Fourier Series

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Showing coefficients from n = -1 to 1

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Function phase-shifted by 0π radians

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Choose Wave category SquareWave SawtoothWave TriangleWave Full-wave Sine Rectified Full-wave Cosine Rectified Half-wave Sine Rectified Half-wave Cosine Rectified

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Component Waves

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$$f(x) =\dfrac{4}{\pi} \sum_{i=1,3,5,...}^\infty \dfrac{1}{n} \sin (\dfrac{n\pi x}{L})$$

Specific Coefficient Formula:

$$C_n = 0, n = 0, \pm 2, \pm 4, ...$$
$$C_n = \frac{1}{j \pi n}, n = \pm 1, \pm 3, ...$$ $$C_n = 0, n = 0$$
$$C_n = \frac{j}{\pi n}, n \neq 0$$ $$C_n = 0.5, n = 0$$
$$C_n = 0, n = \pm 2, \pm 4, ...$$
$$C_n = \frac{-2}{(\pi n)^{2}}, n = \pm 1, \pm 3, ...$$ $$C_n = \frac{-2}{\pi(4n^{2}-1)}$$ $$C_n = \frac{2}{\pi}, n = 0$$
$$C_n = \frac{2}{\pi (4n^{2}-1)}, n = \pm 1, \pm 3, ...$$
$$C_n = \frac{-2}{\pi (4n^{2}-1)}, n = \pm 2, \pm 4, ...$$ $$C_n = \frac{1}{\pi}, n = 0, \pm 2, \pm 4, ...$$
$$C_n = 0, n = \pm 1, \pm 3, ...$$ $$C_n = \frac{-1^{n/2}}{\pi (1-n^{2})}, n = 0, \pm 2, \pm 4, ...$$
$$C_n = 0, n = \pm 1, \pm 3, ...$$

General Coefficient Equation:

$$C_n = \dfrac{1}{T} \int_{0}^{T} f(t) e^{\Big(-j\dfrac{2\pi nt}{T}\Big)} \,dt\ * e^{-j{\omega}_k a }$$