Parameter Integration of Bubble Integral

In summary, the conversation discusses the use of Mathematica's Integrate command to solve a loop integral, which does not give the expected result. The integrand is known to diverge for certain values of the Feynman parameter λ, so the method used involves finding the roots of the expression in the denominator. It is suggested that a deformation of a complex contour may be used to integrate around the branch cut of the logarithm. The solution eventually involves factoring the quadratic argument of the logarithm and integrating by parts.
  • #1
Elmo
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TL;DR Summary
Cant figure out how this PV bubble integral has been solved.
Referring to this link : https://qcdloop.fnal.gov/bubg.pdf
Using Mathematica Integrate command to solve it does not give the result stated here but I am unclear as to how they got to the result in the 4th line.
It is clear that the integrand (1st line) can diverge for certain values of the Feynman parameter λ and this is presumably why they find the roots of the expression in the denominator. I just dont know what they did to solve this loop integral and express the result in terms of the roots.
 
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  • #2
Maybe they used some deformation of a complex contour for the ##\lambda## integral integrating somehow around the branch cut of the ln?
 
  • #3
Just factor the quadratic argument of the logarithm as ##(\lambda-\lambda_1)(\lambda - \lambda_2)##, expand the logarithm into ##\log(\lambda - \lambda_1)+\log(\lambda - \lambda_2)## (modulo constant prefactors) and integrate by parts.
 

1. What is parameter integration of bubble integral?

Parameter integration of bubble integral is a mathematical technique used in fluid dynamics to calculate the volume of a bubble in a fluid. It involves integrating the parameters of the bubble, such as its radius and surface tension, over the entire volume of the bubble.

2. How is parameter integration of bubble integral used in research?

Parameter integration of bubble integral is commonly used in research to study the behavior of bubbles in different fluids. It can be used to determine the stability, growth, and collapse of bubbles, as well as their effect on fluid flow and heat transfer.

3. What are the limitations of parameter integration of bubble integral?

One limitation of parameter integration of bubble integral is that it assumes the bubble is spherical in shape and does not account for irregularities or deformations in the bubble's shape. Additionally, it does not take into account the effects of external forces on the bubble, such as gravity or electric fields.

4. How is parameter integration of bubble integral different from other methods of bubble analysis?

Parameter integration of bubble integral is a more accurate and comprehensive method compared to other techniques, such as the Rayleigh-Plesset equation, which only considers the bubble's radius and pressure. Parameter integration takes into account multiple parameters and can provide a more detailed understanding of bubble behavior.

5. Can parameter integration of bubble integral be applied to other shapes besides spheres?

While parameter integration of bubble integral is commonly used for spherical bubbles, it can also be adapted for other shapes, such as ellipsoids or cylinders. However, the calculations become more complex and may require additional assumptions or approximations.

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