My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
August 5th, 2012, 10:49 PM   #1
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Improper Gamma Integral

Okay, I know that and both diverges. But what about this limit:

?

Is this also divergent? Or is it convergent?

Even if I had a graph of , It would mean a lot to me.

Can anybody give me a graph?(Oh, I know that's not possible )

Any help will be greatly appreciated.

Balarka

.
mathbalarka is offline  
 
August 6th, 2012, 12:34 AM   #2
Senior Member
 
Joined: Aug 2011

Posts: 334
Thanks: 8

Re: Improper Gamma Integral

Hi !

I think that F(n) is not converging because, for n tending to infinity, the Sum is equivalent to Gamma(n) and the Integral is equivalent to Gamma(n)/ln(n))
JJacquelin is offline  
August 6th, 2012, 12:39 AM   #3
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: Improper Gamma Integral

Quote:
Originally Posted by JJacquelin
for n tending to infinity, the Sum is equivalent to Gamma(n) and the Integral is equivalent to Gamma(n)/ln(n))
Wow! Can you prove that (I mean, can you show me the proof?)?
mathbalarka is offline  
August 6th, 2012, 06:51 AM   #4
Senior Member
 
Joined: Aug 2011

Posts: 334
Thanks: 8

Re: Improper Gamma Integral

Quote:
Originally Posted by mathbalarka
Wow! Can you prove that (I mean, can you show me the proof?)?
The proof for the integral requires somme background about polygamma functions (especially asymptotic expansion)
Attached Images
File Type: jpg Equivalent Sum.JPG (16.9 KB, 123 views)
JJacquelin is offline  
August 6th, 2012, 06:53 AM   #5
Senior Member
 
Joined: Aug 2011

Posts: 334
Thanks: 8

Re: Improper Gamma Integral

&............................
Attached Images
File Type: jpg Equivalent Integral.JPG (46.3 KB, 123 views)
JJacquelin is offline  
August 6th, 2012, 10:03 AM   #6
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: Improper Gamma Integral

Ok.

Lots of Thanks,

Balarka

.
mathbalarka is offline  
August 6th, 2012, 10:57 AM   #7
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: Improper Gamma Integral

I found another interesting thing about :

It is well known that:



Now, the euler's reflection formula for the gamma function states that:



Substituting this into (1) gives



From this,



So, F(1-x) diverges for all

I am currently very sleepy now so please forgive me if I am wrong

Balarka

.
mathbalarka is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
gamma, improper, integral



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Improper integral limes5 Calculus 6 April 3rd, 2013 06:11 AM
Improper Integral strammer Real Analysis 4 February 11th, 2013 04:24 PM
Integral by using Gamma function jackson_wang Calculus 9 January 27th, 2012 06:48 AM
Integral by using Gamma function jackson_wang Calculus 3 June 2nd, 2011 09:04 PM
Integral improper. ZardoZ Complex Analysis 10 April 13th, 2011 03:32 AM





Copyright © 2019 My Math Forum. All rights reserved.