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Q.1. Expand in trigonometric Fourier series. Draw three periods and use it to derive the following relations.
Thus the trigonometric Fourier series of the given function is
The graph of the function is shown in the figure(i) At x = 0 , the Fourier series converges to
(ii) At x = π we have
(iii) we have at x = π/2Thus
Hence
Q.2. Expand the given function in Fourier Series and hence find the value of
Given Function
Here, 2l = 2 ⇒ l = 1
Thus, b_{n} = 4/nπ for odd n.= 0 for even n.
Hence the Fourier Series of the given function is
Q.3.
The function shown in the graph is plotted for one period. Sketch it over three periods and use the graph to find the Fourier series of the function shown.
f (x) = (x + π), π < x < 0
x, 0 < x < π
Hence
= 0 for even n
Thus, b_{n} = 2/n for odd n
= 0 for even n
Thus the Fourier series of the given function is
Q.4. Expand the given function in Fourier Series and hence find the value of
Thus Fourier series is
At x = 1, f (x) = 2
Q.5. Find the Fourier series of
Now, for n = 1, 5, 9, 13
for 3, 7,11,15
and for even n.
Thus, a_{n} = 1/nπ for n = 1, 5, 9, 13
for n 3, 7,11,15
We can write the general formula for for even n andfor odd n.
Now, for 2, 6,10,14....
= +1 for n = 4, 8,12,16 ......
= 0 for odd n
Hence b_{n} = 2/nπ for n = 2, 6,10,14....
= 0 for n = 4, 8, 12, 16 ....= 1/nπ for odd n.
We can write the general formula for even nThus b_{n} = 1/nπ , for odd n.
Thus the Fourier series of the given function is
Q.6. Expand the given function in Fourier Series and hence find the value of
The given function is
Here, 2l = 2 ⇒ l = 1
for odd n and a_{n} a = for even n.
Thus the Fourier Series of the given function isPutting x = 0 gives
Q.7. Find the Fourier series of f (x) = x^{2} , 0 < x < 2π
The given function is f (x) = x^{2} , 0 < x < 2π
Thus, b_{n} = 4π/n for all n.
Thus the Fourier series of the given function is
Q.8. Expand the given function in Fourier Series f(x) = π sinπx, 0 < x < 1 and hence find the value of
Here, 2l = 1 ⇒ l = 1/2
For all n
Thus the Fourier Series of the given function is given byAt x = 0 , the function is discontinuous and the fourier series converges to
Q.9. Find the Fourier series of
The given function is
We know that
We see that this formula holds for n ≠ 1.
Hence we directly calculate
Now let us evaluate a_{n} using equation (I)
when n is odd, (n + 1) is even and (n  1) is even. Thus for odd n,
When n is even (n + 1) and (n  1) are odd, hence
Thus a_{n} = 0 for odd n
Now we have
We see that we cannot calculate b_{1} using this formula.
Let us evaluate b_{1} directly
Let us evaluate other b_{n} ’s using equation (ii)
Thus b_{n} = 0 for all n.Hence the Fourier Series of the given function is
In the expanded form
Q.10. Expand the given functions in Fourier Series f(x) = e^{x} , 0 < x < 2 and hence find the value of
f(x) = e^{x} , 0 < x < 2
Hence,
Thus the Fourier Series of the given function is
In the expanded form we can writeAt x = 0, the function is discontinuous and the Fourier series converges to the average of its left hand and right hand limit.
Thus , at x = 0
18 docs24 tests
