How to integrate with branch cuts?

In summary, the speaker is struggling with understanding integration with branch cuts, specifically with the example of \sqrt{x} being multivalued and having 2 branches when a branch cut is taken from 0 to +infty. They are seeking guidance and pointers for understanding this concept and hope to apply it to integrals involving \sqrt{(x-1)(x-2)}. The speaker also mentions dealing with a specific integral involving Bessel functions.
  • #1
krindik
65
1
Hi,

I find integration with branch cuts difficult to grasp.
For example, I can understand that [tex]\sqrt{x}[/tex] 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 [tex]\sqrt{x}[/tex] ?

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

I hope to find answers to integrals like [tex]\sqrt{(x-1){x-2}}[/tex]
 
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  • #2
Did you get un answer for your question?

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
 

FAQ: How to integrate with branch cuts?

1. What is integration with branch cuts?

Integration with branch cuts is a mathematical technique used to evaluate definite integrals that involve functions with branch points or discontinuities. These branch points or cuts can be thought of as "jumps" in the function and can make traditional integration methods difficult to use.

2. How is integration with branch cuts different from regular integration?

Integration with branch cuts involves taking into account the behavior of the function at its branch points or cuts, whereas regular integration does not. This means that the domain of integration may need to be split into multiple regions or the function may need to be transformed in order to properly evaluate the integral.

3. What are some common functions that require integration with branch cuts?

Functions that have logarithmic, trigonometric, or power functions with fractional powers are common examples that require integration with branch cuts. For example, the function f(z) = (1-z)^(-1/2) has a branch cut at z = 1 and thus requires integration with branch cuts.

4. How do you handle branch cuts when integrating complex functions?

The handling of branch cuts when integrating complex functions depends on the specific function and its branch points. In some cases, the branch points can be avoided by choosing a different contour for integration. In other cases, the function may need to be transformed or the domain of integration may need to be split into multiple regions.

5. What are some applications of integration with branch cuts?

Integration with branch cuts is used in many areas of mathematics and physics, such as in the evaluation of certain physical quantities in quantum mechanics and in the computation of complex functions in engineering. It is also used in the study of special functions, such as the elliptic functions and the hypergeometric functions.

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