Multivariate Taylor expansion or else a double integral identity

[tjackson3]

Homework Statement

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 [itex]f(x,y) = \cos\sqrt{x+y}[/itex] about (x,y) = (0,0)

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

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

I vaguely remember there being an identity for when you had an integrand that you can transform [itex]f(x,y) \rightarrow f(x+y)[/itex]. 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

Homework Equations

The Attempt at a Solution

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

 

[lippyka]

.For the identity I'm thinking of, it may just be a generalization of the substitution rule for integrals. When you have something like \int_a^bf(x)dx, you can do a substitution u = x+c to get \int_{a+c}^{b+c}f(u-c)du. This is likely the same concept, but with integrals over two variables.