation

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

and the Weber functions satisfy the equation

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

References

• Template:AS ref
• C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29
• Template:Dlmf
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• Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
  
  my web-site http://himerka.com/
• 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