Matematika/Furje eilutės: Skirtumas tarp puslapio versijų
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200 eilutė:
:<math>b_n=0;</math>
:<math>a_0=\frac{1}{\pi}\int_{-\pi}^{\pi}\phi(\xi) \text{d}\xi=\frac{2}{\pi}\int_{0}^{\pi}\phi(\xi) \text{d}\xi=\frac{2}{\pi}\int_0^l f(x)\cdot \frac{\pi}{l}\text{d}x=\frac{2}{l}\int_0^l x^2 \text{d}x=\frac{2}{l} \frac{x^3}{3}|_0^l=\frac{2}{l}\cdot \frac{l^3}{3}=\frac{2l^2}{3};</math>
:<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)
:<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^{\pi}-\int_0^{\pi}2x \cdot \frac{\sin(nx)}{\frac{n\pi}{l}}\text{d}x\right]=</math>
:<math>=-\frac{4}{n\pi}\int_0^{\pi}x\sin(nx)\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>
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