Quantum Field Theory: Evaluating Integrals on Page 27

[touqra]

I don't understand how Peskin & Schroeder can evaluate the integral on page 27 by having the real axis wrapping around branch cuts just like that. The picture of the contours are on page 28.

 

[mjsd]

I think what they have done is simply completed a loop (like a key-hole contour) but the arc/circular bit dies away as your variable go to infinity so effectively the flat/horizontal bit is same as the two vertical bits (by Cauchy theorem... as no poles inside loop)

describing the loop: first bit is the original bit the flat/horizontal (-R,+R) bit with R eventually taken to infinity, then to complete the loop you need to add a 1/4 of an arc going from +R to +iR, then comes down to avoid the branch cut, go around the pole and goes up again before arch back from +iR to -R.

 

[touqra]

I think what they have done is simply completed a loop (like a key-hole contour) but the arc/circular bit dies away as your variable go to infinity so effectively the flat/horizontal bit is same as the two vertical bits (by Cauchy theorem... as no poles inside loop)

describing the loop: first bit is the original bit the flat/horizontal (-R,+R) bit with R eventually taken to infinity, then to complete the loop you need to add a 1/4 of an arc going from +R to +iR, then comes down to avoid the branch cut, go around the pole and goes up again before arch back from +iR to -R.

Why would the arc or circular bit dies away as the variable goes infinity?

 

[mjsd]

I haven't check this particular example and see if it does goes away... but it usually does and that's why we close the contour in the first place...by the way, I did say "I think"...perhaps you can check that... to prove that you need to look at your integrand and see what happen when R becomes large (ie. when the integration variable expressed in polar form becomes large). Sometimes Jordon's lemma or ML-estimate maybe used to help.

 

[touqra]

I haven't check this particular example and see if it does goes away... but it usually does and that's why we close the contour in the first place...by the way, I did say "I think"...perhaps you can check that... to prove that you need to look at your integrand and see what happen when R becomes large (ie. when the integration variable expressed in polar form becomes large). Sometimes Jordon's lemma or ML-estimate maybe used to help.

I looked up on Jordan's lemma, and yeah the integrand of the semicircular path (excluding the real axis) tends to zero as R goes infinity.
OOOooo contour integrals are so interesting !
Thank you!

Is ML estimate maximum likelihood estimate? How can ML estimate be used here since it is about probability?

 

[mjsd]

when I said ML-estimate I mean the following:
Suppose C is a piecewise smooth curve. If $$h(z)$$ is continuous function on C then
$$\displaystyle{\left|\int_{C} h(z)\, dz\right| \leq \int_{C}|h(z)|\, |dz|}.$$
and if C has length L and $$|h(z)|\leq M$$ on C then
$$\displaystyle{\left|\int_{C} h(z)\, dz\right| \leq ML}$$