# Multivariate Taylor expansion or else a double integral identity

1. Jan 29, 2012

### tjackson3

1. The problem statement, all variables and given/known data

This is part of a larger problem, but in order to take what I believe is the first step, I need to take the Taylor series expansion of $f(x,y) = \cos\sqrt{x+y}$ about (x,y) = (0,0)

On the other hand, the purpose of doing this expansion is to find an asymptotic expression for the integral

$$\int_0^{\pi^2/2}\ ds\int_0^{\pi^2/2}\ e^{x\cos\sqrt{s+t}}\ dt$$

I vaguely remember there being an identity for when you had an integrand that you can transform $f(x,y) \rightarrow f(x+y)$. Possibly the domain had to be square, which it is here. Does anyone know what I'm talking about there?

Edit: This identity allows for reduction to a single integral

2. Relevant equations

3. The attempt at a solution

I think it'd just be $1 + (1/2)f_{xx}(0,0)x^2 + f_{xy}(0,0)xy + (1/2)f_{yy}(0,0)y^2.$ Would that be correct? The first partials are excluded since f has a maximum there

Last edited: Jan 29, 2012