GATE EE 2014 Set 1 — Question 36
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Engineering Mathematics → Calculus → Fourier Series
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Question
Let be a function defined by , where represents the integer part of . (That is, it is the largest integer which is less than or equal to ).
The value of the constant term in the Fourier series expansion of is __________
The value of the constant term in the Fourier series expansion of is __________
Correct answer
0.5 to 0.5
Solution
The function given is , which represents the fractional part of . This is a periodic function.First, let's determine the period, , of the function .
.
Thus, the function is periodic with a fundamental period of .The constant term in the Fourier series expansion of a periodic function is its average value over one period. This term is commonly denoted as in the trigonometric Fourier series .The formula for the constant term is:Substituting the period and the function :In the interval of integration, , the integer part of , denoted by , is equal to 0. Therefore, the function simplifies to for .The integral becomes:Now, we evaluate the definite integral:So, the value of the constant term in the Fourier series expansion of is 0.5.
.
Thus, the function is periodic with a fundamental period of .The constant term in the Fourier series expansion of a periodic function is its average value over one period. This term is commonly denoted as in the trigonometric Fourier series .The formula for the constant term is:Substituting the period and the function :In the interval of integration, , the integer part of , denoted by , is equal to 0. Therefore, the function simplifies to for .The integral becomes:Now, we evaluate the definite integral:So, the value of the constant term in the Fourier series expansion of is 0.5.
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