# Show that $h(z)=\int_{0}^{1}\frac{t^2}{1-tz}dt$ is analytic in a nbh. of zero.

Consider the function $$h(z)=\int_{0}^{1}\frac{t^2}{1-tz}dt$$ whenever the integral exists, $$z \in \mathbb{C}$$. Show that $$h$$ is analytic in a neighborhood of the origin and calculate the power series expansion of $$h$$ centered around the origin.

I have difficulties with this problem. I tried to show that its complex derivative exists but getting nowhere. Any tips?

• you have an analytic integrand around zero, so... – Masacroso Nov 27 '18 at 15:08

More generally, if $$g(z,t)$$ is continuous as a function of $$z$$ and $$t$$ in some region $$U \times V \subseteq \mathbb C^2$$ and analytic in $$z$$ there, then $$\oint_\Gamma g(z,t)\; dt$$ is analytic in $$U$$. This can be seen by approximating the integral as a limit of Riemann sums.
To get the power series, expand the integrand in a geometric series about $$z=0$$ and integrate term-by-term.