# How to get fourier transform from fourier series

• Ahmed Abdullah
In summary, the Fourier transform is a generalization of the Fourier series in the limit as L approaches infinity. It replaces the discrete coefficients A_n with a continuous function F(k)dk and changes the sum to an integral. This allows for the transformation of a discrete variable to a continuous one and solves the issue of A_n becoming F(k)dk.
Ahmed Abdullah
How you get Fourier transform from Fourier series? Do Fourier series becomes Fourier transform as L --> infinity?

http://mathworld.wolfram.com/FourierTransform.html

I don't understand where discrete A sub n becomes continuous F(k)dk ( where F(k) is exactly like A sub n in Fourier series)?
I also have a general question which is;
How to transform a discrete variable to a continuous variable in order to convert a summation to integral?

I think I know the answer now.

" For a function periodic in [-L/2,L/2], Fourier series is
$$f(x) = \sum_{n=-\infty}^{\infty }A_{n}e^{i(2\pi nx{/}L)} \\A_{n} = 1{/}L \int_{-L{/}2}^{L{/}2}f(x)e^{-i(2\pi nx{/}L)}dx.$$"

The part that was bothering me was " The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete $$A_{n}$$ with the continuous F(k)dk while letting n/L->k. Then change the sum to an integral, and the equations become
$$f(x) = \int_{-\infty}^{\infty} F(k)e^{2\pi ikx}dk \\ F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi ikx}dx$$ "
My question was how $$A_{n}$$ becomes $$F(k) d(k)$$. Especially where the dk comes from?
$$k=n{/}L$$; then $$\Delta k = (n+1){/}L -n{/}L =1{/}L$$ , when $$L$$ goes to infinity $$\Delta k$$becomes dk.
So when $$L \rightarrow \infty; A_{n} =F(k)dk$$. Indeed! I am a happy man now :).

Last edited:

## 1. What is the difference between Fourier transform and Fourier series?

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, while the Fourier series is a way of representing a periodic function as a sum of sinusoidal functions. The Fourier transform is typically used for non-periodic functions, while the Fourier series is used for periodic functions.

## 2. How do I calculate the Fourier transform from a Fourier series?

To calculate the Fourier transform from a Fourier series, you can use the formula: F(w) = 2πc(n)e^(-jwnT), where c(n) is the Fourier coefficient and T is the period of the function. You can also use a computer program or online calculator to perform the calculation.

## 3. Can any function be represented by a Fourier series?

No, only functions that are periodic can be represented by a Fourier series. If a function is not periodic, it cannot be expressed as a sum of sinusoidal functions and therefore cannot have a Fourier series representation.

## 4. What is the practical application of the Fourier transform and Fourier series?

The Fourier transform and Fourier series have many practical applications, including signal processing, image processing, and data compression. They are also used in fields such as physics, engineering, and computer science to analyze and model complex systems.

## 5. How do I interpret the results of a Fourier transform or Fourier series?

The results of a Fourier transform or Fourier series show the amplitude and phase of the different frequencies present in the original function. This can help in understanding the behavior of the function and can be used to filter out unwanted frequencies in signal processing applications.

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