Fourier analysis problem (problem number 5)

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26 Chapter 1. THE GENESIS OF FOURIER ANA 5. Verify that f(x) = eins is periodic with period 27 and that 了。 cim ds = { 1 if n = 0, 0 if n +0. Use this fact to prove that if n, m > 1 we have so if n #m, cos na cos mu dr = 1 n = m, 7T - and similarly 1 S. sin nx sin mu dx = { 0 if n #m, 1 n = m. Finally, show that L sin nx cos mx dx = 0 for any n,m. (Hint: Calculate einte-imz + einzeime and ente -ima eina ima] 6. Prove that if f is a twice continuously differentiable function on R a solution of the equation f"(t) + f(t) = 0, then
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