Solving Integrals by Series Expansions

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Discussion Overview

The discussion revolves around the challenges of solving an integral involving the generalized MarcumQ function and a probability density function (PDF) in the context of a PhD research project. Participants explore the implications of using series expansions and the conditions under which convergence can be guaranteed.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant describes the need to solve an integral of the generalized MarcumQ function multiplied by a PDF, noting that while the numerical solution yields a bounded result, the series expansion leads to divergence.
  • Another participant expresses confusion regarding the terms "convergent" and "divergent" in the context of the participant's statements, suggesting a need for clarification.
  • A participant reiterates the initial concern about the potential divergence when swapping the order of integration and summation, indicating that this could be a source of the problem.
  • A further elaboration on the issue includes the need to average a function expandable in power series, questioning the mathematical validity of the resulting series after multiplication with the PDF.
  • One participant questions whether the series for the function F(a) is uniformly convergent for all values of the parameter, suggesting this could be problematic.
  • Several participants inquire about methods to test for uniform convergence, with references to the δ-ε method and its requirements.
  • There is a discussion about the implications of uniform convergence on the expectation of the function, with questions about the relationship between f(x) and Un(x), and whether x is a random variable.
  • Concerns are raised about whether the convergence of the original series will be affected by multiplication with the PDF and whether the integral will remain convergent.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding and concern regarding the convergence of series and integrals, with no clear consensus on the resolution of the issues raised. Multiple competing views on the conditions for convergence remain present throughout the discussion.

Contextual Notes

Participants mention the potential need to double-check calculations and the conditions required for swapping integration and summation, indicating that assumptions about convergence may not hold in all cases.

yfatehi
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In my PhD I need to solve an integral of the generalized MarcumQ function multiplied by a certain probability density function to get the overall event probability. Numerically solution produces bound result as it represents a probability but when trying to use a convergent power expansion of the MarcumQ function the result is divergent.

Is this logic?
 
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Your statement is confusing. What is convergent and what is divergent (your statement sounds like you saying the same thing is both)?

You might supply some details.
 
yfatehi said:
In my PhD I need to solve an integral of the generalized MarcumQ function multiplied by a certain probability density function to get the overall event probability. Numerically solution produces bound result as it represents a probability but when trying to use a convergent power expansion of the MarcumQ function the result is divergent.

Is this logic?

You are not guaranteed to get a convergent series if you swap the order of an integral and an infinite sum, even if the original series was convergent. This could be what is happening, but it could also be that you may just need to double-check your work.
 
Form more clarity
I have a function F(a) expandable in power series. The parameter a is random variable and I must average by multiplying F(a) with the PDF of this variable and integrate from 0 to infinity. Although the PDF function is bound and the expansion of F(a) in series is convrgent, the resulting multiplication and integration is divergent series. This is my issue.
First is it mathematically possible or sure I made some mistake although i made the integration using mathematica
 
Is the series for F(a) uniformly convergent for all values of the parameter? If not that could be a problem.
 
How to test the uniform convergence
 
yfatehi said:
How to test the uniform convergence
If you use the basic δ ε method, then the choice has to be independent of the parameter.
 
If the initial series is uniform convergent in the region from 0 to inf. what is the status of the Expectation? ie E{f(x)}= Sum ( E{Un(x)} )
 
yfatehi said:
If the initial series is uniform convergent in the region from 0 to inf. what is the status of the Expectation? ie E{f(x)}= Sum ( E{Un(x)} )
What is the connection between f(x) and Un(x)? Also is x a random variable?
 
  • #10
f(x)= Sum ( Un(x) ), n from 0 to infinity , x is random positive variable We need to get the average E{f(x)} over x so we first multiply with the Prob. Density function PDF(x) then integrate fom 0 to inf

first if the original series Un(x) is uniform convergent over x from 0 to inf, will its convergence be affected by mutiplication with PDF(x) of any shape?

Second if not will the integral after multiplication be convergent?
 
  • #11

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