Tuesday, January 07, 2014
\(\int_{-\pi}^{\pi}\cos{nx}\cos{mx}\,dx\)
What is the value this integral ?
Surprisingly it's always zero when \(n \not = m\), but when \(n = m\) the value is exactly \(\pi\)!
How so ? Suppose \(n \not = m\), similar steps used to evalue \(\displaystyle \int_{-\pi}^{\pi}\sin{nx}\cos{mx}\,dx\) can be used to integrate and show that the value is zero. When \(n = m\), however, some of the terms "collapse" early into the constant \(\displaystyle \frac{1}{2}\) before the integration, and that's the key step to show why the value ends up as \(\pi\). Rather cool.
