I'm trying to show equation (6.39) on page 189 in Peskin & Schroeder.(adsbygoogle = window.adsbygoogle || []).push({});

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

Show that [tex]\frac{1}{AB}=\int_0^1dxdy\delta(x+y-1)\frac{1}{(xA+yB)^2}[/tex].

2. Relevant equations

Defining property of Dirac delta function:

[tex]\int_{-\infty}^{\infty}dx f(x)\delta(x-x_0)=f(x_0)[/tex]

Also:

[tex]\int_{x_0-\varepsilon}^{x_0+\varepsilon}dx f(x)\delta(x-x_0)=f(x_0)[/tex]

3. The attempt at a solution

Since the defining properties of the delta function requires the limits of integration to be outside the points where the argument of the delta function is zero, I cannot simply plug my expression into any of these relations. If I instead choose the limits of integration [tex]1+\varepsilon[/tex] and [tex]0-\varepsilon[/tex] I can use the second

delta function identity from above to "force" x=1-y as I integrate over dx. So

[tex]\int_{0+\varepsilon}^{1+\varepsilon}dxdy\delta(x+y-1)\frac{1}{(xA+yB)^2}

=\int_{1+\varepsilon}^{1+\varepsilon}dy\frac{1}{[(1-y)A+yB]^2}[/tex]. After a few routine steps

the expression I get goes to [tex]\frac{1}{AB}[/tex] in the limit [tex]\varepsilon\rightarrow 0[/tex], which is the result I want for the integral without the epsilons.

Is this calculation relevant in obtaining the value of [tex]\int_0^1dx (...)[/tex] and how are the two integrals related? I'm getting very confused here.

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# Homework Help: Feynman parameter integral

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