Complex integral:
Let $\gamma: [\pi/2, \pi] \rightarrow \mathbb{C}$ be given by $\gamma(t)=3e^{it}, \pi/2 \le t \le \pi$. Show that $$\left|\int_{\gamma} \frac{dz}{|z^2-1|}\right|\le \frac{3\pi}{16}$$

Question

Complex integral:
Let $\gamma: [\pi/2, \pi] \rightarrow \mathbb{C}$ be given by $\gamma(t)=3e^{it}, \pi/2 \le t \le \pi$. Show that $$\left|\int_{\gamma} \frac{dz}{|z^2-1|}\right|\le \frac{3\pi}{16}$$

raffle_snaffle · Answer

@zzr0ck3r

anonymous · Answer

which complex integration technique are you trying to use?

kirbykirby · Answer

I think I somehow need to use the ML formula (Max * length) ?

kirbykirby · Answer

which provides an upper bound .

anonymous · Answer

That's a pretty difficult integral, I'm not gonna lie. @mathmale

mathmale · Answer

Looks like this is from Complex Variables.  I last studied that in 1969 and remember it all like it happened yesterday.  ;)

kirbykirby · Answer

Yes it is in complex analysis. Wow seriously? How can you remember that from so long :O

kirbykirby · Answer

Do you also remember how to do more complicated integrals, such as those involving "keyhole" contours? @mathmale Maybe you could help me on a second question after (if you can/have time)?

fibonaccichick666 · Answer

isn't that an application of Cauchy integral thm?

fibonaccichick666 · Answer

then put a circle around the singularity and let the radius go to zero?

kirbykirby · Answer

Oh gosh I finally figured it out >_> Didn't actually need to compute the whole integral.

fibonaccichick666 · Answer

yay!