How can I find negative factorial ? (-0.5)!

How can I find negative factorial ? (-0.5)!

any one can help me the code to fined negative factorial as (-0.5)

Answer 1

[the cyclist]

Use the gamma function:

x = -0.5;
gamma(x+1) % gamma(x+1) is equal to factorial(x) 

ans = 1.7725

[esam baker]

Thank you very much, Sir

Answer 2

[John D'Errico]

Simple. Yes, you could use MATLAB, IF you knew that the extension of the function factorial(n) onto the real line is just gamma(n+1). So you might just do this:

gamma(-0.5 + 1)

ans = 1.7725

Or if you want a prettier form...

gamma(sym(-0.5) + 1)

ans = 

Perhaps a tricky way is to use the Euler reflection formula.

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

It tells us that

gamma(z)*gamma(1-z) = pi/sin(pi*z)

Now, recalling the relation between factorial and gamma, and accepting that factorial can indeed be extended onto the real line, we can write that as:

factorial(z-1)*factorial(-z) = pi/sin(pi*z)

Does that help us in any way? When happens when z = 1/2? z=1/2 is actully a truly magic number in our problem.

factorial(1/2 - 1)*factorial(-1/2) = [factorial(-1/2)]^2 = pi/sin(pi/2)

Does that help us?

Since sin(pi/2) = 1, then we see the square of the desired factorial we wanted to compute is just pi.

And therefore?

factorial(-1/2) = sqrt(pi)

Neat. I never even needed to use MATLAB. Pencil and paper was enough. Unless, of course, I was too lazy to do the math. It is always more fun to do the math.

[esam baker]

x = -0.5 is an example of my problem, in practice, I have a loop containing of n Iterations for random negative fractional numbers in Domain (-10,0), thank you very much, Sir

[John D'Errico]

You asked a specific question about a specific value. Answered here.

Since your question was not really what you asked, then all you need to know is the extension onto the real line for the factorial function. I can't answer what you want to know if you don't state your real problem.

[esam baker]

Sir your answer was very helpful to me, I Am very grateful for your reply.

[Bruno Luong]

@John D'Errico "And therefore? factorial(-1/2) = sqrt(pi)"

Do you have to dsicard the negative solution -sqrt(pi) before this conclusion?

Answer 3

[Bruno Luong]

There are actually different ways of extending factorial function, see here

http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html

So rigourously your question is not precise to be answered