Jacobi–Anger expansion
In mathematics, the Jacobi–Anger expansion is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics, and in signal processing. This identity is named after the 19th-century mathematicians Carl Jacobi and Carl Theodor Anger.
The most general identity is given by:
where is the -th Bessel function of the first kind and is the imaginary unit,
Substituting by, we also get:
Using the relation valid for integer, the expansion becomes:The following real-valued variations are often useful as well:
