Assignment on Fourier Analysis

Assignment on Fourier Analysis
1. What is a periodic Function? Provide an example with neat sketch. Determine the period of the function
𝑦𝑦 = 2023𝑠𝑠𝑠𝑠𝑛𝑛202211𝑥𝑥 + 12
2. De�ine Odd and Even Function with �igures. Provide example.
3. From the de�inition of Fourier Series 𝑓𝑓(𝑥𝑥) = 𝑎𝑎0 + ∑ (𝑎𝑎𝑛𝑛𝑐𝑐𝑐𝑐𝑐𝑐 �
𝑛𝑛𝑛𝑛𝑛𝑛
𝐿𝐿 � + 𝑏𝑏𝑛𝑛𝑠𝑠𝑠𝑠𝑠𝑠 ∞
𝑛𝑛=1 �
𝑛𝑛𝑛𝑛𝑛𝑛
𝐿𝐿 �, which is period
over the interval [−𝐿𝐿, +𝐿𝐿], derive the formula for the coef�icients 𝑎𝑎0, 𝑎𝑎𝑛𝑛, 𝑏𝑏𝑛𝑛.
4. Using Euler’s Identities, prove that the Fourier series can be expressed as 𝑓𝑓(𝑥𝑥) = ∑ 𝑐𝑐𝑛𝑛𝑒𝑒
𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
+∞ 𝐿𝐿 𝑛𝑛=−∞
5. De�ine Orthogonal Functions. Using ∫ 𝑠𝑠𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 = 0, 𝑎𝑎𝑎𝑎𝑎𝑎 +𝜋𝜋
−𝜋𝜋 ∫ 𝑐𝑐𝑐𝑐𝑐𝑐 𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 = 0, 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑚𝑚, 𝑛𝑛 ∈ 𝑍𝑍, +𝜋𝜋
−𝜋𝜋 prove
the following identities –
a. ∫−𝜋𝜋
𝜋𝜋 cos 𝑛𝑛𝑛𝑛 cos 𝑚𝑚𝑚𝑚 𝑑𝑑𝑑𝑑 = �
2𝜋𝜋 if 𝑛𝑛 = 𝑚𝑚 = 0
𝜋𝜋 if 𝑛𝑛 = 𝑚𝑚 ≠ 0
0 if 𝑛𝑛 ≠ 𝑚𝑚
b. ∫0
𝜋𝜋 cos 𝑛𝑛𝑛𝑛 cos 𝑚𝑚𝑚𝑚 𝑑𝑑𝑑𝑑 = �
𝜋𝜋 if 𝑛𝑛 = 𝑚𝑚 = 0 𝜋𝜋
2
if 𝑛𝑛 = 𝑚𝑚 ≠ 0
0 if 𝑛𝑛 ≠ 𝑚𝑚
c. ∫−𝜋𝜋
𝜋𝜋 sin 𝑛𝑛𝑛𝑛 sin 𝑚𝑚𝑚𝑚 𝑑𝑑𝑑𝑑 = �
𝜋𝜋 if 𝑛𝑛 = 𝑚𝑚
0 if 𝑛𝑛 ≠ 𝑚𝑚
d. ∫0
𝜋𝜋 sin 𝑛𝑛𝑥𝑥 sin 𝑚𝑚𝑚𝑚 𝑑𝑑𝑑𝑑 = �
𝜋𝜋
2
if 𝑛𝑛 = 𝑚𝑚
0 if 𝑛𝑛 ≠ 𝑚𝑚
e. ∫−𝜋𝜋
𝜋𝜋 sin 𝑛𝑛𝑛𝑛 cos 𝑚𝑚𝑚𝑚 𝑑𝑑𝑑𝑑 = 0
6. Draw sketches and determine the Fourier Series for the following functions.
a. 𝑠𝑠(𝑥𝑥) = 𝑥𝑥
𝜋𝜋
, for −𝜋𝜋 < 𝑥𝑥 < +𝜋𝜋
b. 𝑠𝑠(𝑥𝑥) = 3|sin 𝑥𝑥| for 0 ≤ 𝑥𝑥 < 2𝜋𝜋
c. 𝑠𝑠(𝑥𝑥) = �
2sin 𝑥𝑥 for 0 ≤ 𝑥𝑥 < 𝜋𝜋
0 for 𝜋𝜋 ≤ 𝑥𝑥 < 2𝜋𝜋
d. 𝑠𝑠(𝑥𝑥) = �
1 for 0 ≤ 𝑥𝑥 < 𝜋𝜋
0 for 𝜋𝜋 ≤ 𝑥𝑥 < 𝜋𝜋
e. 𝑠𝑠(𝑥𝑥) = 𝐴𝐴 − 𝐴𝐴𝐴𝐴
𝑃𝑃
for 0 ≤ 𝑥𝑥 < 𝑃𝑃