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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s in the basis of their harmonics. It is useful in physics (for example, to convert between
plane wave In physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
s and cylindrical waves), and in
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
(to describe FM signals). This identity is named after the 19th-century mathematicians
Carl Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants and number theory. Biography Jacobi was ...
and Carl Theodor Anger. The most general identity is given by:Colton & Kress (1998) p. 32.Cuyt ''et al.'' (2008) p. 344. : e^ \equiv \sum_^ i^n\, J_n(z)\, e^, where J_n(z) is the n-th
Bessel function of the first kind Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
and i is the
imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
, i^2=-1. Substituting \theta by \theta-\frac, we also get: : e^ \equiv \sum_^ J_n(z)\, e^. Using the relation J_(z) = (-1)^n\, J_(z), valid for integer n, the expansion becomes: :e^ \equiv J_0(z)\, +\, 2\, \sum_^\, i^n\, J_n(z)\, \cos\, (n \theta).


Real-valued expressions

The following real-valued variations are often useful as well:Abramowitz & Stegun (1965
p. 361, 9.1.42–45
/ref> : \begin \cos(z \cos \theta) &\equiv J_0(z)+2 \sum_^(-1)^n J_(z) \cos(2n \theta), \\ \sin(z \cos \theta) &\equiv -2 \sum_^(-1)^n J_(z) \cos\left left(2n-1\right) \theta\right \\ \cos(z \sin \theta) &\equiv J_0(z)+2 \sum_^ J_(z) \cos(2n \theta), \\ \sin(z \sin \theta) &\equiv 2 \sum_^ J_(z) \sin\left left(2n-1\right) \theta\right \end


See also

*
Plane wave expansion Plane most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface * Plane (mathematics), generalizations of a geometrical plane Plane or planes may also refer to: Biology * Plane ...


Notes


References

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External links

* {{DEFAULTSORT:Jacobi-Anger expansion Special functions Mathematical identities