# From log of a product function to gamma function

• May 10th 2013, 01:49 AM
Katmarn
From log of a product function to gamma function
Hi everyone

I have this expression: Attachment 28318

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: Attachment 28319

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! (Clapping)
• May 10th 2013, 03:24 AM
Katmarn
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
• May 10th 2013, 06:29 AM
Katmarn
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
• May 10th 2013, 03:42 PM
Prove It
Re: From log of a product function to gamma function
Quote:

Originally Posted by Katmarn
Hi everyone

I have this expression: Attachment 28318

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: Attachment 28319

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! (Clapping)

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?
• May 11th 2013, 04:19 AM
Katmarn
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 :(