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I realize this is technically strictly a Math problem, but as it comes up in QFT I was thinking someone here might be more familiar with the theorem. I am trying to prove:

[tex]\lim_{x \to \infty}f(x) + \lim_{x \to -\infty}f(x) = \lim_{\epsilon \to 0^+} \epsilon \int_{-\infty}^{\infty} dx f(x) e^{-\epsilon |x|}[/tex]

It crops up in Weinberg's QFT book (Vol. I) as he explains how to derive the S matrix using path integrals. f(x) is a wavefunction and for the sake of argument I am making two assumptions:

1) [tex]\lim_{x \to \pm \infty}f(x)[/tex] both exist and

2) f(x) is a [tex]C^{\infty}[/tex] function.

(The text actually only requires f(x) to be "sufficiently smooth." I'll happily accept a proof with a [tex]C^{\infty}[/tex] function and worry about other details later.)

The specific text reference is "The Quantum Theory of Fields," Vol 1., by Weinberg. It is equation 9.2.15 on page 388.

Thanks and Merry Christmas everyone!

-Dan

Edit: I find it amusing that, though the Math problem is possibly at undergraduate level that someone thought a Physics reference to Quantum Field Theory would be at an undergraduate level and thus moved this post here.

[tex]\lim_{x \to \infty}f(x) + \lim_{x \to -\infty}f(x) = \lim_{\epsilon \to 0^+} \epsilon \int_{-\infty}^{\infty} dx f(x) e^{-\epsilon |x|}[/tex]

It crops up in Weinberg's QFT book (Vol. I) as he explains how to derive the S matrix using path integrals. f(x) is a wavefunction and for the sake of argument I am making two assumptions:

1) [tex]\lim_{x \to \pm \infty}f(x)[/tex] both exist and

2) f(x) is a [tex]C^{\infty}[/tex] function.

(The text actually only requires f(x) to be "sufficiently smooth." I'll happily accept a proof with a [tex]C^{\infty}[/tex] function and worry about other details later.)

The specific text reference is "The Quantum Theory of Fields," Vol 1., by Weinberg. It is equation 9.2.15 on page 388.

Thanks and Merry Christmas everyone!

-Dan

Edit: I find it amusing that, though the Math problem is possibly at undergraduate level that someone thought a Physics reference to Quantum Field Theory would be at an undergraduate level and thus moved this post here.

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