# Math Help - From log of a product function to gamma function

1. ## From log of a product function to gamma function

Hi everyone

I have this expression:

And I would very much like to express it in an alternative way without the product function (or a sum function for that matter).

Wolframalpha suggest this as an alternative form:

I have tried number examples, and they come up with the same results, so I guess they

Assuming it is correct, I really need a step-by-step derivation of why the two expressions are identical. As a note I can add that I have the following restrictions on the parameters:

0<b<1, T>t>=0

I am aware that the gamma function works as an intrapolation of the faculty function, but not much more...

Here you can see it in wolframalpha: log[ product (1+b*(T-s)), s=0 to t] - Wolfram|Alpha

I hope someone has the time and skill to help me!

2. ## Re: From log of a product function to gamma function

Just found out that it does not work for certain values of b because the gamma function is not defined for negative integers... Hmmm

3. ## Re: From log of a product function to gamma function

By the way, I am using the expression for an estimation. As I without problems can restrict b to 0.5<b<1 it is no problem that the gamme function is not defined for negative integers (and 0). Help as to how wolfram comes up with the beautiful expression is therefore still very much appreciated!

Thank you

4. ## Re: From log of a product function to gamma function

Originally Posted by Katmarn
Hi everyone

I have this expression:

And I would very much like to express it in an alternative way without the product function (or a sum function for that matter).

Wolframalpha suggest this as an alternative form:

I have tried number examples, and they come up with the same results, so I guess they

Assuming it is correct, I really need a step-by-step derivation of why the two expressions are identical. As a note I can add that I have the following restrictions on the parameters:

0<b<1, T>t>=0

I am aware that the gamma function works as an intrapolation of the faculty function, but not much more...

Here you can see it in wolframalpha: log[ product (1+b*(T-s)), s=0 to t] - Wolfram|Alpha

I hope someone has the time and skill to help me!
Well I would start by recalling that \displaystyle \begin{align*} \log{( a \cdot b )} = \log{(a)} + \log{(b)} \end{align*}. Can you do anything with that?

5. ## Re: From log of a product function to gamma function

I am aware of that, and have tried different things, but I still have not been able to show that they are identical