著名的贝塞尔展开公式
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In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion ofexponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (forexample, to convert between plane waves and cylindrical waves), and in signal processing (todescribe FM signals). This identity is named after the 19th-century mathematicians Carl Jacobiand Carl Theodor Anger.
The most general identity is given by:[1][2]
where Jn(z) is the n-th Bessel function. Using the relation
integer n, the expansion becomes:[1][2] valid for
The following real-valued variations are often useful as well:[3]
1.^ a b Colton & Kress (1998) p. 32.
2.^ a b Cuyt et al. (2008) p. 344.
3.^ Abramowitz & Stegun (1965) p. 361, 9.1.42–45 (http://www.math.sfu.ca/~cbm/aands/page_361.htm)Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9" (http://www.math.sfu.ca
/~cbm/aands/page_355.htm) , Handbook of Mathematical Functions with Formulas, Graphs,and Mathematical Tables, New York: Dover, pp. 355, MR0167642 (http://www.77cn.com.cn/mathscinet-getitem?mr=0167642) , ISBN 978-0486612720, http://www.math.sfu.ca
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