1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Multivariate Taylor expansion or else a double integral identity

  1. Jan 29, 2012 #1
    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 [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

    2. Relevant equations

    3. 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
    Last edited: Jan 29, 2012
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted