Question

Find the complex exponential form of the Fourier series for the following function:

Given the function: f(t)={-1 for −π≤t<0 {1 for 0≤t<π To find the complex exponential form of the Fourier series for this function, follow the necessary steps, including determining the period, expressing f(t) as a piecewise function with a common period, calculating the complex Fourier coefficients, and expressing the Fourier series in complex exponential form.

Answer 1

To find the complex exponential form of the Fourier series for the given function, follow the necessary steps including determining the period, expressing f(t) as a piecewise function, calculating the complex Fourier coefficients, and expressing the Fourier series in complex exponential form.

Step-by-step explanation:

To find the complex exponential form of the Fourier series for the given function f(t)={-1 for −π≤t<0 {1 for 0≤t<π, we need to follow the necessary steps:

1. Determine the period of the function.
2. Express f(t) as a piecewise function with a common period.
3. Calculate the complex Fourier coefficients.
4. Express the Fourier series in complex exponential form.

Let's go through each step in detail:

1. The given function has a period of 2π.
2. To express f(t) as a piecewise function with a common period, we can rewrite it as:

f(t) = {-1 for −π≤t<0 {1 for 0≤t<π, for -π≤t<π

1. To calculate the complex Fourier coefficients, we need to find the values of the coefficients a_n and b_n using the formulas:

a_n = (1/π) ∫[−π,π] f(t) cos(nt) dt

b_n = (1/π) ∫[−π,π] f(t) sin(nt) dt

1. The complex exponential form of the Fourier series can be expressed as:

f(t) = ∑[n=-∞ to ∞] c_n e^(int), where c_n = (a_n - i b_n)/2