$(a)$ Show that there is a complex analytic function defined on the set $U=\{z\in\mathbb{C} \ | \ |z|>4\}$ whose derivative is $\displaystyle\frac{z}{(z-1)(z-2)(z-3)}$.

$(b)$ Is there a complex analytic function on $U$ whose derivative is $\displaystyle\frac{z^2}{(z-1)(z-2)(z-3)}$?

The problem clearly reduces to find a analytic function $f(z)$ on $U$ such that $f'(z)=\displaystyle\frac{z}{(z-1)(z-2)(z-3)}$. But if I integrate both sides over the contour of $U$, the RHS is analytic on $U$. So $f(z)=0$. But this result is too absurd to consider. Again same goes for $(b)$, but which is not the case since we have to prove no such $f$ would exist. I considered the CR-equations but the given function is too large to handle for this. What is the best approach for this problem? Any help is appreciated.

  • $\begingroup$ why don't you just use partial fractions..? $\endgroup$ Feb 24, 2020 at 6:42
  • $\begingroup$ it becomes -2/(z-2) + 1/[2(z-1)) + 3/[2(z-3)] $\endgroup$ Feb 24, 2020 at 6:43
  • $\begingroup$ I believe there should be an easier method for this problem, but I can't figure it out. $\endgroup$ Feb 24, 2020 at 6:43
  • $\begingroup$ I used it above. But it yields no good result. $\endgroup$ Feb 24, 2020 at 6:46
  • $\begingroup$ there's a shortcut lol. remove the (z-1) from the denominator, and then plug in z=1 to z/((z-2)(z-3) to get 1/2 $\endgroup$ Feb 24, 2020 at 6:46


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