Jacobi–Anger expansion

Short description: Expansion of exponentials of trigonometric functions in the basis of their harmonics

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:

[math]\displaystyle{ e^{i z \cos \theta} \equiv \sum_{n=-\infty}^{\infty} i^n\, J_n(z)\, e^{i n \theta}, }[/math]

where [math]\displaystyle{ J_n(z) }[/math] is the [math]\displaystyle{ n }[/math]-th Bessel function of the first kind and [math]\displaystyle{ i }[/math] is the imaginary unit, [math]\displaystyle{ i^2=-1. }[/math] Substituting [math]\displaystyle{ \theta }[/math] by [math]\displaystyle{ \theta-\frac{\pi}{2} }[/math], we also get:

[math]\displaystyle{ e^{i z \sin \theta} \equiv \sum_{n=-\infty}^{\infty} J_n(z)\, e^{i n \theta}. }[/math]

Using the relation [math]\displaystyle{ J_{-n}(z) = (-1)^n\, J_{n}(z), }[/math] valid for integer [math]\displaystyle{ n }[/math], the expansion becomes:

[math]\displaystyle{ e^{i z \cos \theta} \equiv J_0(z)\, +\, 2\, \sum_{n=1}^{\infty}\, i^n\, J_n(z)\, \cos\, (n \theta). }[/math]

Real-valued expressions

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

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

Similarly useful expressions from the Sung Series:

[math]\displaystyle{ \begin{align} \sum_{\nu=-\infty}^\infty J_\nu(x) &= 1, \\ \sum_{\nu=-\infty}^\infty J_{2 \nu}(x) &= 1, \\ \sum_{\nu=-\infty}^\infty J_{3 \nu}(x) &= \frac{1}{3} \bigg[1+2\cos{\frac{x\sqrt{3}}{2}}\bigg], \\ \sum_{\nu=-\infty}^\infty J_{4 \nu}(x) &= \cos^2(\frac{x}{2}). \end{align} }[/math]

See also

• Plane wave expansion

References

• Abramowitz, Milton; Stegun, Irene Ann, eds (1983). "Chapter 9". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 355. LCCN 65-12253. ISBN 978-0-486-61272-0. http://www.math.sfu.ca/~cbm/aands/page_355.htm. 
• Colton, David; Kress, Rainer (1998), Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences, 93 (2nd ed.), ISBN 978-3-540-62838-5 
• Handbook of continued fractions for special functions, Springer, 2008, ISBN 978-1-4020-6948-2 

External links

• Weisstein, Eric W.. "Jacobi–Anger expansion". MathWorld — a Wolfram web resource. http://mathworld.wolfram.com/Jacobi-AngerExpansion.html. Retrieved 2008-11-11.