Harmonic Function Values (Complex Analysis)

1. Jun 26, 2012

EC92

1. The problem statement, all variables and given/known data
Let $u$ be a continuous real-valued function in the closure of the unit disk $\mathbb{D}$ that is harmonic in $\mathbb{D}$. Assume that the boundary values of $u$ are given by

$u(e^{it}) = 5- 4 \cos t.$

Furthermore, let $v$ be a harmonic conjugate of $u$ in $\mathbb{D}$ such that $v(0) = 1$. Find $u(1/2)$ and $v(1/2)$.

2. Relevant equations

Harmonic functions fulfill
$u_{xx} + u_{yy} = 0$.
Poisson's Integral Formula (for harmonic functions) tells us that
$u(z) = \frac{1}{2\pi} \int_{0}^{2\pi} \frac{1-|z|^2}{|e^{i\theta}-z|^2}u(e^{i\theta}) d\theta$.
The harmonic conjugate of $u$ is given by the line integral
$v(z) = \operatorname{Im} \int_0 ^z u_x(w) - iu_y(w) dw$.
The harmonic conjugate is itself harmonic, so Poisson's formula applies to it as well.

3. The attempt at a solution

It's easy to find $u(1/2)$: it is
$\frac{1}{2\pi} \int_0^{2\pi} \frac{3/4}{5/4 - \cos\theta} (5-4\cos \theta) d\theta = \frac{1}{2\pi} \int_0 ^{2\pi} 3 d\theta = 3$.

However, I can't figure out how to compute $v(1/2)$; the formulas just seem to give a hopelessly complicated mess.
Any ideas on how to proceed?
Thanks.