In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics.
It is useful in physics (for example, to convert between plane waves and cylindrical waves), and in signal processing (to describe FM signals).
This identity is named after the 19th-century mathematicians Carl Jacobi and Carl Theodor Anger.
The most general identity is given by:[1][2] where
-th Bessel function of the first kind and
is the imaginary unit,
, the expansion becomes:[1][2] The following real-valued variations are often useful as well:[3] Similarly useful expressions from the Sung Series: [4] [5]