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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
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