In
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.]
:
where
is the
-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
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 ...
,
Substituting
by
, we also get:
:
Using the relation
valid for integer
, the expansion becomes:
[
:
]
Real-valued expressions
The following real-valued variations are often useful as well:[Abramowitz & Stegun (1965]
p. 361, 9.1.42–45
/ref>
:
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
*
*
*
External links
*
{{DEFAULTSORT:Jacobi-Anger expansion
Special functions
Mathematical identities