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{2}{l\pi}\int_0^l x\cos\frac{n\pi }x}{l}^2\cos(nx)\text{d}x=\frac{2}{l\pi}\left([\frac{l\cdot x^2\sin\frac(nx)}{n\pi x}|_0^{l}}{n\pi}|_0^l-\frac{l2}{n\pi}\int_0^l\sin\frac{n\pi }x}{l}\sin(nx)\text{d}x \right)]=</math>
:<math>=-\frac{24}{l}\left(\frac{l^2\sin\frac{n\pi l}{l}}{n\pi}+\fracint_0^{l}{n^2\pi^2}x\cossin(nx)\fractext{n\pi d}x}{l}|_0^l \right)=-\frac{24}{ln\pi}\left(-\frac{l^2x\sincos(n\pinx)}{n}|_0^{\pi}+-\frac{l-1}{n^2}\piint_0^2}(\cos\frac{n\pi l}{l}-\cos(nx)\fractext{n\pi \cdot 0d}{l}) x\right)=</math>
:<math>=-\frac{2}{l}\cdot \frac{l4}{n^2\pi^2}(\cosleft(n\pi)-1) =\frac{2}{n^2\pi^2}(\cos(n\pi)-1)=}{n}+\frac{21}{n^2}\pi^2}sin((-1nx)|_0^n-1{\pi}\right)=\begin{cases}</math>
:<math>=\frac{4}{n^2}\cos(n\pi)=\frac{4}{n^2}\cdot (-1)^n=(-1)^n\frac{4}{n^2};</math>
0, \quad \text{kai} \; n \; \text{lyginis}, & \\
:čia pasinaudojome integravimu dalimis <math>\int u(x) v'(x) \mathsf{d}x = u(x)v(x) - \int u'(x) v(x) \mathsf{d}x. </math>
-\frac{4l}{n^2\pi^2}, \quad \text{kai} \; n \; \text{nelyginis}. &
:Furjė eilutė funkcijos <math>f(x)\;</math> yra tokia
\end{cases}</math>
:<math>|x|^2=\frac{l\pi^2}{23}-4\frac{4l}{\pi^2}left(\cos \frac{\picos x}{l1}+-\frac{1}{3^2}\cos\frac{3\pi x(2x)}{l2^2}+\frac{1}{5^2}\cos\frac{5\pi x(3x)}{l3^2}+-... \right).</math>
:Furjė eilutė funkcijos <math>f(x)\;</math> yra tokia
:<math>|x|=\frac{l}{2}-\frac{4l}{\pi^2}(\cos\frac{\pi x}{l}+\frac{1}{3^2}\cos\frac{3\pi x}{l}+\frac{1}{5^2}\cos\frac{5\pi x}{l}+...).</math>
 
==Nuorodos==