📘 fourier analysis

1. **Problem statement:** Expand $f(x) = \cos x$ for $0 < x < \pi$ in a Fourier sine series and determine the values of $f(x)$ at $x=0$ and $x=\pi$ for convergence. 2. **Fourier si

1. **Problem:** Find the Fourier series representation of the function $$f(x) = x$$ for $$-\pi < x < \pi$$, assuming it is $$2\pi$$-periodic. 2. **Formula and rules:** The Fourier

1. **Problem statement:** Find the Fourier cosine integral representation of the function $$f(x) = \begin{cases} \sin x, & 0 \leq x \leq \pi \\ 0, & x > \pi \end{cases}$$

1. **Problem Statement:** Find the Fourier transform of the piecewise function

1. **Problem (a):** Find the Fourier series expansion of $$f(x) = \begin{cases} \pi + x, & -\pi < x < 0 \\ \pi - x, & 0 < x < \pi \end{cases}$$

1. Stating the problem: Given the Fourier transform problem for the function

1. We are asked to find the Fourier cosine and sine transforms of the function $f(x) = e^{-kx}$ with $k > 0$ and $x > 0$. 2. The Fourier cosine transform $F_c(w)$ is defined as: