Fourier Series
The Fourier Series of a function 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(x)} over an interval 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,b]} is an infinite series that defines the function 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} as a sum of sines and cosines.
Definition
where
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 \begin{align} a_0 &= \frac{1}{b-a} \, \int_a^b f(x) \, \mathrm{d}x \\ a_k &= \frac{2}{b-a} \, \int_a^b f(x) \, \cos{\left( \frac{2 \pi}{b-a} \, k \, x \right)} \, \mathrm{d}x \\ b_k &= \frac{2}{b-a} \, \int_a^b f(x) \, \sin{\left( \frac{2 \pi}{b-a} \, k \, x \right)} \, \mathrm{d}x \\ \end{align}}
Where 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-a} represents the period of the function. The resulting series, if plotted beyond 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 \left[ a,b \right]} would be a periodic extension of the function.
Complex Definition
Using Euler's formula, (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 \mathrm{e}^{i \, \theta} = \cos{\theta} + i \, \sin{\theta}} ), we can rewrite the definition above as
where
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 \begin{align} c_k &= \frac{1}{b-a} \, \int_a^b f(x) \, \mathrm{e}^{-i \, \frac{2 \pi}{b-a} \, k \, x} \, \mathrm{d}x \end{align}}
Even and Odd Extension Variants
By default, the Fourier series simply duplicates the function 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} on the interval 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,b]} and shifts it by 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 k \, (b-a)} for 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 k \in \mathbb{Z}} . Alternatively, even and odd periodic extensions can be constructed
Even Extension
In this case, the Fourier series simulates an even function. To accomplish this, we fix the left endpoint at the origin (i.e. 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=0} ) and construct a Fourier series with the property 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(-x) = f(x)} . This series will involve only cosine terms.
where
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 \begin{align} a_0 &= \frac{1}{b} \int_0^b f(x) \, \mathrm{d}x \\ a_k &= \frac{2}{b} \int_0^b f(x) \, \cos{\left( \frac{\pi}{b} \, k \, x \right)} \, \mathrm{d}x \end{align}}
Let 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 g(x)} be the even periodic extension of 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(x)} on the interval 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, b]} such that the image of 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 g} on the interval 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,0]} is the image of 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} on the interval 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 [0,b]} mirrored about the 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 y} -axis.
We will compute the Fourier series of 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 g(x)} . We begin by finding 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_0} :
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 \begin{align} a_0 &= \frac{1}{b - (-b)} \, \int_{-b}^b g(x) \, \mathrm{d}x \\ &= \frac{1}{2b} \, \left( \int_{-b}^0 g(x) \, \mathrm{d}x + \int_0^b g(x) \, \mathrm{d}x \right) \\ &= \frac{1}{2b} \, \left( -\int_{b}^{0} g(-x) \, \mathrm{d}x + \int_{a}^{b} g(x) \, \,\mathrm{d}x \right) & u &= 2a-x & \mathrm{d}u &= -\mathrm{d}x \\ &= \frac{1}{2b} \, \int_{0}^{b} g(-x) + g(x) \,\mathrm{d}x \end{align}}
By the symmetry of 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 g(x)} described above, we know that 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 g(x) = f(x)} for all 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 x \in [a,b]} and that 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 g(-x) = f(x)} for all 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 x \in [-b,0]} .
