Jacobi–Anger expansion explained

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]}

e ^{i} \equiv \sum ^{inf ty} _{n =-inf ty} i ^{n} J _{n(z)} e ^{i} ,

where

J _{n(z)} is the n -th Bessel function of the first kind and i is the imaginary unit, $i^2=-1.$ Substituting $\backslash theta$ by $\backslash theta-\backslash frac$, we also get:e ^{i} \equiv \sum ^{inf ty} _{n =-inf ty} J _{n(z)} e ^{i} .

Using the relation

J _{-n} (z) = ( -1 ) ^{n} J _{n} (z), valid for integer n , the expansion becomes:^{[1] [2]} e ^{i} \equiv J _{0(z)} + 2 \sum ^{inf ty} _{n =1} i ^{n} J _{n(z)} \cos (n \theta).

Real-valued expressions

The following real-valued variations are often useful as well:^[3]

\begin{align} \cos(z \cos \theta) &\equiv J _{0(z)+2} \sum ^{inf ty} _{n =1} ( -1 ) ^{n} J _{2n} (z) \cos( 2 n \theta), \\ \sin(z \cos \theta) &\equiv -2 \sum ^{inf ty} _{n =1} ( -1 ) ^{n} J _{2n-1} (z) \cos\left[\left( 2 n -1 \right) \theta\right], \\ \cos(z \sin \theta) &\equiv J _{0(z)+2} \sum ^{inf ty} _{n =1} J _{2n} (z) \cos( 2 n \theta), \\ \sin(z \sin \theta) &\equiv 2 \sum ^{inf ty} _{n =1} J _{2n-1} (z) \sin\left[\left( 2 n -1 \right) \theta\right]. \end{align}

Similarly useful expressions from the Sung Series:^{[4] [5]}

\begin{align} \sum ^{inf ty} _{\nu =-inf ty} J _{\nu(x)} & = 1 , \\ \sum ^{inf ty} _{\nu =-inf ty} J _{2} (x) & = 1 , \\ \sum ^{inf ty} _{\nu =-inf ty} J _{3} (x) & = (1)/(\left[ 1+2 \cos{ 3) (x\sqrt{ 3)/() }}\right], \\ \sum_^\infty J_(x) &= \cos^2\left(\frac\right).\end

See also

• Plane wave expansion

External links

• Web site: Weisstein . Eric W. . Jacobi–Anger expansion . MathWorld — a Wolfram web resource . 2008-11-11 .

Notes and References

1. Colton & Kress (1998) p. 32.
2. Cuyt et al. (2008) p. 344.
3. Abramowitz & Stegun (1965) p. 361, 9.1.42–45
4. Sung . S. . Hovden . R. . On Infinite Series of Bessel functions of the First Kind . 2022 . math-ph . 2211.01148.
5. Watson . G.N. . A treatise on the theory of bessel functions . Cambridge University Press . 1922.