Oh I see. So it is really like the analytical continuation of the sinc function that you took as an example. Thanks for answering!
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Oh I see. So it is really like the analytical continuation of the sinc function that you took as an example. Thanks for answering!
An example of these class of problems would be:
under what conditions on the parameter a do there exist solutions to
dx/dt= ax/t + bt such that \lim_{t \to 0} x(t)/t^2 exists (where b is a free parameter, which here acts like a perturbation term)
with some straightforward calculus you can show that if a \neq 2 the desired solution exists.
The b-term acts like a perturbation term to the ax/t part so the more general version of the above you would replace the b-term by some very general perturbation.
Ok I (think I) see.
However, in your example, if b=0 then the
a \neq 2condition is not necessary, is it? In this maybe pathological case, the solution is trivially proportional to ta. In this sense, the b-term has in my taste deeper implications than just being a perturbation.Am I missing something trivial or am I putting too much emphasis on an irrelevant configuration?
For b=0 you have a zero solution. That's correct. Casting it in the more general framework you want to find something that holds for all b. So b defines her a space of functions.
More general you would look at
dx/dt= ax/t + g(x,t)
with g in a specific space. So you would want to find results for all g in that specific space.
These problems are motivated by classical problems of the type:
dx/dt = Ax + f(x)
with f(x) in O(x^2)
As always, I focus too much on the pathological cases. Thanks for the clarifications ;)