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Suppose $f$ is an entire function, and take $a,b\in\mathbb{C}$. For $R>\max\{|a|,|b|\},$ estimate $$\int_{|z|=R} \frac{f(z)}{(z-a)(z-b)} dz.$$
Assuming $f$ is bounded, let $R\rightarrow\infty$ and show $f$ is constant.

I'm not sure what it means by "estimate", are we supposed to take an upper and lower bound?

Also, after reading the second part, I suspect that it's actually a derivation of Liouville's Theorem, as the only condition to apply the theorem is that $f$ is an entire bounded function...

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  • $\begingroup$ Are you assume $f$ is bounded and entire or boundedness is a result of integration? $\endgroup$
    – Nosrati
    Commented Sep 28, 2017 at 6:47
  • $\begingroup$ I think $f$ is entire and bounded in the second part. The question was given as is. $\endgroup$ Commented Sep 28, 2017 at 6:50
  • $\begingroup$ If I'm not mistaken, due to residue theorem this should be equal to $2\pi i (f(a) + f(b))$ no? $\endgroup$ Commented Sep 28, 2017 at 6:59
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    $\begingroup$ $2\pi i\dfrac{f(b)-f(a)}{b-a}$ I think! $\endgroup$
    – Nosrati
    Commented Sep 28, 2017 at 7:15
  • $\begingroup$ That's correct. $\endgroup$ Commented Sep 28, 2017 at 7:50

1 Answer 1

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The point is that if $|f| \le B$, your integrand is bounded by $\dfrac{B}{(R-|a|)(R-|b|)}$ so the integral is bounded by $\dfrac{2\pi R B}{(R-|a|)(R-|b|)}$. What happens to this as $R \to \infty$? Compare to the result of the residue theorem.

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