Matematika/Furje eilutės: Skirtumas tarp puslapio versijų

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202 eilutė:
:<math>a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}\phi(\xi) \cos(n\xi)\text{d}\xi=\frac{2}{\pi}\int_0^{\pi}\phi(\xi) \cos(n\xi)\text{d}\xi=\frac{2}{\pi}\cdot\frac{\pi}{l}\int_0^l f(x) \cos\frac{n\pi x}{l} \text{d}x=</math>
:<math>=\frac{2}{l}\int_0^l x^2 \cos\frac{n\pi x}{l} \text{d}x=\frac{2}{l}\left[\frac{x^2\sin\frac{n\pi x}{l}}{\frac{n\pi}{l}}|_0^l-\int_0^l 2x \cdot \frac{\sin\frac{n\pi x}{l}}{\frac{n\pi}{l}}\text{d}x\right]=</math>
:<math>=\frac{2}{l}\cdot \left(-\frac{2}{\frac{n\pi}{l}} \right)\int_0^{l} x \sin\frac{n\pi x}{l}\text{d}x=-\frac{4}{n\pi}\left(-\frac{x\cos(nx)}{n}|_0^{\pi}-\frac{-1}{n}\int_0^{\pi}\cos(nx)\text{d}x\right)=</math>
:<math>=-\frac{4}{n\pi}\left(-\frac{\pi\cos(n\pi)}{n}+\frac{1}{n^2}\sin(nx)|_0^{\pi}\right)=</math>
:<math>=\frac{4}{n^2}\cos(n\pi)=\frac{4}{n^2}\cdot (-1)^n=(-1)^n\frac{4}{n^2};</math>