Special Functions: Valid Gauss Formula Parameters

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Homework Statement



It is known that Euler's integral representation
[PLAIN]http://img12.imageshack.us/img12/5578/euler.png
is valid for Re(c)>Re(b)>0 and |z|<1.

The series (Gauss Formula)
[PLAIN]http://img830.imageshack.us/img830/2365/gaussz.png
on the other hand converges for Re(c-b-a)>0.

For what values of the parameters a, b and c is the Gauss Formula valid? (Think carefully)

The Attempt at a Solution



Anyone help?
 
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c must be bigger than a+b ? sorry i mean x must equal 1 for the integral to become the second gamma expression on the nomenator
 
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Where's an x? Do you mean z? I'm not asked to derive the formula here - just to explain what values of a,b,c it is valid for. I was thinking c cannot be a non-negative integer
 
im sorry ,z must equal 1. the integral is a beta function and is equal to the gamma (the second in the nomenator) of the second function when z = 1
 
ardie said:
im sorry ,z must equal 1. the integral is a beta function and is equal to the gamma (the second in the nomenator) of the second function when z = 1

I'm not asked to derive the formula here - just to explain what values of a,b,c it is valid for. I was thinking c cannot be a non-negative integer
 
yes c has to be positive and bigger than a+b, and z must equal 1
 
but the thing is u already mention this, when u wrote Re(c-b-a)>0.
so i don't see what the question is
 
ardie said:
but the thing is u already mention this, when u wrote Re(c-b-a)>0.
so i don't see what the question is

The full question is this: (I've done part (a) so far which is deriving it...)
[PLAIN]http://img7.imageshack.us/img7/7766/fullqs.png
 
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well b can start from 1 and a from 1 so c must start from 2. I am assuming that the gamma function cannot take negative arguments. last ones just numbercrunching i suppose
 
Do a,b and c have to be integers? And why does a+b have to be less than c?

How do I compute the sum of the power series with b=-1 for part (c)?
 
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the gamma functions are evaluated using the gauss integral method such that:
gamme (n) = factorial (n-1)
you want the factorial to have integer values as input otherwise it becomes really sad and starts to cry
just plug in the numbers to get those values.
factorial (-1) is a mathematical nonsense, hence why c-b-a must be positive for the function to be welldefined
 
ardie said:
the gamma functions are evaluated using the gauss integral method such that:
gamme (n) = factorial (n-1)
you want the factorial to have integer values as input otherwise it becomes really sad and starts to cry
just plug in the numbers to get those values.
factorial (-1) is a mathematical nonsense, hence why c-b-a must be positive for the function to be welldefined

Part (c) says verify it for b=-1 but you've said that b must start from 1 for it to be valid in part (b)? a,b and c can be complex numbers...
 
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yes you can still have b=-1 with a=0 and c=0 and you will still not run into any trouble.
you can't have for example c=10 with b = 10 and a = 5
you also can't have c=-10 with b = 10 and a = 5
 
ardie said:
yes you can still have b=-1 with a=0 and c=0 and you will still not run into any trouble.
you can't have for example c=10 with b = 10 and a = 5
you also can't have c=-10 with b = 10 and a = 5

But I thought we established for part (b) that the formula was only valid for a and b starting from 1 and c>2? This shows it can be valid for other values too...
 
yes i think i made a mistake there again, but trying to minimise c, you can still have c=0 and then move upwards fitting all possible a and b values in
 
ardie said:
yes i think i made a mistake there again, but trying to minimise c, you can still have c=0 and then move upwards fitting all possible a and b values in

So it looks like as long as [tex]c>a+b[/tex] it will be valid?
 
yes... and that c is a positive integer
 
ardie said:
yes... and that c is a positive integer

So for part (c), do I have to show that

[tex]\sum_{n=0}^{\infty} \frac{(a)_n(-1)_n}{(c)_n n!} = \frac{\Gamma(c)\Gamma(c-a+1)}{\Gamma(c-a)\Gamma(c+1)}\;?[/tex]

If so, how?

The definition of the shifted factorial (Pochhammer symbol) is:
[tex](a)_n = \frac{\Gamma(a+n)}{\Gamma(a)}[/tex]
 
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i believe if u do the summation in the power series you will find by induction that both represent the same rational number
 
ardie said:
i believe if u do the summation in the power series you will find by induction that both represent the same rational number

Still struggling with evaluating either side...
 
calculate the LHS and confirm it is equal to the RHS?
 
ardie said:
calculate the LHS and confirm it is equal to the RHS?

How do I calculate the LHS though?
 
ok I am not entirely sure about the notation used in your book, but in my reference books, the a,b,c's on that LHS with the subscripts are pretty hefty recurance relations which I cannot remember off by heart. So you basically just write the numbers in and keep putting new n's. since you have a n! in ur denomenator, youll find that the series converges pretty quickly, maybe after 4-5 terms, then you just calculate and they should be the same