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Jacobi–Anger expansion

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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:

e i z cos ⁡ θ ≡ ∑ n = − ∞ ∞ i n J n ( z ) e i n θ , {\displaystyle e^{iz\cos \theta }\equiv \sum _{n=-\infty }^{\infty }i^{n}\,J_{n}(z)\,e^{in\theta },}

where J n ( z ) {\displaystyle J_{n}(z)} is the n {\displaystyle n} -th Bessel function of the first kind and i {\displaystyle i} is the imaginary unit, i 2 = − 1. {\textstyle i^{2}=-1.} Substituting θ {\textstyle \theta } by θ − π 2 {\textstyle \theta -{\frac {\pi }{2}}} , we also get:

e i z sin ⁡ θ ≡ ∑ n = − ∞ ∞ J n ( z ) e i n θ . {\displaystyle e^{iz\sin \theta }\equiv \sum _{n=-\infty }^{\infty }J_{n}(z)\,e^{in\theta }.}

Using the relation J − n ( z ) = ( − 1 ) n J n ( z ) , {\displaystyle J_{-n}(z)=(-1)^{n}\,J_{n}(z),} valid for integer n {\displaystyle n} , the expansion becomes:

e i z cos ⁡ θ ≡ J 0 ( z ) + 2 ∑ n = 1 ∞ i n J n ( z ) cos ( n θ ) . {\displaystyle e^{iz\cos \theta }\equiv J_{0}(z)\,+\,2\,\sum _{n=1}^{\infty }\,i^{n}\,J_{n}(z)\,\cos \,(n\theta ).}

Real-valued expressions

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

cos ⁡ ( z cos ⁡ θ ) ≡ J 0 ( z ) + 2 ∑ n = 1 ∞ ( − 1 ) n J 2 n ( z ) cos ⁡ ( 2 n θ ) , sin ⁡ ( z cos ⁡ θ ) ≡ − 2 ∑ n = 1 ∞ ( − 1 ) n J 2 n − 1 ( z ) cos ⁡ [ ( 2 n − 1 ) θ ] , cos ⁡ ( z sin ⁡ θ ) ≡ J 0 ( z ) + 2 ∑ n = 1 ∞ J 2 n ( z ) cos ⁡ ( 2 n θ ) , sin ⁡ ( z sin ⁡ θ ) ≡ 2 ∑ n = 1 ∞ J 2 n − 1 ( z ) sin ⁡ [ ( 2 n − 1 ) θ ] . {\displaystyle {\begin{aligned}\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,\\\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.\end{aligned}}}

Similarly useful expressions from the Sung Series:

∑ ν = − ∞ ∞ J ν ( x ) = 1 , ∑ ν = − ∞ ∞ J 2 ν ( x ) = 1 , ∑ ν = − ∞ ∞ J 3 ν ( x ) = 1 3 [ 1 + 2 cos ⁡ x 3 2 ] , ∑ ν = − ∞ ∞ J 4 ν ( x ) = cos 2 ⁡ ( x 2 ) . {\displaystyle {\begin{aligned}\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}}\left,\\\sum _{\nu =-\infty }^{\infty }J_{4\nu }(x)&=\cos ^{2}\left({\frac {x}{2}}\right).\end{aligned}}}

See also

Notes

  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. (2022). "On Infinite Series of Bessel functions of the First Kind". arXiv:2211.01148 .
  5. Watson, G.N. (1922). "A treatise on the theory of bessel functions". Cambridge University Press.

References

External links

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