Question

If the function f(x) is odd, then which of the only coefficient is present?
a) an
b) bn
c) a0
d) everything is present​

Answer 1

If the function f(x) is odd then only the coefficient bₙ is present

Given :

The function f(x) is odd

To find :

If the function f(x) is odd, then which of the only coefficient is present

a) aₙ

b) bₙ

c) a₀

d) everything is present

Solution :

Step 1 of 3 :

Check the function is even or odd

Here it is given that the function f(x) is odd

Step 2 of 3 :

Define Fourier Series of a function

For a function f(x) in the interval ( - c, c)

The Fourier series is given by

$\displaystyle \sf{ f(x) = \frac{a_0}{2} + \sum\limits_{n=1}^{ \infty } a_n \: cos \frac{n\pi x}{c} + \sum\limits_{n=1}^{ \infty } \: b_n \: sin \frac{n\pi x}{c} }$

Where ,

$\displaystyle \sf{ a_0 = \frac{1}{c} \int\limits_{- c}^{c} f(x) \, dx }$

$\displaystyle \sf{ a_n = \frac{1}{c} \int\limits_{- c}^{c} f(x) \: cos\frac{n\pi x}{c} \, dx }$

$\displaystyle \sf{ b_n = \frac{1}{c} \int\limits_{- c}^{c} f(x) \: sin\frac{n\pi x}{c} \, dx }$

Step 3 of 3 :

Find the coefficient which is only present

The given function f(x) is an odd function

Thus we get

$\displaystyle \sf{ a_0 = \frac{1}{c} \int\limits_{- c}^{c} f(x) \, dx } = 0$

$\displaystyle \sf{ \because \: cos \: \frac{n \pi x}{c} \: \: is \: an \: even \: function }$

$\displaystyle \sf{ \therefore \: f(x)cos \: \frac{n \pi x}{c} \: \: is \: an \: odd \: function }$

$\displaystyle \sf{ \therefore \: a_n = \frac{1}{c} \int\limits_{- c}^{c} f(x) \: cos\frac{n\pi x}{c} \, dx } = 0$

Again ,

$\displaystyle \sf{ \because \: sin \: \frac{n \pi x}{c} \: \: is \: an \: odd \: function }$

$\displaystyle \sf{ \therefore \: f(x)sin \: \frac{n \pi x}{c} \: \: is \: an \: even \: function }$

$\displaystyle \sf{ \therefore \: b_n = \frac{1}{c} \int\limits_{- c}^{c} f(x) \: sin\frac{n\pi x}{c} \, dx = \frac{2}{c} \int\limits_{0}^{c} f(x) \: sin\frac{n\pi x}{c} \, dx }$

So when the function f(x) is odd then only the coefficient bₙ is present

Hence the correct option is b) bₙ

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