Talk:Sine of Sum
What we need to do,on this and all similar pages, is to transclude into the parent page the real case and the complex case as separate entities, or make the complex case the main page with the real case as a subpage. Then the proofs which are real-only can be applied to the real case, while those which can be applied to both real and complex can be given a wider scope. Clearly every result needs to be investigated separately, especially when it comes to proving the complex form as a corollary or whatever of the real case.
Advise that we take as first principle the complex definitions of sine and cosine using the Euler formulae, and proving the complex results using those definitions. Then we have everything covered. --prime mover (talk) 19:43, 15 June 2024 (UTC)
- I thought this page was for $a,b\in\R$.
- The complex case is rather uninteresting and a corollary of this real case in view of Identity Theorem.--Usagiop (talk) 20:51, 15 June 2024 (UTC)
- First you have to prove that sine is analytic. --prime mover (talk) 20:53, 15 June 2024 (UTC)
- I think $\sin$ is analytic by definition. --Usagiop (talk) 20:57, 15 June 2024 (UTC)
- And for each fixed $b \in \C$, $a \mapsto \map \sin {a + b}$ is also analytic by definition.--Usagiop (talk) 21:00, 15 June 2024 (UTC)
- I think $\sin$ is analytic by definition. --Usagiop (talk) 20:57, 15 June 2024 (UTC)
- First you have to prove that sine is analytic. --prime mover (talk) 20:53, 15 June 2024 (UTC)
- All of which needs to be taken into account.
- I appreciate how much it bores you, so don't feel pressured into mending all this stuff. Yes it's horribly wrong and we need to fix it, but until either I get round to sorting it out, or someone volunteers to tighten it all up, so that we are bulletproof against snidey comments that we haven't made completely sure of our domain, it is going to have to remain unsorted-out.
- I will get to it in due course. It hasn't mattered much in the last 10 to 12 years or so, so I'm not sure it's going to increase in urgency.
- In the meantime, feel free to plaster disingenuous "explain" template queries over the place so we know exactly what we have to address to make this website tighter.
- Or feel free to do what needs to be done to do that tightening up. As you point out, it's important but it's dull. At least, one approach to it is dull. And so trivial it may not take much longer to implement than it does to invoke an "explain" to alert us to it. --prime mover (talk) 22:48, 15 June 2024 (UTC)