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?

  • 1
    $\begingroup$ you have an analytic integrand around zero, so... $\endgroup$ – 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.

  • $\begingroup$ Ok thats was easy knowing that theorem!! Thanks $\endgroup$ – JustANoob Nov 27 '18 at 15:23

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