Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Continuum sum = Continuum product
I was thinking about making the transformation from continuum sum to continuum product and encountered the following problem.

The continuum sum can be written as a triple integral. If is defined on the line And as its imaginary part goes to infinity it grows slower than decays to zero.

Now the immediate problem is that I think mellin inversion will not work on functions like the logarithm, or any function that gives complex values to the logarithm. and this is how we go from the continuum sum to the continuum product.

Therefore, we'd have to find a more general way of inverting the mellin transform that could work on functions like the logarithm.

Or we could do some algebra and prove the result that

As well as the same result for the integral. With that we can get an expression for the factorial as:

Where psi is the di gamma function and can be shown to equal:


Messages In This Thread
Continuum sum = Continuum product - by tommy1729 - 08/21/2013, 11:27 PM
RE: Continuum sum = Continuum product - by JmsNxn - 08/22/2013, 04:01 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Between addition and product ( pic ) tommy1729 4 5,220 07/10/2016, 07:32 AM
Last Post: Gottfried
  2015 Continuum sum conjecture tommy1729 3 4,745 05/26/2015, 12:24 PM
Last Post: tommy1729
  Another way to continuum sum! JmsNxn 6 8,510 06/06/2014, 05:09 PM
Last Post: MphLee
  Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 6,889 05/06/2014, 09:47 PM
Last Post: tommy1729
  applying continuum sum to interpolate any sequence. JmsNxn 1 3,648 08/18/2013, 08:55 PM
Last Post: tommy1729
  Powerful way to perform continuum sum JmsNxn 7 9,925 08/12/2013, 07:17 PM
Last Post: JmsNxn
  Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 9,071 07/17/2013, 09:46 AM
Last Post: Gottfried
  sexp by continuum product ? tommy1729 6 10,812 06/30/2011, 10:07 PM
Last Post: tommy1729
  product functions tommy1729 5 8,031 06/01/2011, 05:38 PM
Last Post: tommy1729
  continuum sum again tommy1729 0 2,750 02/10/2011, 01:25 PM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)