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Absolute Convergence

  1. Jun 27, 2011 #1
    1. The problem statement, all variables and given/known data

    Let [tex]z,p,q \in \mathbb{C}[/tex] be complex parameters.

    Determine that the Gamma and Beta integrals:
    [tex]\displaystyle \Gamma (z) = \int_0^{\infty} t^{z-1} e^{-t}\;dt[/tex]
    [tex]\displaystyle B(p,q) = \int^1_0 t^{p-1} (1-t)^{q-1}\;dt[/tex]
    converge absolutely for [tex]\text{Re}(z)>0[/tex] and [tex]p,q>0[/tex] respectively and explain why they do.

    3. The attempt at a solution

    How do I show that they converge absolutely and why do they?
     
  2. jcsd
  3. Jun 27, 2011 #2

    lanedance

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    how about using a convergence technique such as a comparison method
     
  4. Jun 28, 2011 #3
    I'm aquainted with such techniques for series but not integrals...
     
  5. Jun 28, 2011 #4

    lanedance

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    well whats your definition of absolute convergence for an integral?
     
  6. Jun 28, 2011 #5
    [tex]\int_A f(x)\;dx[/tex] where f(x) is a real or complex-valued function, converges absolutely if [tex]\int_A |f(x)|\;dx<\infty[/tex] where [tex]A=[a,b][/tex] is a closed bounded interval.
     
    Last edited: Jun 28, 2011
  7. Jun 28, 2011 #6

    lanedance

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    Ok so for the first one, can you convince yourself that the integral over t from 1 to infinity converges?

    That leaves you with the portion from 0 to 1 to prove. For that, take the absolute value.

    The comparison test says
    [tex] |g(x)|>|f(x)| \ \forall x \in I[/tex]

    [tex] \implies \int_I|g(x)|>\int_I |f(x)| [/tex]

    hence if the integral over |g| converges, so does the integral over |f|

    you should be able to use this for both portions if need be
     
  8. Jun 28, 2011 #7

    lanedance

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    for the 2nd the issue is the possibility each blows up too quickly at the boundaries, so I would again separate into 2 and consider each endpoint separately
     
  9. Jun 28, 2011 #8

    lanedance

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