Question

1. Express function Faz) = sin(A sin tr), 0 < x < as a Fourier sine series. λ is a parameter. Hint: use the integral representation for Bessel functions.

Answer 1

Let us consider that f (x) is defined as integrable on an interval [0, L] Fourier Sine series for f on [0, L] is defined to be f (x) = Sigma^infinity_n = 1 b_n Sin (n pi x/L) , where b_n = 2/L integral^L_0 f (x) middot Sin (n pi x/L) dx (a) to find the fourier Sine series for f (x) = Cos (x) on [0, pi/2] Here we have L = pi/2 Therefore f (x) = Sigma^infinity_n = 1 b_n middot Sin (2 nx), where b_n = 4/pi integral^pi/2 f (x) middot Sin (2nx) dx now for f (x) = Cos (x) b_n = 4/pi integral^pi/2 Cos (x) middot Sin (2 nx) dx = 4/pi integral^pi /2_0 1/2 [Sin {2 nx + x} + Sin {2 nx - x}] middot dx Because Sin A Cos B = 1/2 [Sin (A + B) + Sin (A