Principal component regression

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

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

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

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

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 $z^2{y}^{\prime \prime }+z{y}^{\prime }+(z^2-{\nu }^2)y=0$. More precisely, the Anger functions satisfy the equation

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

and the Weber functions satisfy the equation

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

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