Jacobi–Anger expansion

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 z \cos \theta}=\sum_{n=-\infty}^{\infty} i^n\, J_n(z)\, e^{i n \theta}

and

e^{iz \sin \theta} = \sum_{n=-\infty}^\infty J_n(z) e^{in\theta},

where J_n(z) is the n-th Bessel function. Using the relation J_{-n}(z) = (-1)^n\, J_{n}(z), valid for integer n, the expansion becomes:[1][2]

e^{i z \cos \theta}=J_0(z)\, +\, 2\, \sum_{n=1}^{\infty}\, i^n\, J_n(z)\, \cos\, (n \theta).

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

\begin{align} \cos(z \cos \theta) &= J_0(z)+2 \sum_{n=1}^{\infty}(-1)^n J_{2n}(z) \cos(2n \theta), \\ \sin(z \cos \theta) &= -2 \sum_{n=1}^{\infty}(-1)^n J_{2n-1}(z) \cos\left[\left(2n-1\right) \theta\right], \\ \cos(z \sin \theta) &= J_0(z)+2 \sum_{n=1}^{\infty} J_{2n}(z) \cos(2n \theta), \\ \sin(z \sin \theta) &= 2 \sum_{n=1}^{\infty} J_{2n-1}(z) \sin\left[\left(2n-1\right) \theta\right]. \end{align}

Notes

  1. ↑ 1.0 1.1 Colton & Kress (1998) p. 32.
  2. ↑ 2.0 2.1 Cuyt et al. (2008) p. 344.
  3. ↑ Abramowitz & Stegun (1965) p. 361, 9.1.42–45

References

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