# Get dominant terms in asymptotic expansion

I'd like to find the dominant terms of an expression given the conditions $x,y >> 1$:

$c_0 y+\frac{y^2 \left(4 c_0^2 y - 12 c_1 c_2^2+y^2\right)}{24 x^2 \left(\text{c1}^2+\text{c2}^2+y\right)}$

Is there an easy way to do this? I need the dominant terms for a given order O(x,y). It would be helpful to get the "small" contribution as a separate output. Thanks.

• You can try Series[(*expression here*), {x,Infinity,0},{y,Infinity,0}] – QuantumDot Oct 5 '15 at 19:05
• thanks! made an answer for future reference. – anon01 Oct 5 '15 at 19:21
• You really require more information regarding relative sizes of x vs. y. – Daniel Lichtblau Oct 5 '15 at 20:30
• ? I'm interested in your thoughts if you wish to clarify. I consider them to increase parametrically on the same scale. – anon01 Oct 5 '15 at 21:10
• If they vary jointly e.g. y=k*x (or approximately equals) for some unspecified constant k!=0 then you can just replace y with k*x and take the Series. If they vary in an unspecified way e.g. y could be on the order of x^2 or x^(1/2) or Exp[x] then your asymptotic behavior will behave quite differently in different regions of the parametrized space. – Daniel Lichtblau Oct 6 '15 at 14:25

A straightforward example for some function f = f(x,y):
Series[f/.{x->q,y->q},{q,Infinity,2}]

Of course this requires that both variables grow parametrically at the same rate. This also allows you to take different limits: x->1/q for example if one of your variables limits toward zero. A thanks to QuantumDot.