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

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esam baker 2022-8-9

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https://ww2.mathworks.cn/matlabcentral/answers/1776495-how-can-i-find-negative-factorial-0-5

回答： Bruno Luong 2022-8-9

采纳的回答： the cyclist

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

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

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Answer 1

the cyclist 2022-8-9

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https://ww2.mathworks.cn/matlabcentral/answers/1776495-how-can-i-find-negative-factorial-0-5#answer_1023715

编辑：the cyclist 2022-8-9

在 MATLAB Online 中打开

Use the gamma function:

x = -0.5;
gamma(x+1) % gamma(x+1) is equal to factorial(x) 
ans = 1.7725

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esam baker 2022-8-9

Thank you very much, Sir

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Answer 2

John D'Errico 2022-8-9

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https://ww2.mathworks.cn/matlabcentral/answers/1776495-how-can-i-find-negative-factorial-0-5#answer_1023885

编辑：John D'Errico 2022-8-9

在 MATLAB Online 中打开

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.