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Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics PDF Download

Even & Odd function

A function g (x) is said to be even if g (-x) = g (x), so that its graph is symmetrical with respect to vertical axis.

A function h (x) is said to be even ifh (-x) = -h (x).
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Since the definite integral of a function gives the area under the curve of the function between the limits of integration, we have
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Fourier Cosine Series and Fourier Sine Series
Fourier series of an even function of period 2L, is a “Fourier cosine series”
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
with coefficients( note integration from 0 to L)
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Fourier series of an odd function of period 2L, is a “Fourier sine series”
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
 with coefficients Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
NOTE:
 (i) For even function f (x);
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
and Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
(ii) For odd function f (x);
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
The Case of Period 2π

If L = π, and f(x) is even function then
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
with coefficients
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
If  f(x) is odd function then
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
with coefficients
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Sum and Scalar Multiple

(a) The Fourier coefficients of a sum f1 + f2 are the sums of the corresponding Fourier coefficients of f1 and f2.

(b) The Fourier coefficients of cf are c times the corresponding Fourier coefficients of f.


Example 4: Find the Fourier series of the periodic function f (x) as shown in figure:
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics

We have already calculated the Fourier series of the periodic function f (x) as shown in figure:
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
The Fourier series is Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
The function given in the problem can be obtained by adding k to the above function. Thus the Fourier series of a sum k+ /'(x) are the sums of the corresponding Fourier

series of k and /(x).
The Fourier series is f Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics 

Example 5: Find the Fourier series of the periodic function
f (x) = x + π ; ( - π < x < π) having period 2π

Let f(x) = f1 + f2   where f1 = x and f2 = π.

The Fourier coefficient of f2 = π is a0 = π, an= 0 and bn = 0 . The Fourier coefficient of f1 = x is
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Thus the Fourier series of f (x) is
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics

Half-Range Expansion

Half-range expansions are Fourier series. The idea is simple and useful. We could extend f (x) as a function of period L and develop the extended function into a Fourier series. But this series would in general contain both cosine and sine terms.

We can do better and get simpler series. For our given function f (x) we can calculate

Fourier cosine series coefficient(a0 and an ) . This is the even periodic extension f1(x) of f (x) in figure (b).

For our given function f (x) we can calculate Fourier sine series coefficient (bn ). This

is the odd periodic extension f2 (x) off (x) in figure (c).

Both extensions have period 2L . Note that f(x) is given only on half the range, half the

interval of periodicity of length 2L .
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics


Example 6: Find the two half-range expansion of the function f (x) as shown in figure below.
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics

Even periodic extension
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Let us calculate the integral
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics 
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
and Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Hence the first half-range expansion of f (x) is
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
This Fourier cosine series represents the even periodic extension of the given function/ (x), of period 2 L as shown in figure.

Odd periodic extension
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Let us calculate the integral
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics 
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Hence the other half-range expansion of f (x) is
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
This Fourier sine series represents the odd periodic extension of the given function f (x).
of period 2L as shown in figure.

Complex Fourier series

The Fourier series
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
can be written in complex form, which sometimes simplifies calculations.

∵ einx = cos nx + i sin nx and e-inx = cos nx -i sin nx
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Thus Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Lets take Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
where coefficients c0, cn and kn are given by
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
We can also write (2) as (take k= c-n)
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
This is so called complex form of the Fourier series or, complex Fourier series off (x) .
The cn are called complex Fourier series coefficients of f (x).

