Harmonic functions using complex analysis

Question

Suppose we want a harmonic function $f$ in the first quadrant with the boundary condition that for $\arg(z) = \theta$ where $\theta$ is fixed and along $\Im(z) = 0$ we have $f = 1$. Since $\arg(z^{\frac{2\pi}{\theta}}) + 1$ is the imaginary part of a natural logarithm, would this work? The only problem is I don't know how to get a function of $x$ and $y$ if $2\pi/\theta$ is not an integer where $z = x+iy$. Is there another complex analytic function that would work here that satisfied the boundary conditions?