Magnetic dipole transition

2013-06-05T08:03:57Z

128.12.95.13:

In [[mathematics]], a '''radial function''' is a [[function (mathematics)|function]] defined on a [[Euclidean space]] '''R'''''n'' whose value at each point depends only on the distance between that point and the origin. For example, a radial function Φ in two dimensions has the form
:<math>\Phi(x,y) = \varphi(r), \quad r = \sqrt{x^2+y^2}</math>
where φ is a function of a single non-negative real variable. Radial functions are contrasted with [[spherical function]]s, and indeed any decent function on Euclidean space can be decomposed into a series consisting of radial and spherical parts: the [[solid spherical harmonic]] expansion.

A function is radial [[if and only if]] it is invariant under all [[rotation]]s leaving the origin fixed. That is, ''ƒ'' is radial if and only if
:<math>f\circ \rho = f\,</math>
for all {{nowrap|ρ ∈ SO(''n'')}}, the [[special orthogonal group]] in ''n'' dimensions. This characterization of radial functions makes it possible also to define radial [[distribution (mathematics)|distributions]]. These are distributions ''S'' on '''R'''''n'' such that
:<math>S[\phi] = S[\varphi\circ\rho]</math>
for every test function φ and rotation ρ.

Given any (locally integrable) function ''ƒ'', its radial part is given by averaging over spheres centered at the origin. To wit,
:<math>\phi(x) = \frac{1}{\omega_{n-1}}\int_{S^{n-1}} f(rx')\,dx'</math>
where ω''n''−1 is the surface area of the [[N sphere|(''n''−1)-sphere]] ''S''''n''−1, and {{nowrap|1=''r'' = |''x''|}}, {{nowrap|1=''x''′ = ''x''/r}}. It follows essentially by [[Fubini's theorem]] that a locally integrable function has a well-defined radial part at [[almost every]] ''r''.

The [[Fourier transform]] of a radial function is also radial, and so radial functions play a vital role in [[Fourier analysis]]. Furthermore, the Fourier transform of a radial function typically has stronger decay behavior at infinity than non-radial functions: for radial functions bounded in a neighborhood of the origin, the Fourier transform decays faster than ''R''−(''n''−1)/2. The [[Bessel functions]] are a special class of radial function that arise naturally in Fourier analysis as the radial [[eigenfunction]]s of the [[Laplacian]]; as such they appear naturally as the radial portion of the Fourier transform.

==See also==
* [[Radial basis function]]

==References==
*{{citation|last1=Stein|first1=Elias|authorlink1=Elias Stein|first2=Guido|last2=Weiss|authorlink2=Guido Weiss|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=978-0-691-08078-9|location=Princeton, N.J.}}.

{{DEFAULTSORT:Radial Function}}
[[Category:Harmonic analysis]]
[[Category:Rotational symmetry]]
[[Category:Types of functions]]

formulasearchengine - User contributions [en]

Magnetic dipole transition