Let $f:\mathbb R\to\mathbb R$ such that $f(t)=0$ if $|t|\leq 1$ and $f(t)=|t|^\lambda$ if $|t|>1$. Here $\lambda<0$ is a constant.
We consider the following function on $\mathbb C_+=\{x+iy:\;x,y\in\mathbb R,y>0\}$:
For $z=x+iy$, $y>0$ we set $F(x+iy)=\frac1{\pi}\int_{\mathbb R}\frac y{(x-t)^2+y^2}\,f(t)\,dt$. My question is that:
- Does such function $F(x+iy)$ is analytic on $\mathbb C_+$ ?
No. This function is real-valued, so it cannot be analytic since this would violate the Cauchy-Riemann equations. However, it is a harmonic function. Indeed, $F$ is a convolution by the Poisson kernel, which I suggest you look into a little.