Fourier Series

Most of the phenomena that are studied in Engineering and Science are periodic in nature. For instance, current and voltage in an alternating current circuit. These periodic functions could be analysed into their constituent components (fundamentals and harmonics). It can be done by using a process called Fourier analysis. Examples of the Fourier series are trigonometric functions like sin x and cos x with period \[2\pi\] and tan x with period \[\pi\].

In this article, we will discuss the Fourier series, formulas, and uses and applications of the Fourier series.

So, let’s begin with what the Fourier series is.

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What is the Fourier Series?

The Fourier series can be defined as a way of representing a periodic function (possibly infinite) as a sum of sine functions and cosine functions. The Fourier series is known to be a very powerful tool in connection with various problems involving partial differential equations. 

A graph of periodic function f(x) that has a period equal to L exhibits the same pattern for every L unit along the x-axis so that f(x + L) is equal to f(x) for each value of x. If we know what the function looks like over one complete period, therefore we can sketch a graph of the function over a wider interval of x (that may contain many periods). 

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We can use this property of repetition that defines a fundamental spatial frequency k = 2π L to give a first approximation to the periodic pattern f(x): f(x)' c₁ sin(kx +a₁) equals to a₁ cos(kx) + b₁ sin(kx), where symbols with subscript equal to 1 are constants that determine the amplitude and phase of this first approximation.

A much better approximation of the periodic pattern that is function f(x) can be built up by adding an appropriate combination of harmonics to the fundamental (sine-wave) pattern. For example, adding c₂ sin(2kx + c₂) = a₂ cos(2kx) + b₂ sin(2kx) (which is the 2nd harmonic) and c₃ sin(3kx + a₃) = a₃ cos(3kx) + b₃ sin(3kx) (that is the 3rd harmonic). Here, symbols with subscripts are the constants that generally determine the amplitude and phase of each harmonic contribution.

We can even approximate a square-wave pattern with a suitable sum that involves a fundamental sine-wave and a combination of harmonics of this fundamental frequency. This sum is known as the Fourier series.

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Laurent Series Yield Fourier Series (Fourier Theorem)

Arbitrary periodic functions have Fourier series representations which are very difficult to understand. In this section, we are going to prove that periodic analytic functions have such a representation using the Laurent expansions.

Fourier Analysis for Periodic Functions

From Laurent expansions, we can derive the Fourier series representation of analytic functions. The elementary complex analysis can generally be used to derive additional fundamental results in the harmonic analysis including the representation of C∞ periodic functions by the Fourier series, Shannon’s sampling theorem, the representation of rapidly decreasing functions by Fourier integrals, and the ideas are classical and of transcendent beauty.

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A function is known as periodic of period L if f(x+L) is equal to f(x) for all the values of x in the domain of function f. The smallest positive value of L can be known as the fundamental period.

A constant function is a periodic function with an arbitrary period equal to L.

It is easy to verify that if we have the functions f₁, f₂,... f_n which are periodic of period L, then any linear combination, c₁f₁(x) + …. + c_nf_n(x) is also known as periodic. Furthermore, if the infinite series

\[\frac{1}{2} a_{o} + \sum_{n=1}^{\infty} a_{n} cos \frac{n \pi x}{L} + b_{n} sin \frac{n \pi x}{L}\]

consists of 2L - periodic functions that converge for all the values of x, then the function to which it converges will be periodic of period 2L. There are two symmetry properties of functions that will be useful in the study of the Fourier series.

Even and Odd Functions

A function f(x) is called to be an even function if f(-x) is equal to f(x).

The function f(x) is called  to be an odd function if f(-x) is equal to -f(x).

Graphically, even functions have symmetry about the y-axis, whereas odd functions have symmetry around the origin.

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Examples of Even and Odd Functions

Sums of odd powers of x are odd: 5x³−3X

Sums of even powers of x are even: −x⁶ + 4x⁴ + x² − 3

As x is odd, the value of cos x is even.

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The product of any two odd functions is even: x sin x is even

The product of  any two even functions is even: x² cos x is even

The product of an even function and an odd function is odd:  sin x cos x is odd.

Note:

To find the Fourier series, we know from the Fourier series definition that it is sufficient to calculate the integrals that will give the coefficients a_o, a_n, and b_n and plug these values into the big series formula as we know from the Fourier theorem.

What is the Fourier Series Formula?

