How To Solve Fourier Series Problems

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In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions.

Any set of functions that form a complete orthogonal system have a corresponding generalized Fourier series analogous to the Fourier series.

Using the method for a generalized Fourier series, the usual Fourier series involving sines and cosines is obtained by taking and .

Since these functions form a complete orthogonal system over , the Fourier series of a function is given by and , 2, 3, ....

Recall: A function `y = f(t)` is said to be even if `f(-t) = f(t)` for all values of `t`. We can also see that it is an odd function, so we know `a_0= 0` and `a_n= 0`.

The graph of an even function is always symmetrical about the y-axis (i.e. `f(t) = 2 cos πt` We can see from the graph that it is periodic, with period `2pi`. So we will only need to find b `b_n=1/pi int_(-pi)^pif(t)sin nt\ dt` `=1/pi(int_(-pi)^0 -3\ sin nt\ dt` `{: int_0^pi 3\ sin nt\ dt)` `=3/pi([(cos nt)/n]_(-pi)^0 [-(cos nt)/n]_0^pi)` `=3/(pi n)(cos 0-cos(-pi n)` `-cos n pi` `{: cos 0)` `=3/(pi n)(2-cos pi n-cos pi n)` `=6/(pi n)(1-cos pi n)` `=12/(pi n)\ (n\ "odd") or` `=0\ (n\ "even")` is non-zero for n odd, we must also have odd multiples of t within the sine expression (the even ones are multiplied by `0`, so will be `0`). It should closely resemble the square wave we started with.This document derives the Fourier Series coefficients for several functions. $$(t) = \sum\limits_^\infty = \sum\limits_^\infty $$ The average is easily found, $$a_0=A\frac T$$ The other terms follow from $$ = \int\limits_T ,\quad n \ne 0$$ Any interval of one period is allowed but the interval from ) (red) $$\sum\limits_^N $$ as well as the highest frequency harmonic, $a_Ncos(N\omega_0t)$ (dotted magenta).The functions shown here are fairly simple, but the concepts extend to more complex functions. Lower frequency harmonics in the summation are thin dotted blue lines (but harmonics with $a_n0$ are not shown.Fourier series make use of the orthogonality relationships of the sine and cosine functions. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. `12/pi sum_(n=1)^5 (sin(2n-1)t)/((2n-1))` `=12/pi(sin t` ` 1/3 sin 3t` ` 1/5 sin 5t` ` 1/7 sin 7t` `{: 1/9 sin 9t)` (We used Scientific Notebook for the final answer.) Recall that `cos((nπ)/2) = 0` for n odd and ` 1` or `-1` for n even.(See the Helpful Revision page.) So we expect 0 for every odd term. Finding zero coefficients in such problems is time consuming and can be avoided.With knowledge of even and odd functions, a zero coefficient may be predicted without performing the integration.Note that the coefficient of the constant term has been written in a special form compared to the general form for a generalized Fourier series in order to preserve symmetry with the definitions of and . The Fourier cosine coefficient and sine coefficient are implemented in the Wolfram Language as if the function satisfies so-called Dirichlet boundary conditions.

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