Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 17

Assume $L>0$. For any nonzero integer $k$,For $m\ne n$, the product-to-sum identity expresses the integrand as one half the sum of cosines with nonzero frequencies $m-n$ and $m+n$, so the integral is zero. For $m=n>0$, use ${\cos }^{2}u=(1+\cos 2u)/2$ to get $L$. Finally, for $m=n=0$ the integrand is one. The complete Fourier orthogonality formula isThe constant mode has twice the squared norm of a positive-frequency cosine mode.

For period $2$, use the Fourier series conventionThe Fourier coefficients have prefactor $1$ on $[-1,1]$. First, $a_0={\int }_{-1}^1{e}^{x}dx=2\sinh 1$. For $q=n\pi$ the useful antiderivatives areAt the endpoints $\sin (n\pi )=0$ and $\cos (n\pi )=(-1{)}^n$, soConsequently, on $-1<x<1$,The Dirichlet-Jordan convergence theorem justifies equality at every point of continuity of the periodic extension. At odd integers the extension has a jump discontinuity between $e$ and $e^{-1}$, so the series converges to their average $\cosh 1$, regardless of any assigned endpoint value. This is the $\kappa =L=1$ case of the Fourier series of a periodically extended exponential.