| Version 2 |
Version 1 |
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Let $f$ be a Riemann integrable function from $[\pi,\pi]$ to $R$. Then the numbers
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Let $f$ be a Riemann integrable function from $[\pi,\pi]$ to $\sR$. Then the numbers
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| $$a_0^f = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx,$$ |
$$a_0^f = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx,$$ |
| $$a_n^f = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)cos(nx)dx,$$ |
$$a_n^f = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)cos(nx)dx,$$ |
| $$b_n^f = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)sin(nx)dx$$ |
$$b_n^f = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)sin(nx)dx$$ |
| are called the Fourier coefficients of the function $f.$ |
are called the Fourier coefficients of the function $f.$ |
| The trigonometric series |
The trigonometric series |
| $$ a_0^f + \sum_{n=1}^{\infty}(a_n^fcos(nx)+b_n^fsin(nx))$$ is called the trigonometric series of the function $f$, or Fourier series of the function $f.$ |
$$ a_0^f + \sum_{n=1}^{\infty}(a_n^fcos(nx)+b_n^fsin(nx))$$ is called the trigonometric series of the function $f$, or Fourier series of the function $f.$ |
