Question

The Fourier series of an odd periodic function
contains only
a) Odd harmonic b) Even harmonic

c) Cosine terms d) Sine terms

​

Answer 1

Answer:

The Fourier series of an odd periodic function contains only Sine terms i.e. option (d).

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Answer 2

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Hint: A Fourier series is a means of representing a periodic function as a sum of sine and cosine functions (possibly infinite).In such problems, finding zero coefficients is time consuming and can be prevented. With understanding of even and odd functions, without implementing the integration, the zero coefficient can be predicted.

Complete step by step answer:

A function y = f(t) is said to be odd if

f(t) = -f(t) for all values of t. The graph

of an odd function is always symmetrical about the origin.

The above graph has amplitude 1 and period

2π

For an odd function f(t) defined over the range

-L to L (period = 2L)

We can observe that an = 0 for all n

Then we have

an= L Sf(t) cos nπt -dt

L

So, the zero coefficients in this case are: ao = 0

and an 0

The coefficients of bn is given by

bn = f(t) sin not T

-dt

Therefore, the Fourier series of the following

odd function is given by

f1 b f(t) = bn sin n=1

nπt

L

Hence, the Fourier series of an odd periodic

function contains only sine terms.

Hence the correct option is (D).

Note:

Using the Fourier series in various questions makes our task easy, fast, and more productive. We can easily find out the zero coefficient of vast problems by the help of Fourier series even without performing the actual integration. Baron Jean Baptiste Joseph Fourier introduced the idea that a series of harmonically related sines and cosines can represent any periodic function.

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