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Numerical integration methods applicable to a type of definite integral

  1. May 29, 2014 #1
    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 ?
     
  2. jcsd
  3. May 30, 2014 #2

    jasonRF

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    Science Advisor
    Gold Member

    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 [itex]f(x)\approx x^a[/itex] for [itex]a>2.7[/itex] for the integral to exist. That is why generalized Gauss-Laguerre has the n>-1 condition. Conditions as [itex]x\rightarrow \infty[/itex] must also be met of course. The exponential in Gauss-Laguerre also helps with this.

    jason
     
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