Proposition

Let $u(x, y) = e^x\sin y$.

  1. Show that $u$ is harmonic on $\mathbb{C}$.
  2. Find an entire function $f$ such that $\Re(f) = u$.

Solution

1

\(u_{xx} = e^x\sin y\) and \(u_{yy} = -e^x\sin y\). Therefore, \(u_{xx} + u_{yy} = 0\).

2

We will use Theorem 6.8[A first course in complex analysis] to find a harmonic conjugate.

\[\begin{align*} \int_{0}^{y} e^x \sin t dt - \int_{0}^{x} e^t \cos(0) dt &= -e^x \cos t \Big\vert^y_0 - e^t \Big\vert^{x}_0 \\ &= -e^x(\cos y - 1) - (e^x - e^0) \\ &= -e^x \cos y + 1. \end{align*}\]

Let $v(x, y) = -e^x \cos y + 1$. Then $f(x + iy) = u(x, y) + iv(x, y)$ is harmonic on $\mathbb{C}$, thus it is entire.