How can you prove that the square root of pi is -0.5!?

Answer 1

H.T. -- there is nothing wrong with the formulation of the problem.

z! = z factorial = Gamma(z + 1)

For example, see the Gamm function page at Wikipedia at

http://en.wikipedia.org/wiki/Gamma_function

which includes stating that Gamma(1/2) = sqrt(pi) and so is the value of -0.5 factorial.

Gamma(z) = integral as t varies from 0 to positive infinity of
t^(z - 1) e^(-t) dt

To prove it, use integration by parts to show that Gamma(z + 1) = z Gamma(z), then show that Gamma(1) = 1, and therefore Gamma(n + 1) = n Gamma(n) = ... = n! Gamma(1) = n! for positive integers n [that is from the article]. This justifies extending the definition of factorial so that (-0.5)! = Gamma(1/2) = integral(t from 0 to +inf.){ t^(-1/2) e^(-t) dt)

Use the substitution t = u^2, dt = 2u du to get
(-0.5)! = integral(u from 0 to +inf.){ (1/u) e^(- u^2) 2u du }
= 2 K where K = integral(u from 0 to +inf.){ e^(- u^2) du }

A standard trick computes (2 K)^2 as a double integral of e^-(x^2 + y^2) dx dy over the plane, which reduces to something nice using polar coordinates: integral(theta from 0 to 2 pi)(integral r from 0 to +inf.){ e^-(r^2) r dr dtheta } = 2 pi (1/2) = pi, so 2K = sqrt(pi).

Answer 2

You can't, because it isn't.

If you square (-0.5), you get 0.25. That's not pi. The actual square root of pi is roughly 1.77245.

Answer 3

if the sqrt of pi is -.5!,
then pi equals ( -0.5! )²

I dunno how the factorial of decimal numbers work,
but using a calculator,
(-0.5) ! = 1.77245385
which is equal to the square root of pi.

Or perhaps you could have a look at this site on Non-integer factorials
http://en.wikipedia.org/wiki/Factorial#Non-integer_factorials

Answer 4

That is not possible because Pi has no exact value the values that are used mathematically such as 3.14 are approximations.

Answer 5

we know that,
∫(0 to π/2)sin^pθ cos^qθ dθ = {Γ(p+1/2 Γ(q+1)/2} / 2 Γ(p+q+2)/2,
.
putting p=q=0, we get ∫(0 to π/2)dθ = { Γ1/2 Γ1/2} / 2 Γ1
π/2 = {Γ1/2}^2
Γ1/2 = root(π)

Answer 6

Dan D is the only one that understood the question and got it right.

Answer 7

use a scientific calculator!

but according to my scientific calculator, the answer is 1.772453851

i used a Sharp scientific calculator model EL-531WH

hope this helps

Answer 8

just the formulation of your question reveals that the very concept is null and void from the very beginning.

So knock it off. You cannot get anywhere with this logic.

Answer 9

on my graphing calculator i get
1.7724538509055

Answer 10

I cant , it isnt