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Defining $\sin$ by means of the exponential function or by power series, it is straightforward to show that $\sin$ satisfies: $$y''=-y, \ \ \ \ y(0)=0, \ \ \ \ y'(0)=1.$$ There is an elementary proof that there is a unique solution to the above on the reals. I wonder if the argument can be extended to, or if there is another elementary proof that shows there is a unique solution in the complex plane.

David Gao
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Sam
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  • Is permanence of analytic relations "elementary"...? – Andrew D. Hwang Mar 04 '24 at 15:01
  • Which part of that proof are you afraid doesn't apply to $\Bbb C$? – Arthur Mar 04 '24 at 15:02
  • @Arthur that proof doesn’t apply word for word. For instance, there it is used that if a sum of squares vanishes then each term vanishes. Of course this is modifiable. – peek-a-boo Mar 04 '24 at 15:09
  • actually scratch that, the above Energy-proof doesn’t seem easily modifiable. I’d resort to uniqueness of analytic continuation. Or just directly use the equation to recursively figure out $y^{(n)}(0)$ for all $n\geq 0$, and then check the resulting power series converges, and so we’re done. – peek-a-boo Mar 04 '24 at 15:34
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    what does elementary mean in this context - since you extend an ODE to $\mathbb C$ you automatically have to assume the general theory of differentiability of complex functions for the question to make sense, so in particular power series etc; in other words when you talk about complex analysis, power series are as elementary as it gets which is definitely not the case for real analysis, where real analyticity is quite a subtle phenomenon – Conrad Mar 04 '24 at 16:07
  • @peek-a-boo I addressed my comment specifically to the OP, in the hopes that they would have a closer look and give us some insight into their thoughts on the problem. – Arthur Mar 04 '24 at 17:39
  • @Arthur precisely the part that peek-a-boo mentioned. – Sam Mar 05 '24 at 11:12
  • @AndrewD.Hwang yes (: – Sam Mar 05 '24 at 11:13
  • @peek-a-boo could you elaborate on how to "recursively figure out $y^{(n)}(0)$ for all $n\ge 0$, and then check the resulting power series converges"? – Sam Mar 05 '24 at 11:14
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    Look at your ODE. If you know $y(0)$, then you automatically know $y’’(0)$, which then automatically tells you $y^{(4)}(0)$, and so on. Since you know both $y(0),y’(0)$, you can keep jumping in steps of two to figure out all the derivatives at the origin. – peek-a-boo Mar 05 '24 at 13:47

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