(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine the first three terms of the Laurent expansion in z of

[itex]f(z)=\int_0^1 dx_1..dx_4 \frac{\delta(1-x_1-x_2-x_3-x_4)}{(x_1 x_2 a + x_3 x_4 b)^{2+z}},\quad a,b>0[/itex]

2. The attempt at a solution

I tried expanding around z = -2.

[itex]f(z)=\sum_{n=-\infty}^\infty a_n (z-(-2))^n[/itex]

For the a_0 term this is easy:

[itex]a_0 = \frac{1}{2\pi i} \oint_\gamma \int_0^1 \frac{dx_1 .. dx_4 dz}{C(\lbrace x_i \rbrace)^{z+2}(z+2)}, \quad C=(x_1 x_2 a + x_3 x_4 b)[/itex]

[itex]= \frac{1}{2\pi i} \int_0^1 .. \oint_\gamma \frac{du}{C^u u} = 1[/itex]

However for the a_1 term I get this:

[itex]a_1 = -\int_0^1 \delta(1-x_1-x_2-x_3-x_4) \ln(x_1 x_2 a + x_3 x_4 b) dx_1 .. dx_4[/itex]

and for a_2 the same with -ln(.) -> (1/2) ln^2(.). These are horrible integrals! Am I doing something wrong or is there any trick or substitution Im missing?

I'm thankful for any suggestion!

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# Homework Help: Laurent series of an integral

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