• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Another way to continuum sum! JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 08/29/2013, 04:05 PM Well if one way to continuum sum wasn't enough, I found another one! Let's start: $\mathcal{L} f(x) = \frac{1}{\Gamma(x)}\int_0^\infty e^{-t} f(t) t^{x-1}dt$ $\mathcal{L}^{-1} f(x) = \frac{e^x}{2 \pi i} \int_{c - i \infty}^{c+ i \infty} \Gamma(t) f(t) x^{-t} dt$ If $\frac{\Delta f(x)}{\Delta x} = f(x) - f(x-1)$ then: $\frac{\Delta}{\Delta x} \mathcal{L} f(x) = \mathcal{L} \frac{d f(x)}{dx}$ The method for continuum sum follows from this! $\sum_a^b f(y) \, \Delta y = (\mathcal{L} \int \mathcal{L}^{-1} f(x))\,|_{x=a}^{x=b}$ « Next Oldest | Next Newest »

 Messages In This Thread Another way to continuum sum! - by JmsNxn - 08/29/2013, 04:05 PM RE: Another way to continuum sum! - by tommy1729 - 08/29/2013, 10:04 PM RE: Another way to continuum sum! - by JmsNxn - 08/29/2013, 11:50 PM RE: Another way to continuum sum! - by tommy1729 - 06/05/2014, 10:57 PM RE: Another way to continuum sum! - by MphLee - 06/06/2014, 11:21 AM RE: Another way to continuum sum! - by JmsNxn - 06/06/2014, 04:00 PM RE: Another way to continuum sum! - by MphLee - 06/06/2014, 05:09 PM

 Possibly Related Threads... Thread Author Replies Views Last Post 2015 Continuum sum conjecture tommy1729 3 3,734 05/26/2015, 12:24 PM Last Post: tommy1729 Continuum sum = Continuum product tommy1729 1 2,885 08/22/2013, 04:01 PM Last Post: JmsNxn applying continuum sum to interpolate any sequence. JmsNxn 1 3,083 08/18/2013, 08:55 PM Last Post: tommy1729 Powerful way to perform continuum sum JmsNxn 7 8,481 08/12/2013, 07:17 PM Last Post: JmsNxn sexp by continuum product ? tommy1729 6 9,252 06/30/2011, 10:07 PM Last Post: tommy1729 continuum sum again tommy1729 0 2,411 02/10/2011, 01:25 PM Last Post: tommy1729 Continuum sums -- a big problem and some interesting observations mike3 17 21,299 10/12/2010, 10:41 AM Last Post: Ansus Continuum sum - a new hope kobi_78 10 13,980 06/13/2010, 11:23 PM Last Post: sheldonison New tetration method based on continuum sum and exp-series mike3 16 22,835 05/02/2010, 09:58 AM Last Post: andydude Borel summation and other continuation/summability methods for continuum sums mike3 2 5,517 12/30/2009, 09:51 PM Last Post: mike3

Users browsing this thread: 1 Guest(s)