Evaluate the formula for cn in Fourier :integral of e^kx dx = e^kx /k :unless k=0: Type your formula for c0 and cn (n>0) into the indicated spaces.
Then rewrite the Fourier series in terms of sines and cosines. Simplify as far as possible (eliminate the zero terms).
To evaluate the formula for c0 and cn in the Fourier series, we need to find the coefficients of the series. The Fourier series represents a periodic function as an infinite sum of sines and cosines.
1) Formula for c0:
For c0, we use the formula:
c0 = (1 / L) * ∫(f(x) dx)
where L is the period of the function and f(x) is the periodic function.
However, in this case, the formula for c0 is not provided in the question. To proceed with the evaluation of the Fourier series, we will need the specific function for which we are calculating the series.
2) Formula for cn (n > 0):
For cn (n > 0), we use the formula:
cn = (2 / L) * ∫(f(x) * cos(nωx) dx)
where ω = 2π / L and L is the period of the function.
Again, you need to have the specific function f(x) to properly evaluate cn.
To rewrite the Fourier series in terms of sines and cosines, we use the formulas:
f(x) = a0/2 + ∑(an * cos(nωx) + bn * sin(nωx))
where n is the coefficient index and ω = 2π / L.
3) Simplifying the Fourier series:
To simplify the Fourier series and eliminate zero terms, we need to find the values of an and bn. These coefficients can be calculated using the formulas:
an = (2 / L) * ∫(f(x) * cos(nωx) dx)
bn = (2 / L) * ∫(f(x) * sin(nωx) dx)
By evaluating these integrals for each coefficient index n and using the given function, you can calculate the values of an and bn. Then substitute these values into the rewritten Fourier series expression to obtain the simplified representation of the function in terms of sines and cosines.