Find the fourier series of the function 𝑓(𝑥) = 0, − 𝜋 < 𝑥 < 0= 𝑥2, 0 < 𝑥 < 𝜋

To find the Fourier series of the function 𝑓(𝑥) = 0, − 𝜋 < 𝑥 < 0= 𝑥2, 0 < 𝑥 < 𝜋, we can follow these steps:

Step 1: Determine the period of the function. Since the function is defined differently for two intervals, we need to find the common period. In this case, the function is periodic with a period of 2𝜋, as it repeats every 2𝜋 units.

Step 2: Express the function as an odd or even function. For the given function, 𝑓(𝑥), we can see that it is an even function because it is symmetric about the y-axis. This means that 𝑓(−𝑥) = 𝑓(𝑥).

Step 3: Calculate the Fourier coefficients. To find the Fourier coefficients, we can use the formulas: 𝑎0 = (1/𝑇) ∫[−𝑇/2, 𝑇/2] 𝑓(𝑥) 𝑑𝑥 𝑎𝑛 = (2/𝑇) ∫[−𝑇/2, 𝑇/2] 𝑓(𝑥) cos(𝑛𝜔𝑥) 𝑑𝑥 𝑏𝑛 = (2/𝑇) ∫[−𝑇/2, 𝑇/2] 𝑓(𝑥) sin(𝑛𝜔𝑥) 𝑑𝑥

Since 𝑓(𝑥) = 0 for −𝜋 < 𝑥 < 0, the integral for 𝑎0 and 𝑎𝑛 will be 0. We only need to calculate 𝑏𝑛.

Step 4: Calculate the Fourier series. The Fourier series for an even function is given by: 𝑓(𝑥) = 𝑎0/2 + ∑[𝑛=1, ∞] (𝑎𝑛 cos(𝑛𝜔𝑥) + 𝑏𝑛 sin(𝑛𝜔𝑥))

Since 𝑎0 = 0, the Fourier series simplifies to: 𝑓(𝑥) = ∑[𝑛=1, ∞] 𝑏𝑛 sin(𝑛𝜔𝑥)

Step 5: Calculate the 𝑏𝑛 coefficients. To calculate the 𝑏𝑛 coefficients, we can use the formula: 𝑏𝑛 = (2/𝑇) ∫[−𝑇/2, 𝑇/2] 𝑓(𝑥) sin(𝑛𝜔𝑥) 𝑑𝑥

For the given function, 𝑓(𝑥) = 𝑥^2 for 0 < 𝑥 < 𝜋, the integral becomes: 𝑏𝑛 = (2/2𝜋) ∫[0, 𝜋] 𝑥^2 sin(𝑛𝑥) 𝑑𝑥

Step 6: Evaluate the integral and simplify the Fourier series. By evaluating the integral and simplifying the Fourier series, we can obtain the final expression for the Fourier series of the given function 𝑓(𝑥).

Note: The specific calculations for the 𝑏𝑛 coefficients and the final expression of the Fourier series will depend on the values obtained in Step 5.