Question
For specific weight funciton \(w(x)=1\) and domain \(x\in(-L,L)\), then: [Important inner products]
Answer
- \(\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \cos\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \cos\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}\)
- \(\displaystyle\left\langle \sin\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\sin\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}\)
- \(\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=0\)
Question
For specific weight funciton \(w(x)=1\) and domain \(x\in(-L,L)\), then: [Important inner products]
Question
For specific weight funciton \(w(x)=1\) and domain \(x\in(-L,L)\), then: [Important inner products]
Answer
- \(\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \cos\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \cos\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}\)
- \(\displaystyle\left\langle \sin\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\sin\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}\)
- \(\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=0\)
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Important Inner Products For specific weight funciton \(w(x)=1\) and domain \(x\in(-L,L)\), then - \(\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \cos\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \cos\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}\) - \(\displaystyle\left\langle \sin\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\sin\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=L\cdot\delta_{nm}\) - \(\displaystyle\left\langle \cos\left(\frac{n\pi x}{L}\right), \sin\left(\frac{m\pi x}{L}\right)\right\rangle=\int^{L}_{-L}\cos\left(\frac{n\pi x}{L}\right)\ \sin\left(\frac{m\pi x}{L}\right)\cdot1dx=0\) Summary
| status | not learned | | measured difficulty | 37% [default] | | last interval [days] | |
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| repetition number in this series | 0 | | memorised on | | | scheduled repetition | |
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