Chapter 7.9, Problem 25P

Suppose that $f(x)$ and its derivative $f^{\prime }(x)$ are both expanded in Fourier series on $(-\pi ,\pi ).$ Call the coefficients in the $f(x)$ series $a_n$ and $b_n$ and the coefficients in the $f^{\prime }(x)$ series $a_n^{\prime }$ and $b_n^{\prime }.$ Write the integral for $a_n$ [equation (5.9)] and integrate it by parts to get an integral of $f^{\prime }(x)\sin nx.$ Recognize this integral in terms of $b_n^{\prime }$ [equation (5.10) for and so show that $b_n^{\prime }=-n{a}_n.$ (In the integration by parts, the integrated term is zero because $f(\pi )=f(-\pi )$ since $f$ is continuous sketch several periods.). Find a similar relation for $a_n^{\prime }$ and $b_n.$ Now show that this is the result you get by differentiating the $f(x)$ series term by term. Thus you have shown that the Fourier series for $f^{\prime }(x)$ is correctly given by differentiating the $f(x)$ series term by term (assuming that $f^{\prime }(x)$ is expandable in a Fourier series).