Schlömilch's series is a Fourier series type expansion of twice continuously differentiable function in the interval ( 0 , π ) {\displaystyle (0,\pi )} in terms of the Bessel function of the first kind, named after the German mathematician Oskar Schlömilch, who derived the series in 1857. The real-valued function f ( x ) {\displaystyle f(x)} has the following expansion:
f ( x ) = a 0 + ∑ n = 1 ∞ a n J 0 ( n x ) , {\displaystyle f(x)=a_{0}+\sum _{n=1}^{\infty }a_{n}J_{0}(nx),}
where
a 0 = f ( 0 ) + 1 π ∫ 0 π ∫ 0 π / 2 u f ′ ( u sin θ ) d θ d u , a n = 2 π ∫ 0 π ∫ 0 π / 2 u cos n u f ′ ( u sin θ ) d θ d u . {\displaystyle {\begin{aligned}a_{0}&=f(0)+{\frac {1}{\pi }}\int _{0}^{\pi }\int _{0}^{\pi /2}uf'(u\sin \theta )\ d\theta \ du,\\a_{n}&={\frac {2}{\pi }}\int _{0}^{\pi }\int _{0}^{\pi /2}u\cos nu\ f'(u\sin \theta )\ d\theta \ du.\end{aligned}}}
Examples Some examples of Schlömilch's series are the following:
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