The discussion focuses on the integration steps required to resolve the Yukawa potential in quantum field theory. The process involves transforming to spherical coordinates, integrating with respect to the variable \( u = \cos \theta \), and manipulating exponential terms to simplify the integrals. Specifically, the third step requires changing the integration variable from \( k \) to \( -k \), which necessitates adjusting the differential accordingly. The final steps include factorizing the denominator and solving the complex integral.
PREREQUISITESPhysicists, mathematicians, and students studying quantum field theory, particularly those interested in advanced integration techniques and the Yukawa potential.
hello sir,ChrisVer said:1st step: go to "spherical coordinates" and integrating out the \phi... u = \cos \theta
2nd step: Integrating wrt to u, will give a difference of e^{i~something} - e^{-i~something} \propto \sin (something). In fact I wouldn't ever write it in terms of sin...
3rd step: uses that he has two integrals one with the exponential with + and the other with the exponential with - ... then changes the integration variable of the one from k to -k, and gets this result.
4th step: factorizes the denominator.
5th step and then final: solves the (complex) integral
If you want to see a step in more details, you can ask for a specific one.
Muh. Fauzi M. said:i can't get the 3rd step. is change the integration variable, change the value from k to -k in that term, including the dk -> d(-k)?