Fourier Series Formula

5/5 - (1 bình chọn)

Fourier Series Formula

The Fourier series formula gives an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. It is used to decompose any periodic function or periodic signal into the sum of a set of simple oscillating functions, namely sines and cosines. Let us understand the Fourier series formula using solved examples.

What Are Fourier Series Formulas?

Fourier series makes use of the orthogonal relationships of the cosine and sine functions. Fourier series formula for a function is given as,

Let us have a look at a few solved examples on the Fourier series formula to understand the concept better.

Examples on Fourier Series Formulas

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

Solution:

Applying the Fourier series formula, we know that f(x) =

FAQs on Fourier Series Formulas

What Is Meant by the Fourier Series?

A Fourier series presents an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier Series makes use of the orthogonality relationships of the sine and cosine functions.

What Is the Fourier Series Formula?

The fourier series formula of the function f(x) in the interval [-L, L], i.e. -L ≤ x ≤ L is given by:
f(x) = A_0 + ∑_{n = 1}^{∞} A_n cos(nπx/L) + ∑_{n = 1}^{∞} B_n sin(nπx/L)

What Is the Application of the Fourier Series Formula?

Fourier series describes a periodic signal in terms of cosine and sine waves. Thus, it models any arbitrary periodic signal with a combination of sines and cosines.

How To Solve a Fourier Series Using a Fourier Series Formula?

The steps for solving a Fourier series are given below:
Step 1: Multiply the given function by sine or cosine, then integrate
Step 2: Estimate for n=0, n=1, etc., to get the value of coefficients.
Step 3: Finally, substituting all the coefficients in the Fourier formula.

What Are the 2 Types of Fourier Series Formula?

The two types of Fourier series formulas are trigonometric series formula and exponential series formula.

Fourier Series

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. 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. Examples of successive approximations to common functions using Fourier series are illustrated above.

In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. 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. For example, using orthogonality of the roots of a Bessel function of the first kind gives a so-called Fourier-Bessel series.

The computation of the (usual) Fourier series is based on the integral identities

Definition of Fourier Series and Typical Examples

Baron Jean Baptiste Joseph Fourier introduced the idea that any periodic function can be represented by a series of sines and cosines which are harmonically related.

To consider this idea in more detail, we need to introduce some definitions and common terms.

Basic Definitions

Solved Problems

Click or tap a problem to see the solution.

Fourier Series Formula

Many of the phenomena studied in the domain of Engineering and Science are periodic in nature. For example current and voltage existing in an alternating current circuit. We can analyze these periodic functions into their constituent components by using a process called Fourier analysis. In this article, we will discuss the Fourier series and Fourier Series Formula. Let us begin learning!

Fourier Series Formula

What is the Fourier Series?

Periodic functions occur frequently in the problems studied during engineering problem-solving. Their representation in terms of simple periodic functions like sine and cosine functions, which are leading towards the Fourier series. Fourier series is a very powerful and versatile tool in connection with the partial differential equations.

A Fourier series is nothing but the expansion of a periodic function f(x) with the terms of an infinite sum of sins and cosine values. Fourier series is making use of the orthogonal relationships of the sine and cosine functions. A difficult thing to understand here is to motivate the fact that arbitrary periodic functions have Fourier series representations.

Fourier Analysis for Periodic Functions

The Fourier series representation of analytic functions has been derived from the Laurent expansions. We use the elementary complex analysis to derive additional fundamental results in the harmonic analysis, which includes representation of the periodic functions by the Fourier series.

The representation of rapidly decreasing functions by Fourier integrals, and Shannon’s sampling theorem. The ideas are classical and of transcendent beauty.

The trigonometric functions sin x and cos x are examples of periodic functions with fundamental period 2π and tan x is periodic with fundamental period \pi.

Therefore a Fourier series is a method to represent a periodic function as a sum of sine and cosine functions possibly till infinity. It is analogous to the famous Taylor series, which represents functions as possibly infinite sums of monomial terms.

For the functions that are not periodic, the Fourier series is replaced by the Fourier transform. For the functions of two variables that are periodic in both variables, the trigonometric basis in the Fourier series is replaced by the spherical harmonics.

Solved Examples

Putting all above values in the equation, we get the expansion of the above function:

Fourier Series

Sine and cosine waves can make other functions!

