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I have a rather trivial question regarding the notation connected with simple asymptotic formulas. Let the solutions of an equation asymptotically be given by $$z_n=iy^2_n+O\,(1/y^2_n),\quad |n|\rightarrow\infty,$$ where $$y_n=(n+1)\pi,\quad n=0,1,2,\ldots$$ Suppose now that we would like to refer to the solution for some specific value of $n$ explicitly. How do we do this in a mathematically correct way? To clarify, say we are interested in the solution for $n=0$. Shall we write

  1. $\quad z_0=iy^2_0+O\,(1/y^2_0),\quad y_0=\pi$
  2. $\quad z_0=i\pi^2+O\,(1/\pi^2)$
  3. $\quad z_0=i\pi^2$

Which of these is common and correct? I would rule out 3. since it neglects the asymptotic notation completely. Many thanks!

Trevor3
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  • Asymptotic formulas are just that: asymptotic. You can’t refer to them meaningfully for $n=0$, as $0$ is quite a long way away from $\infty$ – FShrike Dec 28 '21 at 13:56
  • Thanks, I agree. The reason why I am asking is that I need to split the solutions and their indexing into two branches, one for $n=0$ and the other for $n=1,2,3,\ldots$. So I need to write down a formula corresponding to each of these cases. Maybe there is another was of doing this. – Trevor3 Dec 28 '21 at 14:27
  • If you are aware of an explicit formula for the $\mathcal{O}$ function, then you can use it. It is highly likely however that the $\mathcal{O}$ arose from some inequalities, and is not known precisely. In this case, the formula $z_n=\cdots$ is of no use to you for small $n$, and it is only useful to you for $n$ large. If you are trying to compute these solutions exactly, do not use this formula. If you can control the error, and know the $\mathcal{O}$ constant, you'll be ok. I'm really not sure what the question is – FShrike Dec 28 '21 at 14:30

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