Numerical integration methods applicable to a type of definite integral

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SUMMARY

The discussion focuses on the application of numerical integration methods for the definite integral of the form ∫xnf(x) dx from 0 to infinity, particularly when n can take negative non-integer values such as -3.7. The Gauss-Laguerre quadrature method is considered, but its applicability is limited by the condition that n must be greater than -1 due to singularities at x=0. To ensure convergence, the function f(x) must behave like f(x) ≈ xa for a > 2.7 when n = -3.7. The discussion emphasizes the importance of meeting convergence conditions as x approaches infinity.

PREREQUISITES
  • Understanding of numerical integration techniques, specifically Gauss-Laguerre quadrature.
  • Familiarity with definite integrals and their convergence criteria.
  • Knowledge of singularities in mathematical functions.
  • Basic concepts of asymptotic behavior of functions as x approaches infinity.
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  • Research the specifics of the Gauss-Laguerre quadrature method and its implementation.
  • Study convergence criteria for integrals involving singularities.
  • Explore alternative numerical integration methods suitable for handling singularities.
  • Investigate the behavior of functions near singular points and their implications for integration.
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Mathematicians, computational scientists, and software developers working on numerical integration problems, particularly those dealing with definite integrals that include singularities and non-integer powers.

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Numerical integration methods applicable to a type of definite integrl

Hey, so I've been working on a program to numerically integrate an integral of the form

∫xnf(x) dx, LIM(0 to INF.)

Here n can go to negative non integral values, say -3.7 etc. and f(x)
is a function of sin, cos and x's.
I want to know which numerical integration method I should be using for this
type of definite integral. I was looking at Gauss-Laguerre quadrature method, but I don't
know it it will be applicable, given the constraint on n to be > -1

(http://en.wikipedia.org/wiki/Gauss–Laguerre_quadrature#Generalized_Gauss.E2.80.93Laguerre_quadrature)

Also, there is a singularity at x=0, which will affect it.
Can anyone given any tips on how to handle this ?
 
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Note that if the integral doesn't converge then no numerical algorithm will help you. If n=-3.7, then near zero your f needs to look like f(x)\approx x^a for a>2.7 for the integral to exist. That is why generalized Gauss-Laguerre has the n>-1 condition. Conditions as x\rightarrow \infty must also be met of course. The exponential in Gauss-Laguerre also helps with this.

jason
 

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