Find the Fourier series of the below function:𝑓(𝑥) = 𝑥3 −𝜋 < 𝑥 < 𝜋

To find the Fourier series of the function 𝑓(𝑥) = 𝑥^3 on the interval −𝜋 < 𝑥 < 𝜋, we can follow these steps:

Step 1: Determine the period of the function. Since the function is defined on the interval −𝜋 < 𝑥 < 𝜋, the period is 2𝜋.

Step 2: Express the function as an odd periodic extension. Since the function is odd (𝑓(−𝑥) = −𝑓(𝑥)), we can extend it periodically by reflecting it about the y-axis. This means that the function will repeat itself every 2𝜋 units.

Step 3: Calculate the Fourier coefficients. The Fourier coefficients can be calculated using the formula: 𝑎𝑛 = (1/𝜋) ∫[−𝜋,𝜋] 𝑓(𝑥) cos(𝑛𝑥) 𝑑𝑥 𝑏𝑛 = (1/𝜋) ∫[−𝜋,𝜋] 𝑓(𝑥) sin(𝑛𝑥) 𝑑𝑥

For the given function 𝑓(𝑥) = 𝑥^3, the Fourier coefficients can be calculated as follows: 𝑎0 = (1/𝜋) ∫[−𝜋,𝜋] 𝑥^3 𝑑𝑥 𝑎𝑛 = (1/𝜋) ∫[−𝜋,𝜋] 𝑥^3 cos(𝑛𝑥) 𝑑𝑥 𝑏𝑛 = (1/𝜋) ∫[−𝜋,𝜋] 𝑥^3 sin(𝑛𝑥) 𝑑𝑥

Step 4: Simplify the integrals and calculate the coefficients. By evaluating the integrals, we can find the values of the Fourier coefficients 𝑎0, 𝑎𝑛, and 𝑏𝑛.

Step 5: Write the Fourier series. The Fourier series of the function 𝑓(𝑥) = 𝑥^3 can be written as: 𝑓(𝑥) = 𝑎0/2 + ∑[𝑛=1,∞] (𝑎𝑛 cos(𝑛𝑥) + 𝑏𝑛 sin(𝑛𝑥))

By substituting the calculated values of 𝑎0, 𝑎𝑛, and 𝑏𝑛 into the Fourier series equation, we can obtain the final expression for the Fourier series of 𝑓(𝑥) = 𝑥^3 on the interval −𝜋 < 𝑥 < 𝜋.