This is called the Dirichlet problem for the half-plane. Let so that
The exponentials were chosen over the hyperbolic trig functions because one of the exponetials tends to zero as its argument tends to infinity, while both h-trig functions go to infinity.
To satisfy the limit condition, we must set
Since we are dealing with above, only needs to take on nonnegative values. This is reflected in the integrals in the solution below:
The Fourier coefficients are: