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).
well, you know that
e^iz = cos(z) + i sin(z)
...
To evaluate the formula for c0 and cn in Fourier series, we need to use the given integral:
∫ e^kx dx = e^kx / k, unless k = 0.
In a Fourier series, we are looking to express a function as a sum of sines and cosines. The general formula for the Fourier coefficient cn is:
cn = (1/L) ∫ f(x) * cos(nπx/L) dx, for cosine terms, and
cn = (1/L) ∫ f(x) * sin(nπx/L) dx, for sine terms,
where L is the period of the function and f(x) is the function being represented.
In this case, we have the integral ∫ e^kx dx, and we need to find the values of c0 and cn.
To find c0, we substitute n = 0 into the formula:
c0 = (1/L) ∫ f(x) dx.
In this case, we don't have f(x) explicitly given, but we do have the integral ∫ e^kx dx, which represents f(x). So, for c0, we can simply substitute k = 0 into the given integral:
c0 = ∫ e^0 dx = ∫ 1 dx = x + C.
For cn, we can substitute n > 0 into the formula:
cn = (1/L) ∫ f(x) * cos(nπx/L) dx, for cosine terms, and
cn = (1/L) ∫ f(x) * sin(nπx/L) dx, for sine terms.
In this case, we have f(x) = e^kx, where k is a constant (not equal to zero). So, substituting f(x) and k into the formulas, we get:
cn = (1/L) ∫ e^kx * cos(nπx/L) dx, for cosine terms, and
cn = (1/L) ∫ e^kx * sin(nπx/L) dx, for sine terms.
To rewrite the Fourier series in terms of sines and cosines, we use the formula:
f(x) = c0/2 + Σ(cn * cos(nπx/L) + bn * sin(nπx/L)).
In this case, since we have both cosine and sine terms in the original function, we would include both cosine and sine terms in the Fourier series.
Simplifying the Fourier series involves eliminating the zero terms. Since cn and bn represent the coefficients for the cosine and sine terms, respectively, we need to remove terms where cn = 0 or bn = 0 (if any).
By substituting the values of c0, cn, and bn determined from the integrals and rearranging the terms, the Fourier series can be simplified further.
Note: It would be helpful to have the specific function or interval given to provide a more accurate and meaningful evaluation of the Fourier series.