I recently came across something interesting while reading Mary L Boas text on Mathematical Methods.
Has it ever bothered you, why the power series of a pretty function like 
converges only for 
, even though there's nothing strange which happens at 
? For example, if you take 
, it's easy to see that there's a singularity at 
. But what's the catch here? What's wrong with the power series of
at ? This apparent conundrum is actually trivial when one looks at it from the complex perspective. Consider the complex function,
Since 
is infinite when 
, isn't analytic in any region containing . Now, there's a theorem that says that, since is analytic at points other than 
, it converges inside the circle centered at the origin and extending to the singularity at . From this, one can clearly see that when 
, and the corresponding power series also converges only when 
.
The lesson here is that looking at things from a complex perspective might give important insights in the real case.
3 comments:
Hold yer horses! What do you mean 1/(1+x^2) 'converges'? That it has a limit as x -> 1 ? But _of course_ it has a limit, and it is 1/2! Why should you even think of the complex plane? I do not also understand what is the problem with it as x -> 1, why should you _suspect_ anything wrong at all, in the first place?
That, or I need more sleep :-)
Ok, I meant the power series of 1/(1+x^2). I should be more careful with my wording. I've edited it now.
Ah, peace :-)
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