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 \begin{align} a_0 &= \frac{1}{2b} \int_0^b f(x) + f(x) \, \mathrm{d}x \\ &= \frac{\cancel{2}}{\cancel{2}b} \int_0^b f(x) \, \mathrm{d}x \\ &= \frac{1}{b} \int_0^b f(x) \, \mathrm{d}x \end{align}}
Next we find 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_k}
:
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 \begin{align} a_k &= \frac{2}{b-(-b)} \int_{-b}^b g(x) \, \cos{ \left( \frac{2\pi}{b-(-b)} k \, x \right) } \,\mathrm{d}x \\ &= \frac{\cancel{2}}{\cancel{2}b)} \, \int_{-b}^b g(x) \, \cos{ \left( \frac{\cancel{2}\pi}{\cancel{2}b)} \, k \, x \right) } \,\mathrm{d}x \\ &= \frac{1}{b} \, \int_{-b}^b g(x) \, \cos{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \\ &= \frac{1}{b} \, \left( \int_{-b}^{0} g(x) \, \cos{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x + \int_{0}^{b} g(x) \, \cos{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \right) \\ &= \frac{1}{b} \, \left( -\int_{b}^{0} g(-x) \, \cos{ \left( \frac{\pi}{b} \, k \, (-x) \right) } \,\mathrm{d}x + \int_{0}^{b} g(x) \, \cos{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \right) \\ &= \frac{1}{b} \, \int_{0}^{b} g(-x) \, \cos{ \left( - \frac{\pi}{b} \, k \, x \right) } + g(x) \, \cos{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \\ &= \frac{1}{b} \, \int_{0}^{b} f(x) \, \cos{ \left( \frac{\pi}{b} \, k \, x \right) } + f(x) \, \cos{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \\ &= \frac{2}{b} \, \int_{0}^{b} f(x) \, \cos{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \end{align}}
Finally, we find 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_k} with the same method as above:
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 \begin{align} b_k &= \frac{2}{b-(-b)} \int_{-b}^b g(x) \, \sin{ \left( \frac{2\pi}{b-(-b)} k \, x \right) } \,\mathrm{d}x \\ &= \frac{\cancel{2}}{\cancel{2}b} \int_{-b}^b g(x) \, \sin{ \left( \frac{\cancel{2}\pi}{\cancel{2}b} \, k \, x \right) } \,\mathrm{d}x \\ &= \frac{1}{b} \, \int_{-b}^b g(x) \, \sin{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \\ \end{align}}
Observe that 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 g(x)} is an even function by definition, 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 \sin{ \left( \frac{\pi}{b} \, k \, x \right) }} is an odd function. Therefore their product is odd, and by symmetry, the integral of their product over the symmetric interval 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,b]} is 0. Hence all sine terms in the series cancel.
Odd Extension
In this case, the Fourier series simulates an odd function. To accomplish this, we fix the left endpoint at the origin (i.e. 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=0} ) and construct a Fourier series with the property 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(-x) = -f(x)} . This series will involve only sine terms.
where
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 \begin{align} b_k &= \frac{1}{b} \int_0^b f(x) \, \sin{\left( \frac{\pi}{b} \, k \, x \right)} \, \mathrm{d}x \end{align}}
Let 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 g(x)} be the odd periodic extension of 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(x)} on the interval 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, b]} such that the image of 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 g} on the interval 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,0]} is the image of 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} on the interval 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 [0,b]} rotated 180 degrees about the origin
We will compute the Fourier series of 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 g(x)} . We notice right away that 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_0 = 0} 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 a_k = 0} because they are integrals of an odd function over a symmetric interval. Therefore, only the 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_k} terms need to be computed:
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 \begin{align} b_k &= \frac{2}{b-(-b)} \int_{-b}^b g(x) \, \sin{ \left( \frac{2\pi}{b-(-b)} k \, x \right) } \,\mathrm{d}x \\ &= \frac{\cancel{2}}{\cancel{2}b} \int_{-b}^b g(x) \, \sin{ \left( \frac{\cancel{2}\pi}{\cancel{2}b} \, k \, x \right) } \,\mathrm{d}x \\ &= \frac{1}{b} \, \int_{-b}^b g(x) \, \sin{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \\ &= \frac{1}{b} \, \left( \int_{-b}^{0} g(x) \, \sin{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x + \int_{0}^{b} g(x) \, \sin{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \right) \\ &= \frac{1}{b} \, \left( -\int_{b}^{0} g(-x) \, \sin{ \left( \frac{\pi}{b} \, k \, (-x) \right) } \,\mathrm{d}x + \int_{0}^{b} g(x) \, \sin{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \right) \\ &= \frac{1}{b} \, \int_{0}^{b} g(-x) \, \sin{ \left( -\frac{\pi}{b} \, k \, x \right) } + g(x) \, \sin{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \\ \end{align}}
This time, observe that both 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 g(x)} 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 \sin{x}} are odd functions, so 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 g(-x) = -g(x)} 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 \sin{(-x)} = -\sin{x}} .
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 \begin{align} b_k &= \frac{1}{b} \, \int_{0}^{b} f(x) \, \sin{ \left( \frac{\pi}{b} \, k \, x \right) } + f(x) \, \sin{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \\ &= \frac{2}{b} \, \int_{0}^{b} f(x) \, \sin{ \left( \frac{\pi}{b} \, k \, x \right) } \,\mathrm{d}x \\ \end{align}}
Simplified Interval
On the interval 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 [ -\pi, \pi]} , the formulas simplify to