Homework Help: Need help solving an equation involving the Gamma function.

1. May 5, 2010

Reid

1. The problem statement, all variables and given/known data

I am to solve the below stated equations for the variables $$\beta_{1}$$, $$\beta_{2}$$, and $$\eta$$, 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

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

3. The attempt at a solution

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

$$\frac{\beta_{1}\Gamma \left( 1+\frac{1}{\beta_{1}}\right)}{\lambda_{1}} \left( \frac{t \Gamma \left( 1+ \frac{1}{\beta_{1}} \right)}{\lambda_{1}}\right)^{\beta_{1}-1} exp \left( \left(-\frac{t \Gamma \left( 1+\frac{1}{\beta_{1}}\right}{\lambda_{1}} \right)^{\beta_{1}} \right)$$ $$= \frac{\beta_{2}\Gamma \left( 1+\frac{1}{\beta_{2}}\right)}{\lambda_{2}} \left( \frac{t \Gamma \left( 1+ \frac{1}{\beta_{1}} \right)}{\lambda_{2}}\right)^{\beta_{2}-1} exp \left( \left(-\frac{t \Gamma \left( 1+\frac{1}{\beta_{2}}\right}{\lambda_{2}} \right)^{\beta_{2}} \right)$$

Would be so happy for any help!