Untitled Quiz

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

What is the Fourier coefficient represented by?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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