Continue to Site

Welcome to EDAboard.com

Welcome to our site! EDAboard.com is an international Electronics Discussion Forum focused on EDA software, circuits, schematics, books, theory, papers, asic, pld, 8051, DSP, Network, RF, Analog Design, PCB, Service Manuals... and a whole lot more! To participate you need to register. Registration is free. Click here to register now.

How to derive the factorial of (1/2) ?

Status
Not open for further replies.

amriths04

Full Member level 5
Joined
Jul 15, 2006
Messages
263
Helped
23
Reputation
46
Reaction score
9
Trophy points
1,298
Activity points
2,819
i know that (1/2)! = 0.5*√pi
but i want to know how that value was got (gamma functions? if so how)?
 

Re: factorial of (1/2)

amriths04 said:
i know that (1/2)! = 0.5*√pi
but i want to know how that value was got (gamma functions? if so how)?

Yes the value comes from the gamma function.

The factorial n! coincides with the gamma function at positive integer values. So if one equates the factorial function on all positive reals with the gamma function, then one can say that

(1/2)! = sqrt(pi)/2

even though the factorial function was originally only defined for the natural numbers.

Added after 14 minutes:

Noooo I'am wrong. It's Gamma(n+1) = n! and one does not equate the factorial with the gamma function. Sorry.

Added after 26 minutes:

The general definition of a factorial is

x! = Gamma(x+1)

By a little integration you can prove that

Gamma(1/2) = sqrt(pi)
Gamma(x) = (x-1) Gamma(x-1)

So the idea seems to be that

(1/2)! = Gamma(3/2) = 1/2 * Gamma(1/2)
 

Re: factorial of (1/2)

yes, i am aware that Gamma(1/2) = sqrt(pi).
but i want to prove the above mathematically.

that is how ∫(t^-0.5)*(e^-t)dt between 0 and inf = sqrt(pi) ??
 

Re: factorial of (1/2)

amriths04 said:
i know that (1/2)! = 0.5*√pi
but i want to know how that value was got (gamma functions? if so how)?

Damn. This is how it goes. Just calculate

Gamma(3/2)

by integrating and then just use the general definition

x! = Gamma(x+1)

for all real x.

Added after 8 minutes:

amriths04 said:
yes, i am aware that Gamma(1/2) = sqrt(pi).
but i want to prove the above mathematically.

that is how ∫(t^-0.5)*(e^-t)dt between 0 and inf = sqrt(pi) ??

I see. hmm.

Added after 25 minutes:

amriths04 said:
yes, i am aware that Gamma(1/2) = sqrt(pi).
but i want to prove the above mathematically.

that is how ∫(t^-0.5)*(e^-t)dt between 0 and inf = sqrt(pi) ??

Can't say that I see how to do the integral by just looking at it. But you can see how the integral comes
from a property of the Gamma function called the 'Euler reflection formula':

G(x)G(x-1) = pi / sin(pi*x)

by pluging in x=1/2.
 

factorial of (1/2)

change the variable x=z^2.
obtain the new integral and define it i.
calculate i^2 . so it will a double integral
change the variables of double integral from Cartesian to polar.
calculate i^2 . it will be pi.
now i=sqrt(pi)
 

    amriths04

    Points: 2
    Helpful Answer Positive Rating
Re: factorial of (1/2)

amir81 said:
change the variable x=z^2.
obtain the new integral and define it i.
calculate i^2 . so it will a double integral
change the variables of double integral from Cartesian to polar.
calculate i^2 . it will be pi.
now i=sqrt(pi)

Yes check out

https://en.wikipedia.org/wiki/Gaussian_integral

there you find how it's done.
 

Status
Not open for further replies.

Part and Inventory Search

Welcome to EDABoard.com

Sponsor

Back
Top