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 compute the coefficients for each term in the series.
The Fourier series representation of a function f(x) with period T is given by:
f(x) = c0 + Σ(cn*cos(nωt) + dn*sin(nωt))
where ω = 2π/T and n is the harmonic number.
To find c0, we can use the following formula:
c0 = (1/T) * ∫[0,T] f(x) dx
In this case, the given formula is ∫ e^kx dx = e^kx /k if k ≠ 0.
So, to find c0, we substitute f(x) = e^kx /k into the formula and integrate:
c0 = (1/T) * ∫[0,T] (e^kx /k) dx
Integrating e^kx gives us (e^kx)/k, so the integral becomes:
c0 = (1/T) * [(e^kx)/k]^T_0
Evaluating this expression at the upper and lower limits, we get:
c0 = (1/T) * [(e^(kT))/k - (e^0)/k]
Simplifying further, we have:
c0 = (1/T) * [(e^(kT) - 1)/k]
To find cn for n > 0, we can use the following formula:
cn = (2/T) * ∫[0,T] f(x)*cos(nωt) dx
In this case, we substitute f(x) = e^kx /k into the formula and integrate:
cn = (2/T) * ∫[0,T] (e^kx /k) * cos(nωt) dx
We can split the integral into two parts:
cn = (2/T) * (1/k) * ∫[0,T] e^kx * cos(nωt) dx
Using integration by parts, we can evaluate this integral:
cn = (2/T) * (1/k) * [(e^kx * cos(nωt))/(k+nω) - (nω * e^kx * sin(nωt))/(k+nω)^2]^T_0
Evaluating this expression at the upper and lower limits, we get:
cn = (2/T) * (1/k) * [(e^(kT) * cos(nωT))/(k+nω) - (nω * e^0 * sin(0))/(k+nω)^2]
Simplifying further, we have:
cn = (2/T) * (1/k) * [(e^(kT) * cos(nωT))/(k+nω)]
To rewrite the Fourier series in terms of sines and cosines, we can use Euler's formula:
e^(iθ) = cos(θ) + i*sin(θ)
Substituting this into the formulas for c0 and cn, we have:
c0 = (1/T) * [(e^(kT) - 1)/k] = (1/T) * [(cos(kωT) + i*sin(kωT) - 1)/k] = (1/T) * [cos(kωT)/k - 1/k]
cn = (2/T) * (1/k) * [(e^(kT) * cos(nωT))/(k+nω)] = (2/T) * (1/k) * [(cos(kωT) + i*sin(kωT)) * cos(nωT))/(k+nω)] = (2/T) * (1/k) * [(cos(kωT) * cos(nωT))/(k+nω) + i*sin(kωT) * cos(nωT))/(k+nω)]
Note: The terms involving the imaginary unit (i) will be eliminated as they correspond to the sine terms in the Fourier series.
Finally, to eliminate zero terms, we need to determine the conditions under which the cosine terms are equal to zero. If kωT = (2m+1)π/2 for some integer m, then cos(kωT) = 0, which means the corresponding coefficient will be zero in the Fourier series.