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Fourier Series and Laplace Series


In Fourier series and Laplace series first we will discuss about the topic of Fourier series, and later we will go on laplace series, Fourier series was formulated by a Jean-Baptiste Fourier. he showed that an imaginary periodic function can be written as a sum of cosine and sine function. And in other we can say that Fourier series divides or decompose periodic function or periodic signal into the sum of sine’s and cosines that are also called complex exponential. The study related to Fourier series is comes under Fourier analysis. Fourier introduce this series to solve heat equation in a metal plate,

Before Fourier’s work there was no solution to measure heat equation in general way. Eigen solution is the solution from which the heat source and Fourier was working on a supervision of cosine and sine wave give a model for difficult heat source and this supervision is called Fourier series. Fourier series have many applications such as electrical engineering, acoustics, vibration analysis, signal processing, image processing and in many more.

Let’s take a Fourier series example. Let x(t) be a periodic function with Period T


where n is an Integer.

After discussing Fourier series in Fourier series and Laplace series now I am going to explain Laplace series, to you. Laplace series and transform was first introduced by a great mathematician Pierre-Simon Laplace. If he wants then he can easily use this theory but he used it in his Probability Theory. Laplace series can be easily understand if we write it in terms of real spherical harmonics which are the angular portion of a Set that provide the solution to Laplace equation that makes series explanation.

Fourier series as well as Laplace series is also used in same application and Laplace Transform is widely used in integral transform. Both the series are also used for integral equation. Laplace transform is closely related to the Fourier Transform, but there is one difference between Fourier and Laplace transform, Fourier transform takes a function or signal like a series of modes of vibration but Laplace transform resolve or introduce a function into its instance.

Now let’s talk about their comparison how they differ to each other. The Laplace series(LS) is better for the small times or can say that superior for small times and Fourier series(FS) is better for large times or superior for large times. First we take series of Fourier and Laplace than transform that and both are used to solve differential equation but question is arises that which one is best, it is not define yet both plays an important roles in their respected condition whenever their corresponding equation, series is comes. Both the transform Is used for different-2 purpose, Laplace transform is used when we deal with initial value and the Fourier transform is useful when we deal with boundary-value problems.

Fourier Sine Transform

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For transformation of odd Functions the Fourier Transform sine of continuous Fourier transform is used.
The general form of the Fourier transform is given as:
F (x) = 1 / √ (2 п) -∞∫∞ f (t) exp (- j ω t) dt,
Basically Fourier sine transform is the general case of the continuous Fourier transform which came into existence during transformation of the odd Functions. The ‘sin’ function is an odd function while the ‘cos’ function is an even function. If the generalized form defined above is an odd function then the product of ‘f (t) cos ωt’ will also be odd functions while the product of ‘f (t) sin ωt’ will be an even function.
The generalized integral form of a Fourier transform over an interval that is symmetric about the origin is - ∞ to + ∞.
The Fourier sine transform of an odd function is given as:
F(ω) = - i √(2 / п) 0∫∞ f (t) sin (ω t) dt,
Here the transformed function F (ω) will also be an odd function. The analysis of general inverse Fourier transformation will give the second Fourier transform of sine.

Fourier Cosine Transform

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Fourier transform of odd and even Functions is solved as per the rules for the continuous Fourier Transform. The Fourier cosine transform is the result of the continuous Fourier transform of even Functions.
While performing the transformation of the even functions the fourier transform cosine is the result.
The generalized form for Fourier transform is written as
G (x) = 1 / √(2 п) -∞ g (t) exp (- j ω t) dt,
Fourier cosine transform is the special case of the continuous Fourier transform. ‘Sin’ and ‘cos’ functions are considered as Odd and Even Functions respectively. If the generalized form above defined is an even function then the product of ‘f (t) cos ω t’ would be an even function and the product of ‘f (t) sin ω t’ would be an odd function.
The integral form of Fourier transform over an interval would be symmetric about the origin that is - ∞ to + ∞. The second Integration of this Fourier transform will eliminate zero and the first transformation will give the Fourier cosine transform of an even function;
G (ω) = √(2 / п) 0 g (t) cos (ω t) dt,
The second Fourier transform of ‘cos’ can be derived from the same. Here this transformed form G (ω) will show an even function.
This is how we perform fourier transform of cos.