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Questions and Answers

What is the condition for a function f(x) to be an even function?

f(-x) = -f(x)

f(-x) = f(x) (correct)

f(x) = 0

f(x) = x

What is the Fourier coefficient represented by?

b_n

a_0 (correct)

c_n

a_n

What is the type of function represented by f(-x) = -f(x)?

Periodic function

Even function

Odd function (correct)

Dirichlet function

What is the condition for a function f(x) to be an odd function?

f(-x) = -f(x) Signup and view all the answers

What is the value of the Fourier coefficient a0 if f(x) = -x for -π< x< 0?

-(π^2)/2 Signup and view all the answers

What is the type of function that has no cosine terms in its Fourier expansion?

Odd function Signup and view all the answers

What is the condition for the integrability of a function f(x) in a Fourier series?

Dirichlet condition Signup and view all the answers

What is the value of ∫f(x) dx between the limits -a to a if f(x) is an even function?

2∫f(x) dx Signup and view all the answers

What is the condition for the Fourier series expansion of f(x) given by f(x) = a0/2 + ∑(an cos nx + bn sin nx)?

The function f(x) must satisfy certain conditions in the interval c1≤ x≤ c2. Signup and view all the answers

If the periodic function f(x) is even, what is the form of its Fourier expansion?

a0/2 + ∑(an cos nx + bn sin nx) Signup and view all the answers

What is the form of the Fourier coefficient an for an even function f(x)?

2/l ∫f(x) cos(nπx/l) dx Signup and view all the answers

What is the period of cos nx where n is a positive integer?

2π/n Signup and view all the answers

What is the Fourier coefficient a0 for the function f(x) = x for 0 < x < π?

π Signup and view all the answers

If the function f(x) = -π in the interval –π < x < 0, what is the coefficient a0?

π^2/3 Signup and view all the answers

If the function f(x) = x sin x in –π < x < π, what is the Fourier coefficient bn?

0 Signup and view all the answers

For the cosine series, which of the Fourier coefficients will vanish?

an Signup and view all the answers

What is the value of bn in the Fourier series of the function f(x) = x^3 in –π< x< π?

0 Signup and view all the answers

What type of function is F(x) = x cos x in –π< x< π?

odd function Signup and view all the answers

If f(-x) = -f(x), then what type of function is f(x)?

odd function Signup and view all the answers

What is the formula for finding the Fourier coefficient a_0 in Harmonic analysis?

(2/N)Σ y cos nx Signup and view all the answers

What is the term a1cos x+ b1 sin x called in Fourier Series expansion?

second harmonic Signup and view all the answers

What type of function is F(x) = x sin x in –π< x< π?

even function Signup and view all the answers

Study Notes

Fourier Series

A function with period 2π is cos x.
The Fourier coefficient a_0 is given by 1/π ∫f(x) dx between the limits c and c+2π.
If f(x) = -x for -π < x < 0, then a_0 = - (π^2)/2.

Even and Odd Functions

A function f(x) is even if f(-x) = f(x).
A function f(x) is odd if f(-x) = -f(x).
Examples of odd functions: sin x, x^3.
An even function has a_0 = 0 and an odd function has a_0 = 0.

Fourier Expansion

The Fourier series of f(x) is given by a_0 /2 + ∑ (an cos nx + bn sin nx).
If f(x) is even, the Fourier expansion contains no sine terms.
If f(x) is odd, the Fourier expansion contains no cosine terms.
The Fourier expansion of an even function is of the form a_0 /2 + ∑ an cos nx.
The Fourier expansion of an odd function is of the form ∑ bn sin nx.

Dirichlet Conditions

A function f(x) has a finite number of maxima and minima in an interval.
A function f(x) has a finite number of discontinuities in an interval.

Fourier Coefficients

a_0 is given by 1/π ∫f(x) dx between the limits -a to a if f(x) is even.
a_n is given by 2/l ∫f(x) sin(nπx/l) dx if f(x) is even.
b_n is given by 2/l ∫f(x) cos(nπx/l) dx if f(x) is odd.
If f(x) is even, then a_0 is not zero, and a_n is zero.
If f(x) is odd, then a_0 is zero, and b_n is not zero.

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