The formula of the Fourier series for a function is given as

\[f(x) = \frac{1}{2} a_{o} + \sum_{n=1}^{\infty} a_{n} cos \, nx + \sum_{n=1}^{\infty} b_{n} sin \, nx\]

where, 

• \[a_{o} = \frac{1}{\pi} \int_{-\pi}^{\pi} f_{x}dx\]

• \[a_{n} = \frac{1}{\pi} \int_{-\pi}^{\pi} f_{x} cos \, nx dx\]

• \[b_{n} = \frac{1}{\pi} \int_{-\pi}^{\pi} f_{x}sin \, nx dx\]

• n = 1, 2, 3……

Uses and Application of Fourier Series

A Fourier (that can be pronounced for-YAY) series is a specific type of infinite mathematical series that involves trigonometric functions. Fourier series are the ones that are used in applied mathematics, and especially in the field of physics and electronics, to express periodic functions such as those that comprise communications signal waveforms.

It is analogous to a Taylor series, that represents functions as possibly infinite sums of the monomial terms. 

Solved Examples

Example 1: Expand the function f(x) = e^x in the interval [ – π , π ] using Fourier series formula.

Solution:

We know that f(x)= \[\frac{1}{2} a_{o} + \sum_{n=1}^{\infty} a_{n} cos \, nx +\sum_{n=1}^{\infty} b_{n} sin \, nx\]

\[a_{o} = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{x} dx\]

\[= \frac{e^{\pi} - e^{-\pi}}{2\pi}\]

\[a_{n} = \frac{1}{\pi} \int_{-\pi}^{\pi} e^{x} cos(nx)dx\]

\[= \frac{1}{\pi} \frac{e^{x}}{1+n^{2}} [cos(nx) + nsin(nx)]_{-\pi}^{\pi}\]

\[=  \frac{1}{\pi(1+n^{2})} [e^{\pi}(-1)^{n} - e^{-\pi}(-1)^{n}\]

\[b_{n} = \frac{1}{\pi} \int e^{x} sin(nx)dx\]

\[= \frac{e^{x}}{\pi(1+n^{2})} [sin(nx) - ncos(nx)]_{-\pi}^{\pi}\]

\[=  \frac{1}{\pi(1+n^{2})} [e^{\pi}(-1(-1)^{n}) - e^{-\pi}(-n)(-1)^{n}\]

Putting all the above values in the equation, we get:

f(x)=ex= \[\frac{e^{\pi} - e^{-\pi}}{2{\pi}} + \sum_{n=1}^{\infty} \frac{(-1)^{n} (e^{\pi} - e^{-\pi})}{\pi(1+n^{2})}[cos \, nx - n\, sin\, nx]\]

Example 2: What will be the Fourier series of the function f(x)=1−x² in the interval [−1,1]?

Solution:

We know that, the fourier series of the function f(x) in the interval

[-L , L], i.e. -L \[\leq x \leq\] L is written as:

\[f(x) = A_{o} + \sum_{n=1}^{\infty} A_{n}. cos(\frac{n \pi x}{L}) + \sum_{n=1}^{\infty}B_{n}. sin (\frac{n \pi x}{L})\]

Here,\[A_{o} =\frac{1}{2L}.\int_{-L}^{L} f(x) dx\]

\[A_{n} = \frac{1}{1}. \int_{-1}^{1} f(x) cos(\frac{n \pi x}{1})dx\], n>0

\[B_{n} = \frac{1}{1}. \int_{-1}^{1} f(x) sin(\frac{n \pi x}{1})dx\], n>0 

Now, by applying the formula for f(x) in the interval [-1,1]:

\[f(x) = \frac{1}{2.1}. \int_{-1}^{1}(1-x^{2})dx + \sum_{n=1}^{\infty} \frac{1}{1} . \int_{-1}^{1} (1-x^{2}) cos (\frac{n \pi x}{1})dx . cos (\frac{n \pi x}{1})dx + \sum_{n=1}^{\infty} \frac{1}{1} . \int_{-1}^{1} (1-x^{2}) sin (\frac{n \pi x}{1})dx . sin (\frac{n \pi x}{1})dx\]

Now simplifying the definite integrals

= \[ \frac{1}{2.1} (\frac{4}{3}) + \displaystyle\sum\limits_{n=1}^\infty \frac{1}{1} (- \frac{4(-1)^{n}}{\pi^{2} n^{2}}) cos (\frac{n\pi x}{1}) + \displaystyle\sum\limits_{n=1}^\infty \frac{1}{1} . O. sin (\frac{n\pi x}{1} )\]

\[= \frac{2}{3} + \displaystyle\sum\limits_{n=1}^\infty - \frac{4(-1)^{n}cos(\pi n x)}{\pi^{2}n^{2}} \]