Compute the Fourier Series of e^x on the Interval [-pi,pi]?

e^x represents an Exponential Function. Fourier series can be thought of as an expansion of any periodic function in the form of an inestimable sum of sine and cosine Functions. Perpendicular associations of two trigonometric functions: sine and cosine is of prime use in Fourier series Transformations. To compute the Fourier series of e to the power x on the interval minus pi,p we will follow steps shown below:

f (x) = e^x,
Where, x belongs to range [−pi, pi],
e₀ = 1 / pi –_pi ∫ ^pi (e^x) dx = (E ^pi - E ^–pi) / pi,
As we know that hyperbolic representation of sine function is given as:
Sin hx = (e^x - e^-x) / 2,

Using this formula we can write:
(e ^pi - e ^–pi) / pi = (2 sin (h * pi)) / pi,

If exponential function e^x is multiplied to cosine and sine functions of Trigonometry we get following Fourier series:
e₀ = (1 / n) * –_pi ∫ ^pi (e^x) cos (n x) dx = (1 / pi) * [(1 / n) * e ^x * sin (n x)_–pi] ^pi – (1 / n) * –_pi ∫ ^pi (e^x) sin (n x) dx.

On solving above equation we get following result:
((- 1)n + 1 * (E ^-pi - E ^pi)) / (pi (1 + n²)),
e₀ = (1 / n) * _–pi ∫ ^pi (e^x) sin (n x) dx = (1 / pi) * [(-1 / n) * e x * sin (n x)_–pi] ^pi + (1 / n) * –_pi ∫ ^pi (ex) cos (n x) dx,

On solving above equation we get following result:
(n * (- 1)ⁿ * (e ^-pi - e ^pi)) / (pi (1 + n²)).