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determination of Fourier coefficients
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(Derivation)
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Suppose that the real function $f$ may be presented as sum of the Fourier series:
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(1) |
Therefore, $f$ is periodic with period $2\pi$ . For expressing the Fourier coefficients $a_m$ and $b_m$ with the function itself, we first multiply the series (1) by $\cos{nx}$ ($n \in \mathbb{Z}$ ) and integrate from $-\pi$ to
$\pi$ . Supposing that we can integrate termwise, we may write
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(2) |
When $n = 0$ , the equation (2) reads
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(3) |
since in the sum of the right hand side, only the first addend is distinct from zero.
When $n$ is a positive integer, we use the product formulas of the trigonometric identities, getting $$\int_{-\pi}^\pi\cos{mx}\cos{nx}\,dx = \frac{1}{2}\int_{-\pi}^\pi[\cos(m-n)x+\cos(m+n)x]\,dx,$$ $$\int_{-\pi}^\pi\sin{mx}\cos{nx}\,dx = \frac{1}{2}\int_{-\pi}^\pi[\sin(m-n)x+\sin(m+n)x]\,dx.$$ The latter expression vanishes
always, since the sine is an odd function. If $m \neq n$ , the former equals zero because the antiderivative consists of sine terms which vanish at multiples of $\pi$ ; only in the case $m = n$ we obtain from it a non-zero result $\pi$ . Then (2) reads
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(4) |
to which we can include as a special case the equation (3).
By multiplying (1) by $\sin{nx}$ and integrating termwise, one obtains similarly
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(5) |
The equations (4) and (5) imply the formulas $$a_n \;=\; \frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos{nx}\,dx \quad (n = 0,\,1,\,2,\,\ldots)$$ and $$b_n \;=\; \frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin{nx}\,dx \quad (n = 1,\,2,\,3,\,\ldots)$$ for finding the values of the Fourier coefficients of $f$ .
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"determination of Fourier coefficients" is owned by pahio.
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Cross-references: formulas, imply, multiples, terms, antiderivative, odd function, sine, vanishes, expression, trigonometric identities, product formulas, integer, positive, right hand side, equation, integrate, series, function, Fourier coefficients, period, periodic, Fourier series, sum, real function
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This is version 4 of determination of Fourier coefficients, born on 2008-09-12, modified 2009-05-04.
Object id is 11022, canonical name is DeterminationOfFourierCoefficients.
Accessed 1000 times total.
Classification:
| AMS MSC: | 42A16 (Fourier analysis :: Fourier analysis in one variable :: Fourier coefficients, Fourier series of functions with special properties, special Fourier series) | | | 26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type) |
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Pending Errata and Addenda
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