Author: kanaksindhu

Assistant Professor in Mathematics, Dept of Science and Humanities, Dr Mahalingam College of Engineering and Technology, Pollachi

LxT’s on Fourier series

kanaksindhu

Any Periodic function can be expanded as Fourier series. A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. Examples of successive approximations to common functions using Fourier series are illustrated above.

In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions.

Any set of functions that form a complete orthogonal system have a corresponding generalized Fourier series analogous to the Fourier series. For example, using orthogonality of the roots of a Bessel function of the first kind gives a so-called Fourier-Bessel series.

Lbd on Fourier series

kanaksindhu

LbD 1.1

Fourier series can be expanded for what type of functions

(a) Periodic Functions
(b) Aperiodic functions
(c) Minimal functions
(d) Maximal functions

LbD 1.2

The constant term for an odd function in the Fourier series is
(a) -1
(b) 1
(c) 0
(d) 2

LbD 1.3

If a Fourier series consists of only sine terms, such a series is called
(a) sine series
(b) cosine series
(c) Fourier series
(d) Infinite series