# What is difference between Fourier series and Fourier integral?

## What is difference between Fourier series and Fourier integral?

5 Answers. The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials.

### Can you integrate a Fourier series?

The theorem for integration of Fourier series term by term is simple so there it is. Supposef(x) is piecewise smooth then the Fourier sine series of the function can be integrated term by term and the result is a convergent infinite series that will converge to the integral of f(x) .

#### What is Fourier complex integral?

we can write the Fourier series of the function in complex form: f(x)=a02+∞∑n=1(ancosnx+bnsinnx)=a02+∞∑n=1(aneinx+e−inx2+bneinx−e−inx2i)=a02+∞∑n=1an−ibn2einx+∞∑n=1an+ibn2e−inx=∞∑n=−∞cneinx. The complex form of Fourier series is algebraically simpler and more symmetric.

Why Fourier integral is used?

The straightforward application of the Fourier integral to determine the response of a linear invariable circuit to an arbitrary impressed force is reviewed. When a Fourier integral representation of the impressed force exists and the system starts from rest, the problem is routine.

What is Fourier integral transform?

The Fourier transform is an extension of the Fourier series that results when the period of the represented function is lengthened and allowed to approach infinity. Due to the properties of sine and cosine, it is possible to recover the amplitude of each wave in a Fourier series using an integral.

## What are the 2 types of Fourier series?

Fourier series is of two types- trigonometric series and exponential series.

### How to find the integration of a Fourier series?

Integration of Fourier Series Let g(x) be a 2π -periodic piecewise continuous function on the interval [−π,π]. Then this function can be integrated term by term on this interval. The Fourier series for g(x) is given by

#### Can a Fourier series be a function of period L?

Provided L is finite, we still have a Fourier series, representing a function of period L. Our main interest in taking L infinite is that we would like to represent a nonperiodic function, for example a localized wave packet, in terms of plane-wave components.

Which is the complex form of the Fourier transform?

Discussion: pointwise convergence of Fourier integrals and series Heuristics In the previous Lecture 14we wrote Fourier series in the complex form

How to recover the amplitude of a Fourier series?

Due to the properties of sine and cosine, it is possible to recover the amplitude of each wave in a Fourier series using an integral. In many cases it is desirable to use Euler’s formula, which states that e 2πiθ = cos(2πθ) + i sin(2πθ), to write Fourier series in terms of the basic waves e 2πiθ.