I have a complicated function which I intend to predict its asymptotic behavior (specially I have a $f(r)$ which I want to know its asymptotic behavior in $r\rightarrow \infty$). I know that the command Limit
gives me the numerical limit of the function in the special limits instead, I need a function which simulate the behavior of the original function in the limits in the best possible way. Let me get help from a toy example:
We know the limit of Sin[x]
for x->0
is 0 namely
Limit[Sin[x], x -> 0]
(*0*)
but we know it behaves just as x
function in small x
s. The following plot shows this fact:
Plot[{Sin[x], x}, {x, 0, 4}, PlotLegends -> "Expressions"]
So we say x
function simulates the asymptotic behavior of Sin[x]
. I need to find such a thing for my complicated function. Note that I don't want use fitting or something else, I just want to reach such a function by manipulating (for example choosing terms whit high growth rate and removing other terms) my original function (what was not possible for simple Sin[x]
function). Is there any command in Mathematica to do this?
Addendum
This is the closed form of my function:
f[re_] :=
1/(8 Sqrt[2]
mp re \[Gamma] (M^2 \[Beta] + mp^2 \[Gamma]) Sqrt[\[Beta] + (
mp^2 \[Gamma])/M^2]) E^(-2 re^2 \[Gamma]) M^2 norm^2 \[Pi]^(
3/2) (E^((M \[Alpha] - 2 mp re \[Gamma])^2/(
2 (M^2 \[Beta] +
mp^2 \[Gamma]))) ((-1 + E^((4 M mp re \[Alpha] \[Gamma])/(
M^2 \[Beta] + mp^2 \[Gamma]))) M \[Alpha] +
2 (1 + E^((4 M mp re \[Alpha] \[Gamma])/(
M^2 \[Beta] + mp^2 \[Gamma]))) mp re \[Gamma]) +
E^((M \[Alpha] - 2 mp re \[Gamma])^2/(
2 (M^2 \[Beta] + mp^2 \[Gamma]))) (M \[Alpha] -
2 mp re \[Gamma]) Erf[(M \[Alpha] - 2 mp re \[Gamma])/(
Sqrt[2] M Sqrt[\[Beta] + (mp^2 \[Gamma])/M^2])] -
E^((M \[Alpha] + 2 mp re \[Gamma])^2/(
2 (M^2 \[Beta] + mp^2 \[Gamma]))) (M \[Alpha] +
2 mp re \[Gamma]) Erf[(M \[Alpha] + 2 mp re \[Gamma])/(
Sqrt[2] M Sqrt[\[Beta] + (mp^2 \[Gamma])/M^2])]);
where all parameters are real and positive. I want to find asymptotic behavior of f[re]
when re
goes to infinity.
Asymptotic
. Use it asAsymptotic[f, r -> Infinity]
. $\endgroup$Asymptotic[Sin[x], x -> 0]
evaluates tox
;Asymptotic[Sin[x], x -> 0, SeriesTermGoal -> 3]
evaluates tox - x^3/6
andAsymptotic[Sin[x], x -> Infinity]
evaluates toSin[x]
$\endgroup$