Need help solving an equation involving the Gamma function.

[Reid]

Homework Statement

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?

Homework Equations

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

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!

 

[mighty2000]

Thank you for reaching out for help with your equations. The gamma function can definitely be tricky to deal with, but it is possible to find a solution to your problem. Here are some suggestions for how to approach it:

1. Use numerical methods: One way to solve for the variables \beta_{1}, \beta_{2}, and \eta is to use numerical methods such as Newton's method or the bisection method. These methods involve iteratively guessing values for the variables and using the equations to refine those guesses until a solution is found. This approach may be easier to implement than trying to find a closed-form solution.

2. Simplify the equations: You may be able to simplify the equations by using properties of the gamma function. For example, you could try using the identity \Gamma(x+1) = x\Gamma(x) to simplify the expressions involving the gamma function.

3. Use approximations: Depending on the values of the variables and the accuracy you need, it may be possible to use approximations to find a solution. For example, if the values of \beta_{1} and \beta_{2} are close to 1, you could use the approximation \Gamma(1+x) \approx 1+x to simplify the equations.

4. Seek help from a mathematician: If you are still having trouble finding a solution, it may be helpful to seek assistance from a mathematician who has experience working with the gamma function and solving equations involving it. They may be able to provide insights or techniques that you haven't considered.

I hope these suggestions are helpful and that you are able to find a solution to your equations. Good luck with your work!