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In [[mathematics]], the '''Jacobi–Anger expansion''' (or '''Jacobi–Anger identity''') is an expansion of exponentials of [[trigonometric function]]s in the basis of their harmonics. It is useful in physics (for example, to convert between [[plane wave]]s and [[cylindrical wave]]s), and in [[signal processing]] (to describe [[frequency modulation|FM]] signals). This identity is named after the 19th-century mathematicians [[Carl Gustav Jacob Jacobi|Carl Jacobi]] and [[Carl Theodor Anger]].

The most general identity is given by:<ref name=Colton_Kress_32>Colton & Kress (1998) p. 32.</ref><ref name=Cuyt_et_al_344>Cuyt ''et al.'' (2008) p. 344.</ref>

:<math>e^{i z \cos \theta}=\sum_{n=-\infty}^{\infty} i^n\, J_n(z)\, e^{i n \theta}</math>

and

:<math>e^{iz \sin \theta} = \sum_{n=-\infty}^\infty J_n(z) e^{in\theta},</math>

where <math>J_n(z)</math> is the ''n''-th [[Bessel function]]. Using the relation <math>J_{-n}(z) = (-1)^n\, J_{n}(z),</math> valid for integer ''n'', the expansion becomes:<ref name=Colton_Kress_32/><ref name=Cuyt_et_al_344/>

:<math>e^{i z \cos \theta}=J_0(z)\, +\, 2\, \sum_{n=1}^{\infty}\, i^n\, J_n(z)\, \cos\, (n \theta).</math>

The following real-valued variations are often useful as well:<ref>Abramowitz & Stegun (1965) [http://www.math.sfu.ca/~cbm/aands/page_361.htm p. 361, 9.1.42–45]</ref>

:<math>
\begin{align}
  \cos(z \cos \theta) &= J_0(z)+2 \sum_{n=1}^{\infty}(-1)^n J_{2n}(z) \cos(2n \theta),
  \\
  \sin(z \cos \theta) &= -2 \sum_{n=1}^{\infty}(-1)^n J_{2n-1}(z) \cos\left[\left(2n-1\right) \theta\right],
  \\
  \cos(z \sin \theta) &= J_0(z)+2 \sum_{n=1}^{\infty} J_{2n}(z) \cos(2n \theta),
  \\
  \sin(z \sin \theta) &= 2 \sum_{n=1}^{\infty} J_{2n-1}(z) \sin\left[\left(2n-1\right) \theta\right].
\end{align}
</math>

== Notes ==
{{reflist}}

== References ==
*{{Abramowitz_Stegun_ref2|9|355}}
* {{ citation
 | title=Inverse acoustic and electromagnetic scattering theory
 | series=Applied Mathematical Sciences
 | volume=93
 | last1=Colton
 | first1=David
 | last2=Kress
 | first2=Rainer
 | edition=2nd
 | year=1998
 | isbn=978-3-540-62838-5
}}
* {{ citation
 | title=Handbook of continued fractions for special functions
 | first1=Annie
 | last1=Cuyt
 | first2=Vigdis
 | last2=Petersen
 | first3=Brigitte
 | last3=Verdonk
 | first4=Haakon
 | last4=Waadeland
 | first5=William B.
 | last5=Jones
 | publisher=Springer
 | year=2008
 | isbn=978-1-4020-6948-2
}}

== External links ==
* {{cite web | last=Weisstein | first=Eric W. | title=Jacobi–Anger expansion | publisher= MathWorld — a Wolfram web resource | url=http://mathworld.wolfram.com/Jacobi-AngerExpansion.html | accessdate=2008-11-11 }}

{{DEFAULTSORT:Jacobi-Anger expansion}}
[[Category:Special functions]]
[[Category:Mathematical identities]]