EMA51002 – Analytical Mathematics
Module 5 FOURIER SERIES
PART-A BTL
1. State the sufficient condition for a function f(x) to be expressed as a Fourier series. 1
2. Does 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥 possess a Fourier expansion? 1
3. Find the value of 𝑏𝑛 for 𝑓(𝑥) = |𝑥| when −𝜋 < 𝑥 < 𝜋 2
4. Find the Fourier constant bn for x sin x in −𝜋 < 𝑥 < 𝜋, when expressed as a Fourier series. 2
5. State the Dirichlet’s conditions 1
6. Find bn of the Fourier series for the function f(x) = x2 , – < x < 2
7 Find the value of 𝑎𝑜 for the Fourier series 𝑓(𝑧) = 𝑥 2 in the interval (0, 2𝜋) 2
8 What do you mean by Harmonic analysis? 1
9. Find the constant term in the expression of 𝑐𝑜𝑠 2 𝑥 as a Fourier series in the interval (−𝜋, 𝜋). 2
10. Find 𝑎0 for 𝑓(𝑥) = 𝑒 𝑥 , in the interval 0 < 𝑥 < 𝜋 2
PART – B
𝑎𝑥
1. Find the Fourier series 𝑓(𝑥) = 𝑒 in0 ≤ 𝑥 ≤ 2𝜋. 3
Find the Fourier series of period 2 for f(x) = x in 0< x <2 . 3
2.
3. Find the value of 𝑏𝑛 for 𝑓(𝑥) = |𝑥| when −𝜋 < 𝑥 < 𝜋 3
1 1
5. Find the Fourier series of period 2 for f(x) = x2 in – < x < and hence show that 14 + 24 + 3
1 𝜋4
+. . . =
34 90
6. Find the Fourier series expansions of 𝑓(𝑥) = 𝑥 2 + 𝑥 in (−𝜋, 𝜋) of periodicity2𝜋. 3
7 Find the Fourier series of the function 𝑓(𝑥) = 𝑥 in the interval (−𝜋, 𝜋) and 𝑓(𝑥) = 𝑓(𝑥 + 2𝜋) 3
8 Obtain a cosine series for 𝑓(𝑥) = 𝑒 𝑥 , 0 < 𝑥 < 𝜋.
9 Find the half range sine series of 𝑓(𝑥) = 𝑥 in the interval (0, 𝜋). 3
10 Develop half range cosine series 𝑓(𝑥) = 𝑥 2 in the interval (0, 𝜋). 3
Part - C
1 Expand 𝑓(𝑥) = 𝑥 as Fourier series in the interval (0, 2𝜋) 3
2 Develop a Fourier series for the function 𝑓(𝑥) = (𝜋 − 𝑥)2 in the interval (0,2𝜋) and deduce 3
1 1 1
the sum as 1 + 22 + 32 + 42 + ⋯
1 1
3 Find the Fourier series of period 2 for f(x) = x2 in – < x < and hence show that 14 + 24 + 3
1 𝜋4
+. . . =
34 90
4 Find the Fourier series for the function 𝑓(𝑥) = |𝑥| when −𝜋 < 𝑥 < 𝜋 . hence show that 1 + 3
1 1 𝜋2
+ +. . . =
32 52 8
1 3
Find the half range cosine series f(x)= x2 in the interval (0, 𝜋)and deduce the sum 1 − 22 +
1
5 +. ..
32
6 Find the half range Fourier cosine series for 𝑓(𝑥) = 𝑥(𝜋 − 𝑥) in the interval (0, 𝜋) 3
Module 5 FOURIER SERIES
PART-A BTL
1. State the sufficient condition for a function f(x) to be expressed as a Fourier series. 1
2. Does 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥 possess a Fourier expansion? 1
3. Find the value of 𝑏𝑛 for 𝑓(𝑥) = |𝑥| when −𝜋 < 𝑥 < 𝜋 2
4. Find the Fourier constant bn for x sin x in −𝜋 < 𝑥 < 𝜋, when expressed as a Fourier series. 2
5. State the Dirichlet’s conditions 1
6. Find bn of the Fourier series for the function f(x) = x2 , – < x < 2
7 Find the value of 𝑎𝑜 for the Fourier series 𝑓(𝑧) = 𝑥 2 in the interval (0, 2𝜋) 2
8 What do you mean by Harmonic analysis? 1
9. Find the constant term in the expression of 𝑐𝑜𝑠 2 𝑥 as a Fourier series in the interval (−𝜋, 𝜋). 2
10. Find 𝑎0 for 𝑓(𝑥) = 𝑒 𝑥 , in the interval 0 < 𝑥 < 𝜋 2
PART – B
𝑎𝑥
1. Find the Fourier series 𝑓(𝑥) = 𝑒 in0 ≤ 𝑥 ≤ 2𝜋. 3
Find the Fourier series of period 2 for f(x) = x in 0< x <2 . 3
2.
3. Find the value of 𝑏𝑛 for 𝑓(𝑥) = |𝑥| when −𝜋 < 𝑥 < 𝜋 3
1 1
5. Find the Fourier series of period 2 for f(x) = x2 in – < x < and hence show that 14 + 24 + 3
1 𝜋4
+. . . =
34 90
6. Find the Fourier series expansions of 𝑓(𝑥) = 𝑥 2 + 𝑥 in (−𝜋, 𝜋) of periodicity2𝜋. 3
7 Find the Fourier series of the function 𝑓(𝑥) = 𝑥 in the interval (−𝜋, 𝜋) and 𝑓(𝑥) = 𝑓(𝑥 + 2𝜋) 3
8 Obtain a cosine series for 𝑓(𝑥) = 𝑒 𝑥 , 0 < 𝑥 < 𝜋.
9 Find the half range sine series of 𝑓(𝑥) = 𝑥 in the interval (0, 𝜋). 3
10 Develop half range cosine series 𝑓(𝑥) = 𝑥 2 in the interval (0, 𝜋). 3
Part - C
1 Expand 𝑓(𝑥) = 𝑥 as Fourier series in the interval (0, 2𝜋) 3
2 Develop a Fourier series for the function 𝑓(𝑥) = (𝜋 − 𝑥)2 in the interval (0,2𝜋) and deduce 3
1 1 1
the sum as 1 + 22 + 32 + 42 + ⋯
1 1
3 Find the Fourier series of period 2 for f(x) = x2 in – < x < and hence show that 14 + 24 + 3
1 𝜋4
+. . . =
34 90
4 Find the Fourier series for the function 𝑓(𝑥) = |𝑥| when −𝜋 < 𝑥 < 𝜋 . hence show that 1 + 3
1 1 𝜋2
+ +. . . =
32 52 8
1 3
Find the half range cosine series f(x)= x2 in the interval (0, 𝜋)and deduce the sum 1 − 22 +
1
5 +. ..
32
6 Find the half range Fourier cosine series for 𝑓(𝑥) = 𝑥(𝜋 − 𝑥) in the interval (0, 𝜋) 3