Here two different sine waves add together to make a new wave:

Square Wave

Can we use sine waves to make a square wave?

Our target is this square wave:

And if we could add infinite sine waves in that pattern we would have a square wave!

So we can say that:

a square wave = sin(x) + sin(3x)/3 + sin(5x)/5 + … (infinitely)

That is the idea of a Fourier series.

By adding infinite sine (and or cosine) waves we can make other functions, even if they are a bit weird.

Finding the Coefficients

How did we know to use sin(3x)/3, sin(5x)/5, etc?

There are formulas!

First let us write down a full series of sines and cosines, with a name for all coefficients:

We can often find that area just by sketching and using basic calculations, but other times we may need to use Integration Rules.

So this is what we do:

  • Take our target function, multiply it by sine (or cosine) and integrate (find the area)
  • Do that for n=0, n=1, etc to calculate each coefficient
  • And after we calculate all coefficients, we put them into the series formula above.

Let us see how to do each step and then assemble the result at the end!

When n is even the areas cancel for a result of zero.

In conclusion:

  • Think about each coefficient, sketch the functions and see if you can find a pattern,
  • put it all together into the series formula at the end

And when you are done go over to:

and see if you got it right!

Why not try it with “sin((2n-1)*x)/(2n-1)”, the 2n−1 neatly gives odd values, and see if you get a square wave.

Other Functions

Of course we can use this for many other functions!

But we must be able to work out all the coefficients, which in practice means that we work out the area of:

  • the function
  • the function times sine
  • the function times cosine

But as we saw above we can use tricks like breaking the function into pieces, using common sense, geometry and calculus to help us.

Here are a few well known ones:

WaveSeriesFourier Series Grapher
Square Wavesin(x) + sin(3x)/3 + sin(5x)/5 + …sin((2n−1)*x)/(2n−1)
Sawtoothsin(x) + sin(2x)/2 + sin(3x)/3 + …sin(n*x)/n
Pulsesin(x) + sin(2x) + sin(3x) + …sin(n*x)*0.1
Trianglesin(x) − sin(3x)/9 + sin(5x)/25 − …sin((2n−1)*x)*(−1)^n/(2n−1)^2

Footnote. Different versions of the formula!

On this page we used the general formula:

Fourier Analysis for Periodic Functions

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

A function is periodic of period L if f(x+L) = f(x) for all x in the domain of f. The smallest positive value of L is called the fundamental period.

The trigonometric functions sin x and cos x are examples of periodic functions with fundamental period 2π and tan x is periodic with fundamental period π. A constant function is a periodic function with arbitrary period L.

It is easy to verify that if the functions f1, . . . , fn are periodic of period L, then any linear combination

Applications

A Fourier Series has many applications in mathematical analysis as it is defined as the sum of multiple sines and cosines. Thus, it can be easily differentiated and integrated, which usually analyses the functions such as saw waves which are periodic signals in experimentation. It also provides an analytical approach to solve the discontinuity problem. In calculus, this helps in solving complex differential equations.

Frequently Asked Questions – FAQs

What is the Fourier Series formula?

The formula for the fourier series of the function f(x) in the interval [-L, L], i.e. -L ≤ x ≤ L is given by:
f(x) = A_0 + ∑_{n = 1}^{∞} A_n cos(nπx/L) + ∑_{n = 1}^{∞} B_n sin(nπx/L)

What is the Fourier series used for?

Fourier series is used to describe a periodic signal in terms of cosine and sine waves. In other other words, it allows us to model any arbitrary periodic signal with a combination of sines and cosines.

How do you solve a Fourier series?

The steps to be followed for solving a Fourier series are given below:
Step 1: Multiply the given function by sine or cosine, then integrate
Step 2: Estimate for n=0, n=1, etc., to get the value of coefficients.
Step 3: Finally, substituting all the coefficients in Fourier formula.

What are the 2 types of Fourier series?

The two types of Fourier series are trigonometric series and exponential series.

What is meant by the Fourier series?

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier Series makes use of the orthogonality relationships of the sine and cosine functions.

Math Formulas ⭐️⭐️⭐️⭐️⭐

Hãy bình luận đầu tiên

Để lại một phản hồi

Thư điện tử của bạn sẽ không được hiện thị công khai.


*