$$v = e^x siny$$ $$\frac{\partial v}{\partial x} = e^x siny \hspace{1cm} \frac{\partial v}{\partial y} = e^x cosy$$ $$\frac{\partial^2 v}{\partial x^2} = e^x siny \hspace{1cm} \frac{\partial^2 v}{\partial y^2} = -sinye^x$$ $$\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = e^x siny + (-e^x siny) = e^x siny - e^x siny$$ $$\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$$

$\therefore$ v is a harmonic function.

$u_{x} = v_{y} \\ \therefore u_{x} = e^x cosy \\ u = \int uxdr = e^x cosy$

$f(z) = u + iv = e^x csy + i(e^x siny) = e^z + c$ (x = z & y = 0)

$f(z) = e^{(x + iy)} = e^x * e^{iy} = e^x (cosy + isiny) = e^x cosy + ie^x siny$

$\therefore v = e^x siny$ is harmonic conjugate.