# Question about the Borwein fast algorithm for certain values of Gamma

#### mesa

I am reading through Borwein's paper, "Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind" and have a question.

If we look at his algorithm's we see they are of this general form:

Gamma(1/2)=2^(-1/4)AG[1]{2^(-1/2)∑[1]}^(-1/2)

I have been able to run through everything up until the curly brackets, {}. Is this a multiplier? Or is it something else entirely?

#### Greg Bernhardt

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

#### mesa

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
It's the first time this has happened using PF so can't really complain.

Dr. Borwein figured out a way to use the arithmetic geometric mean (which he refers to simply as AG method) to take the difference of the squares of an and bn in a summation to get some pretty astounding decimal approximations for certain values of the gamma function.

The primary issue is it seems to be extremely limited in the number of values that can be used to calculate gammas and with the exception of some extraordinary mathematical gymnastics the number of values that can be used is countable.

Regardless, it is a wonderful piece of work!

#### HallsofIvy

Homework Helper
I am reading through Borwein's paper, "Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind" and have a question.

If we look at his algorithm's we see they are of this general form:

Gamma(1/2)=2^(-1/4)AG[1]{2^(-1/2)∑[1]}^(-1/2)

I have been able to run through everything up until the curly brackets, {}. Is this a multiplier? Or is it something else entirely?
I'm not clear on what "$\Sigma[1]$" or "AG[1]" mean. But ignoring that, because the curly brackets are to the "-1/2" power, this is
$$\Gamma(1/2)= \frac{1}{16}\frac{AG[1]}{\sqrt{\frac{1}{\sqrt{2}}\Sigma[1]}}$$

#### mesa

I'm not clear on what "$\Sigma[1]$" or "AG[1]" mean. But ignoring that, because the curly brackets are to the "-1/2" power, this is
$$\Gamma(1/2)= \frac{1}{16}\frac{AG[1]}{\sqrt{\frac{1}{\sqrt{2}}\Sigma[1]}}$$
As far as 'AG' Borwein uses it as short hand for his summation that takes the difference of the squares of an and bn from the arithmetic geometric mean operation inside a summation that is taken away from a value that is dependent on the input for gamma.

The general operation of the identity is based on calculated variable 'N' tables provided by the author to compute certain values of gamma we can 'look up' to insert to get the components of AG[N] and ∑[N]. I did not look into exactly how the author calculated these tables, the process can be daunting and does not yield much value (if someone thinks otherwise please chime in!).

Overall the scope of his identity seems very limited but for the values it can calculate the decimal approximation for accuracy is astounding and grows radically with each step.

On another note, I figured the same for the curly brackets just representing () or [], I just haven't seen them used before, is this a common thing?

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