Write down a periodic function in the designated field and the calculator will compute its Fourier series by displaying calculations.
"The enlargement of the periodic characteristic in phrases of infinite sums of sines and cosines is referred to as Fourier series."
check the given method that indicates the periodic feature f(x) in the c programming language \(-L\le \:x\le \:L\:\)
$$ f\left(x\right)=a_0+\sum _{n=1}^{\infty \:}a_n\cdot \cos \left(\frac{n\pi x}{L}\right)+\sum _{n=1}^{\infty \:}b_n\cdot \sin \left(\frac{n\pi x}{L}\right) $$
Where ;
$$ a_0=\frac{1}{2L}\cdot \int _{-L}^Lf\left(x\right)dx $$
$$ a_n=\frac{1}{L}\cdot \int _{-L}^Lf\left(x\right)\cos \left(\frac{n\pi x}{L}\right)dx,\:\quad \:n>0 $$
$$ b_n=\frac{1}{L}\cdot \int _{-L}^Lf\left(x\right)\sin \left(\frac{n\pi x}{L}\right)dx,\:\quad \:n>0 $$
With the assist of the Fourier coefficients calculator, you could without problems locate values towards those coefficients.
Calculate the Fourier collection of the characteristic given underneath:
$$ f(x) = x^2 \text{ for } -\pi \leq x \leq \pi $$
Solution:
The function is:
$$ f(x) = x^2 $$
First, we determine the coefficients \(a_0\), \(a_n\), and \(b_n\):
The formula for \(a_0\) is:
$$ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx $$
Substitute \(f(x) = x^2\):
$$ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \, dx $$
Since \(x^2\) is even, we can simplify:
$$ a_0 = \frac{2}{\pi} \int_{0}^{\pi} x^2 \, dx $$
$$ a_0 = \frac{2}{\pi} \cdot \left[ \frac{x^3}{3} \right]_0^\pi $$
$$ a_0 = \frac{2}{\pi} \cdot \frac{\pi^3}{3} = \frac{2\pi^2}{3} $$
The formula for \(a_n\) is:
$$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx $$
Substitute \(f(x) = x^2\):
$$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \cos(nx) \, dx $$
Since \(x^2 \cos(nx)\) is even:
$$ a_n = \frac{2}{\pi} \int_{0}^{\pi} x^2 \cos(nx) \, dx $$
Using integration by parts (or an online integral calculator):
$$ a_n = \frac{4(-1)^n}{n^2} $$
The formula for \(b_n\) is:
$$ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx $$
Since \(x^2 \sin(nx)\) is odd:
$$ b_n = 0 $$
Combining the results, the Fourier series is:
$$ f(x) = \frac{2\pi^2}{3} + \sum_{n=1}^\infty \frac{4(-1)^n}{n^2} \cos(nx) $$
Even here, a Fourier series calculator can help simplify such calculations effectively.
Every time you come across complicated capabilities, our unfastened online fourier series calculator is here that will help you out in determining accurate outcomes. you may get a proper situation of the calculations by means of the usage of our calculator.
Input:
Output: The Fourier growth calculator calculates:
A Fourier series illustrates a recurring pattern as an accumulation of sin waves and cosine waves. It helps in studying vibratory patterns, pulse signals, and sinusoidal figures in various areas such as engineering and physics.
It decomposes intricate repetitive signals into basic trigonometric elements, proving advantageous in the fields of wave analysis, sound science, and the domain of electric circuit.
No, the Fourier series applies only to periodic functions. But the Fourier transform extends this concept to non-periodic functions.
Engineers study things like electric circuits, vibrations, and heat to make better ways to talk and control things.
The Fourier series deals with repeating patterns, and the Fourier transform changes non-repeating noise into different tone bits.
This device is used in audio condensing, image enhancement, quantum study, and solving change-related problems in physics and engineering.
It turns hard sounds into easy wavy patterns, which is good for making music, removing noise, and helping computers understand what people say.
Yes, the Fourier series can encapsulate any piecewise continuous function within its period, although certain jumps may result in the Gibbs phenomenon.
Gibbs overshoot deals with imprecise function estimates near a break using Fourier series.
It turns hard math problems into simpler math problems, helping people solve them, especially when dealing with things like machines and building designs.
These are parts of a mathematical expression that explains how much a function bands or turns strange.
This method can break down images into different frequencies, which makes them sharper, easier to see the edges, and cleaner with less disruption.
The Fourier coefficients determine the magnitude of sinusoidal and rectangular oscillations within the summation, delineating the impact of each frequency on the equation.
The concept can be applied to non-continuous functions when using Fourier transform, which examines how these functions behave over an extending or unlimited interval.
This technology is very important in fields such as telephone calls and sending digital info. It is also used in medical devices such as MRI and CT scans.
