How to prove the value of Gamma(1/4)?

  • Thread starter Thread starter bit188
  • Start date Start date
  • Tags Tags
    Value
Click For Summary

Homework Help Overview

The discussion revolves around proving the value of Gamma(1/4), with the original poster expressing difficulty in integrating the function properly. The Gamma function is defined through an improper integral, and the poster aims to establish its approximate value.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss various integration techniques, including integration by substitution and integration by parts. There are questions about the validity of seeking an exact value versus an approximate one, and some participants suggest numerical integration as a viable approach.

Discussion Status

The discussion is ongoing, with participants exploring different perspectives on whether an exact value can be obtained or if numerical methods are more appropriate. Some guidance has been offered regarding the nature of the Gamma function and its properties, but no consensus has been reached on the approach to take.

Contextual Notes

There is mention of the limitations of expressing Gamma(1/4) in terms of common transcendental functions, and the original poster's self-imposed challenge to prove its value adds a layer of complexity to the discussion.

bit188
Messages
45
Reaction score
0

Homework Statement


Self-given problem; I want to prove that Gamma (1/4) is approxiamately equal to 3.625, but can't seem to integrate it properly...

Gamma(z) = (integral between infinity and 0) (t^z-1)(e^-t) dt
(http://www.wikipedia.org/wiki/Gamma_function)

Homework Equations


Gamma(n+1) = n Gamma (n)

The Attempt at a Solution


Tried an integration by substitution and an integration by parts, and no luck!

Thanks for your help!
 
Last edited:
Physics news on Phys.org
So you need to integrate
\int_0^\infty t^{-3/4}e^{-t}dt

What do you mean by "approxiamately equal to 3.625". If that is really what you want to get that, then use a numerical integration.
 
2. That is not the only relevant bit of information, but hey, what the heck (it needs to be an analytic continuation).

3. What makes you think anything but a numerical approximation will work?
 
No, I want to get the real value :P

When I do an integration by parts, I get

(-3/4)(t^6/4) + (3/4)(t^-1/2)(e^-t), which is not the right answer...
 
\Gamma\left(\frac{1}{4}\right) cannot be expressed in terms of values of common transcendental functions.
 
Really? Well, I suppose I'd better learn more math, then. >_<

Thanks anyhow!
 
You sure you don't just want it approximately?

Numerical integration would be fine for a few decimal places >.< good enough lol
 
Nah... I'm pretty intrigued by this particular function. I'm interested for nonpractical reasons. Thanks though! ^_^
 

Similar threads

  • · Replies 5 ·
Replies
5
Views
2K
Replies
1
Views
2K
  • · Replies 8 ·
Replies
8
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 13 ·
Replies
13
Views
3K
Replies
9
Views
2K
  • · Replies 5 ·
Replies
5
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 6 ·
Replies
6
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K