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Important Inner Products

For specific weight funciton \(w(x)=1\) and domain \(x\in(-L,L)\), then
- \(\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \cos\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \cos\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}\)
- \(\displaystyle\left\langle \sin\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\sin\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}\)
- \(\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=0\)

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