How to integrate with branch cuts?

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    Branch Integration
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SUMMARY

This discussion focuses on the challenges of integrating functions with branch cuts, specifically the square root function and its implications for integration over the entire real line. The user highlights the complexity of defining branches for multivalued functions like \(\sqrt{x}\) when integrating from \(-\infty\) to \(+\infty\). Additionally, the integral involving Bessel functions, \(\int_0^{\infty } \frac{(\text{BesselJ}[0,\zeta a]-\text{BesselJ}[0,\zeta b])^2}{\zeta \sqrt{\zeta ^2-k^2}} \, d\zeta\), is presented as a specific case requiring further exploration of branch cuts.

PREREQUISITES
  • Understanding of complex analysis, particularly branch cuts and multivalued functions.
  • Familiarity with integration techniques involving improper integrals.
  • Knowledge of Bessel functions and their properties.
  • Experience with mathematical notation and integral calculus.
NEXT STEPS
  • Research the concept of branch cuts in complex analysis.
  • Study the properties and applications of Bessel functions, particularly \(\text{BesselJ}\).
  • Learn techniques for evaluating improper integrals involving multivalued functions.
  • Explore examples of integrating functions with branch cuts to understand practical applications.
USEFUL FOR

Mathematicians, physicists, and engineering students who are dealing with complex integrals, particularly those involving multivalued functions and Bessel functions.

krindik
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Hi,

I find integration with branch cuts difficult to grasp.
For example, I can understand that \sqrt{x} is mutivalued and has 2 branches if we take a branch cut from 0 to +infty. But given it to be integrated from -infty to +infty what is the the meaning of taking a branch of \sqrt{x} ?

Could u pls give me some pointers to look or guide me with this example?

I hope to find answers to integrals like \sqrt{(x-1){x-2}}
 
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I'm dealing with this integral:
\int_0^{\infty } \frac{(\text{BesselJ}[0,\zeta a]-\text{BesselJ}[0,\zeta b])^2}{\zeta \sqrt{\zeta ^2-k^2}} \, d\zeta
 

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