Transforms and its Applications: UNIT 3: Fourier Series
Fourier Series: Definition, Euler formulas
Here, we express a non‒sinusoidal periodic function into a a fundamental and its harmonics, a series of sines and cosines of an angle and its multiples of the form.
FOURIER SERIES
Periodic
functions occur frequently in engineering problems. Such periodic functions are
often complicated. It is therefore desirable to represent these in terms of the
simple periodic functions of sine and cosine.
Definition: Fourier Series:
Here, we express a non‒sinusoidal periodic function into a a fundamental and its harmonics, a series of sines and cosines of an angle and its multiples of the form.
a0/2 + a1 cosx + a2 cos 2x + … + an cos nx + …
+
b1 sinx + b2 sin 2x + … + bn sin nx + …
=
a0/2 + 
cos nx
+ bn sin nx
is
called the Fourier series, where
a0, a1, a2, ….
an, ... ... b1, b2, …. bn, are
constants.
EULER'S FORMULA FOR THE FOURIER COEFFICIENTS
If
a function f(x) defined in c
<x<c+2π can be expanded as the infinite trigonometric series,
Formula (1), (2) and (3) are known as the
Euler formulas.
Note:
Only. if the constant term is taken as a0/2, formula (2) is true for
n = 0.
Useful Integrals to establish Euler formulae :
To
establish Euler formulae, the following integrals will be required.
Transforms and its Applications: UNIT 3: Fourier Series : Tag: Engineering mathematics, Maths : - Fourier Series: Definition, Euler formulas
Transforms and its Applications: UNIT 3: Fourier Series
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