Talk:Jordan's Lemma

I'm a bit puzzled by this.

If we can prove this:

$\ds \size {\int_{C_r} \map f z \rd z} \le \frac \pi {2 a} \paren {\max_{0 \mathop \le \theta \mathop \le \pi} \size {\map g {r e^{i \theta} } } }$

why do we then weaken it to this:

$\ds \size {\int_{C_r} \map f z \rd z} \le \frac \pi a \paren {\max_{0 \mathop \le \theta \mathop \le \pi} \size {\map g {r e^{i \theta} } } }$

?

Is there something I'm missing? --prime mover (talk) 16:25, 28 May 2021 (UTC)

There may well be a missing factor of $2$ somewhere. I wrote this quite a while ago so I'll try to work it through again. Caliburn (talk) 17:11, 28 May 2021 (UTC)

I believe I found my mistake. That estimate of the sine function only holds on $\closedint 0 {\pi/2}$, so you have to do some playing with symmetry first. That estimate probably warrants a page in itself. I'll try to see to this in due time. Caliburn (talk) 17:13, 28 May 2021 (UTC)

Cleared it up, let me know if there's any more errors. Caliburn (talk) 14:14, 31 May 2021 (UTC)