Civil Engineering (CE) Exam  >  Civil Engineering (CE) Questions  >  Let \(f\left( x \right) = \;\left\{ {\begin{a... Start Learning for Free
Let \(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} { - π }&{if\;}&{ - π
be a periodic function of period 2π. The coefficient of sin 5x in the Fourier series expansion of f(x) in the interval [-π, π] is
  • a)
    4/5
  • b)
    5/4
  • c)
    4/3
  • d)
    3/4
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Let \(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} { - π ...
That's not a complete function. Could you please provide the rest of it?
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Community Answer
Let \(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} { - π ...
Concept:
Let f(x) is a periodic function defined in (C, C + 2L) with period 2L, then the Fourier series of f(x) is

Where the Fourier series coefficients a0, an, and bn are given by
  • If f(x) is an odd function, then only bn exists where a0 and bn are zero.
  • If f(x) is an even function, then both a0 and an exists where bn is zero.
Calculation:
\(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} { - π }&{if\;}&{ - π
Fundamental period = 2π
The given function is an odd function and hence only bn exists.

Here L = 2π

Now, the Fourier series of f(x) is,

The coefficient of sin 5x = b5
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Let \(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} { - π }&{if\;}&{ - πbe a periodic function of period 2π. The coefficient of sin 5x in the Fourier series expansion of f(x) in the interval [-π, π] isa)4/5b)5/4c)4/3d)3/4Correct answer is option 'A'. Can you explain this answer?
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