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Need help solving an equation involving the Gamma function.

  1. May 5, 2010 #1
    1. The problem statement, all variables and given/known data

    I am to solve the below stated equations for the variables [tex]\beta_{1}[/tex], [tex]\beta_{2}[/tex], and [tex]\eta[/tex], the rest are known given a positive noninteger value. But the problem is that I don't know how to deal with the gamma function as can be seen in my attempt. Should I use some approximation? Or is it possible to find a solution?

    2. Relevant equations

    [tex]\lambda_{1} = \eta \Gamma\left(1+\frac{1}{\beta_{1}}\right)[/tex]
    [tex]\lambda_{2} = \eta \Gamma\left(1+\frac{1}{\beta_{2}}\right)[/tex]
    [tex]\frac{\beta_{1}}{\eta}\left(\frac{t}{\eta}\right)^{\beta_{1}-1} exp\left(\left(-\frac{t}{\eta}\right)^{\beta_{1}}\right) = \frac{\beta_{2}}{\eta}\left(\frac{t}{\eta}\right)^{\beta_{2}-1} exp\left(\left(-\frac{t}{\eta}\right)^{\beta_{2}}\right) [/tex]

    3. The attempt at a solution

    I use [tex] \eta = \frac{\lambda_{1}}{\Gamma \left( 1+\frac{1}{\beta_{1}}\right)} [/tex] and [tex] \eta = \frac{\lambda_{2}}{\Gamma \left( 1+\frac{1}{\beta_{2}}\right)} [/tex] to insert in the last equation in secton 2 to get

    [tex] \frac{\beta_{1}\Gamma \left( 1+\frac{1}{\beta_{1}}\right)}{\lambda_{1}} \left( \frac{t \Gamma \left( 1+ \frac{1}{\beta_{1}} \right)}{\lambda_{1}}\right)^{\beta_{1}-1} exp \left( \left(-\frac{t \Gamma \left( 1+\frac{1}{\beta_{1}}\right}{\lambda_{1}} \right)^{\beta_{1}} \right) [/tex] [tex] = \frac{\beta_{2}\Gamma \left( 1+\frac{1}{\beta_{2}}\right)}{\lambda_{2}} \left( \frac{t \Gamma \left( 1+ \frac{1}{\beta_{1}} \right)}{\lambda_{2}}\right)^{\beta_{2}-1} exp \left( \left(-\frac{t \Gamma \left( 1+\frac{1}{\beta_{2}}\right}{\lambda_{2}} \right)^{\beta_{2}} \right) [/tex]

    Would be so happy for any help!
     
  2. jcsd
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