GATE EE 2016 Set 2 — Question 20
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Engineering Mathematics → Calculus → Fourier Series
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Question
Let be a real, periodic function satisfying . The general form of its Fourier series representation would be
Correct answer
(B) f(x) = Σₖ₌₁^(∞) bₖ sin(kx)
Solution
A function is defined as odd if . For an odd periodic function, the Fourier series coefficients are calculated as follows:
1.The constant term because the integral of an odd function over a symmetric interval is zero.
2.The cosine coefficients because the product of an odd function and an even function is an odd function, and its integral over a symmetric interval is zero.
3.The sine coefficients are generally non-zero because the product of two odd functions is an even function.
Therefore, the general form of the Fourier series for an odd function contains only sine terms: .Continue learning with Success Tracker
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