Difference between revisions of "Fourier series"
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:<math> f(t) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty}[a_n \cos(\omega_n t) + b_n \sin(\omega_n t)] </math> | :<math> f(t) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty}[a_n \cos(\omega_n t) + b_n \sin(\omega_n t)] </math> | ||
where, ''n'' is an integer and | where, ''n'' is an integer and | ||
| − | :<math> \omega_n = n\frac{2\pi}{T}</math> | + | :<math> \omega_n = n\frac{2\pi}{T}</math> |
| − | :<math>a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt</math> | + | :<math>a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt</math> |
| − | :<math>b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt </math> | + | :<math>b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt </math> |
Fourier series can be generalized to [[Fourier transformation]]s for non-perodic, piecewise continuous, [[square integrable function]]s. | Fourier series can be generalized to [[Fourier transformation]]s for non-perodic, piecewise continuous, [[square integrable function]]s. | ||
[[category: mathematics]] | [[category: mathematics]] | ||
Revision as of 00:43, March 12, 2008
Fourier series express a piecewise continous, periodic function as a linear combination of orthogonal Sine and Cosine functions.
The Fourier series of a function f(t) is of the form:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty}[a_n \cos(\omega_n t) + b_n \sin(\omega_n t)] }
where, n is an integer and
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_n = n\frac{2\pi}{T}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \cos(\omega_n t)\, dt}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n = \frac{2}{T} \int_{t_1}^{t_2} f(t) \sin(\omega_n t)\, dt }
Fourier series can be generalized to Fourier transformations for non-perodic, piecewise continuous, square integrable functions.