Practical Session 8: Fourier Series for Arbitrary Period. Half-Range Expansions

This practical session focuses on Fourier series for functions with arbitrary period and half-range expansions. Ten problems guide students through computing coefficients for functions with period 2L, using symmetry to simplify work, and deriving Fourier cosine and sine series on a half-interval. Applications include summing infinite series, such as the Leibniz formula for pi and the Basel problem, using both direct substitution and Parseval's identity. Convergence behavior at endpoints is examined for different extensions. The session also connects Fourier series to physical problems, specifically the heat equation with Dirichlet boundary conditions. Each problem includes a complete solution, making it suitable for classroom use or independent study.