Decimal factorials

"4.5!  is the gamma function of 4.5"

Not quite.   Gamma(n) = (n-1)!

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Gamma(4.5) = 11.632   which is between 3! = 6 and 4! = 24

Because with decimals we use the gamma function:

https://www.wikiwand.com/en/Gamma_function

Because it is the gamma funtion.

4! that is 4 factorial = 24

5! that is 5 factorial = 120

Now 4.5 factorial does not makes sense but it you have a curve where (4,24) is one point and (5,120) is the next point, you could join this up to get a smooth curve and that curve is the gamma function,   :)

4.5!  is the gamma function of 4.5

Thanks Alan,

I stand corrected :))

Why is 0.5 ! = (√π )/2 ?

\(\boxed{~ \begin{array}{rcll} n! &=& \Gamma(n+1) \end{array} ~}\)

\(\begin{array}{rcll} 0.5! &=& \Gamma(0.5+1) \\ \end{array}\)

\(\boxed{~ \begin{array}{rcll} \Gamma(x+1) &=& x\cdot \Gamma(x) \end{array} ~}\)

\(\begin{array}{rcll} \Gamma(0.5+1) &=& 0.5\cdot \Gamma(0.5) \end{array}\)

\(\boxed{~ \begin{array}{rcll} \Gamma(0.5) &=& \sqrt{\pi} \end{array} ~}\)

\(\begin{array}{rcll} 0.5\cdot \Gamma(0.5) &=& 0.5\cdot \sqrt{\pi}\\ &=& \dfrac{\sqrt{\pi}} {2} \\\\ \mathbf{0.5!} & \mathbf{=} & \mathbf{ \dfrac{\sqrt{\pi}} {2} } \end{array}\)

Thanks, heureka.....I followed what you did....but....I have  question..... in your fifth step....you state that :

Γ(0.5)  = √ pi

How is this determined????

Also, does this imply that

Γ(0.5)  = Γ( -0.5 + 1) = -0.5 * Γ(-0.5)   ?????

Is it possible to apply the gamma function to a negative real number???.......

This might help answer your question Chris:

It isn't defined for values of t = 0, -1, -2, ... etc though, where you get division by zero.

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OK......thanks, Alan  !!!!!