is $g$ is a harmonic?harmonic polynomial?

Question

Let $u$ be a real valued harmonic function on $\mathbb{C}$ and

$$g:\mathbb{R}^2\to\mathbb{R},~~~~~g(x,y)=\int_{0}^{2\pi} u(e^{i\theta}(x+iy))\sin\theta d\theta$$

Then is $g$ is a harmonic?harmonic polynomial? Could any one help me how to proceed?Thank you.

Answer 1

Differentiation under the integral gives $$\Delta g(x,y) = \int_0^{2\pi}e^{2i\theta}\sin(\theta)[u''(e^{i\theta}(x+iy)) - u''(e^{i\theta}(x+iy))]d\theta = 0$$ Thus $g$ is harmonic.