For a function of period 2L
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics


Example 7: Find the complex Fourier series of the periodic function
f (x) = ex; ( - π < x < π), having period 2π

Let Fourier series is f(x) = Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Thus Fourier series is Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Let us derive the real Fourier series
∵ (1 + in) einx = (1 + in) (cos nx + i sin nx) = (cos nx - nsin nx) + i (cos nx + sin nx)
∵ n varies from -∞ to + ∞, equation (1) has corresponding term with -n instead of n .
Thus

∵ (1 -in) e-inx = (1 -in) (cos nx - i sin nx) = (cos nx-n sin nx) - i (cos nx+sin nx)
Let’s add these two expressions;
(1 + in) einx + (1 - in)e-inx = 2 (cos nx - n sin nx), n = 1,2,3........
For n = 0, Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics   
Thus
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics


Example 8: Consider the periodic function f(t) with time period T as shown in the figure below.
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
The spikes, located at Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics where n = 0,±1,±2,..., are Dirac-delta function of strength +1 . Find the amplitudes an in the Fourier expansion of
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics

Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics and Range: [-1,1] hence 2L = 2.
Comparing with Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics


Approximation by Trignometric Polynomials

Fourier series have major applications in approximation theory, that is, the approximation of functions by simpler functions.

Let f (x)be a periodic function, of period 2π for simplicity that can be represented by a Fourier series. Then the Nth partial sum of the series is an approximation to f (x) :
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
We have to see whether (1) is the "best” approximation to f by a trignometric polynomial of degree N, that is , by a function of the form
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
where "best” means that the "error” of approximation is minimum.
The total square error of F relative to / on the interval -π < x < π is given by
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
The function F is a good approximation to f but |f - F| is large at a point of discontinuity x0.

Minimum square error

The total square error of F relative to f on the interval - π < x < n is minimum if and only if the coefficients of f(x) are the Fourier coefficients of f(x). This minimum value E* is given by
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
From (3) we can see that E* cannot increase as N increases, but may decrease. Hence with increasing N the partial sums of the Fourier series of f yields better and better approximations to f.

Parseval’s Identity

Since E* > 0 and equation (3) holds for every N, we obtain
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Now Parseval’s Identity is
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics


Example 9: Compute the total square error of F with N = 3 relative to
f (x) = x + π    (- π < x < π)
on the interval - π < x < π .

Fourier coefficients are a0 = π, a= 0 and bn = Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Its Fourier series is given by
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Hence,
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics
Although |f (x) - f(x)| is large at x = ± π . where f is discontinuous, F approximates f quite well on the whole interval.

The document Even, Odd functions & Half-Range Expansion | Mathematical Models - Physics is a part of the Physics Course Mathematical Models.
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FAQs on Even, Odd functions & Half-Range Expansion - Mathematical Models - Physics

1. What is a complex Fourier series?
Ans. A complex Fourier series is a mathematical representation of a periodic function using complex exponentials. It is a way to decompose a periodic function into a sum of infinitely many complex sinusoidal functions with different frequencies, amplitudes, and phases.
2. How does the approximation of a function by trigonometric polynomials work?
Ans. The approximation of a function by trigonometric polynomials involves representing a given function as a sum of trigonometric functions, such as sine and cosine. These trigonometric functions are combined with different coefficients to create a polynomial that closely approximates the original function.
3. What are even and odd functions in the context of Fourier series?
Ans. In the context of Fourier series, an even function is a function that satisfies the condition f(x) = f(-x) for all x in its domain. This means that the function is symmetric about the y-axis. On the other hand, an odd function is a function that satisfies the condition f(x) = -f(-x) for all x in its domain. This means that the function is symmetric about the origin.
4. What is the concept of half-range expansion in Fourier series?
Ans. The concept of half-range expansion in Fourier series involves representing a periodic function defined on a symmetric interval as a series of sine or cosine functions over a half of that interval. By using appropriate formulas and techniques, the Fourier series can be simplified and calculated more efficiently for such functions.
5. How can the concepts of complex Fourier series, approximation by trigonometric polynomials, even and odd functions, and half-range expansion be applied in IIT JAM exam?
Ans. In the IIT JAM exam, questions related to complex Fourier series, approximation by trigonometric polynomials, even and odd functions, and half-range expansion can be asked in the Mathematics section. It is important to have a solid understanding of these concepts and their applications to solve such questions effectively. Familiarity with the properties and formulas associated with these topics will be beneficial in tackling the relevant problems in the exam.
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