What is the Fourier series? Easily Explained with Examples

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What is Fourier series

Jean-Baptiste Joseph Fourier (1768-1830) introduced the equation solving a heat equation. He is an important contributor to the study trigonometric. Fourier did research that the trigonometric series could be in use to represent different equations. In other words, harmonically related sin and cosine can represent periodic functions. Sine and cosines are shown in the graph through waves. To form a new wave one can add Different waves.

Definition

The function f (a) is said to be periodic only when f(a+x)= f(a), for all values of x.

A function f (a) can be said to be piecewise continuous if the function is continuous on its interval [x,y] with some finite discontinuities.

A function is said to be piecewise smooth only if both the function and its derivative are continuous.

See Also: What is Integral? Everything you need to know about Integration

Fourier Series Partial Sum

fN (a) is the notation of the partial sum of this series’ of a function f(a) with the interval [−π,π].

A function, let it be f(a) can have a period X, if f(a+X) = f(a). Let us assume that the function f (a) has a period of 2pie, then the function’s behaviour lies on [-π, π].

fN (a)=x0/2+Summation of n=1-N (xncosna+ynsinna)

Fourier Series Convergence

Here you will find the Series Convergence below.

Pointwise

If the function f (a) is a smooth function with the interval [-π, π], then for all values of a that belongs to the interval, the equation is-

lim fN (a0) = {f(a0),if f(a)is continuous on[−π,π]

N→infinity

lim fN(a0)=(f(a0−0)+f(a0+0))/2,if f(a)has a jump discontinuity at a0

N→infinity

Uniform

The partial sums fN (a) are uniformly convergent to the function f(a), when the convergence speed of fN(a) does not depends on the value of a.

lim [max a∈[−π,π] |f(x)−fN(x)|]=0

N→infinity

See Also: How to Evaluate Limits? Complete Guide [Easy Explanation]

The Series

  • If the same function with a period of 2π is integrable on the interval, then the end value is finite. This type of integration is Dirichlet Integral.
  • The function should also have a single value, finite maxima, and minima, and should be continuous.

After satisfying these two conditions, we can say that function f(a) has a Fourier series. And it converges with the function.

At the discontinuity, the series converges to

lim 1/2[f (a0−ε)−f(a0+ε)]

ε→0

Then it will be

f(a)=x0/2+Summation from n=1 to infinity {xncosna+ynsinna}

Here the coefficients are integrals

x0=(1/π)π∫-πf(a)da, xn=(1/π)π∫-πf(a)cosnada, yn=(1/π)π∫-πf(a)sinnada

We can use Alternative of this series. The variable xn and yn by are replaced by cn and φn or cn and θn.

cn=√a2n+b2n, tanφn=xnyn, tanθn= xnyn

The alternative equation formed is

f (a) =x0/2+ Summation from n=1 to infinity cnsin(nx+φn)

Or

f(x)=x0/2+∞∑n=1 cncos(na+θn)

Even and Odd Functions

The even function f(a) with 2π as a period in the series’ expansion do not have a sin equation.

f (a)=x0/2+∞∑n=1xncosna, where the value of the coefficients are

x0=2/ππ∫0f(a) da,xn=2/ππ∫0f(a)cosnada.

The odd function f(a) with 2π as a period in the Fourier series expansion do not have a cos equation.

f(a)=∞∑n=1ynsinna, where the value of the coefficients are

yn=2/ππ∫0f(a)sinnada.

See Also: What are Definite Integrals? Complete Guide [Easy Explanation]

Note 1

If f (a) becomes a piecewise continuous function, then Euler Fourier formulas exist for these functions. The integrals in the formulas are definite, and even if they are improper, the integrals will always converge. The function f (a) does not always need to be piecewise continuous to have this series, but it needs to be periodic. Piecewise continuous helps in finding the Fourier coefficients, without them finding the Fourier coefficients is not guaranteed, because of divergent improper integrals.

Note 2

The function f(a) is equal to its Fourier series only when the function is continuous. If the function f(a) is piece-wise continuous, then it will not be equal with its Fourier series at every discontinuity. Furthermore, the functions need to be continuous from – ∞ to ∞.

Must Know: Trigonometric Integrals

Note 3

The function f(a) is said to be periodic only when f(a+x)= f(a), for all values of x. In that case, x becomes the period of a. Periods are not unique and very multiple of the period is another period of the function. One of the special cases is the constant function. Each constant function is a periodic function and has an arbitrary period, but it does not have a fundamental period. Fundamental periods are absent because these can be very small real numbers.

Note 4

Definite integrals of Euler-Fourier formulas are integrated over an interval of 2z. However, during the integration, the limit of the integral is -z to z.

Example 1

Let the function f(x) be 2π-periodic and suppose the presentation:

f(a)=x0/2 + ∞∑n=1{xncosnx+ynsinna}

Calculate the coefficients x0, xn, and yn.

Solution

Steps to follow:

  • Define x0 by integrating the Fourier series on the interval of [-π, π]
  • Find the value of the integration when n is greater than zero
  • Put those values in the 1st step to obtain the value of x0
  • For finding the values of xn multiplycosma, and then integrate
  • Similarly, for the value of yn, multiply sinma, and then integrate

See Also: Top Important Integration Rules to Simplify Calculus

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