Just to be clear, the fact that the function $g$ is periodic is throwing me off as to how I even integrate it. I'm probably overthinking - I guarantee you it's that, and not that I don't know how to find integrals.
2026-03-25 17:53:08.1774461188
Wave equation on infinite line with piecewise $2\pi$-periodic i.c.
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So far you have $u(x,t)=\frac{1}{2a}(G(x+at)-G(x-at)$ where $G$ is an antiderivative of $g$. A periodic function whose integral over its period is $0$ has a periodic antiderivative. In this case, it's $|x|$ restricted to $[-\pi,\pi]$ and extended periodically. There are various ways to express this function in a formula, for example $G(x) = \operatorname{dist}(x, 2\pi \mathbb{Z})$ which means the distance from $x$ to the nearest element of the set $2\pi \mathbb{Z}$.