Beyond Factorials!

We all know that factorials are defined for any positive integer ‘n’ as $n!=n\times(n-1)\times(n-2)....\times 2\times1$ . Following a pattern, we can also say $0!=1$ . But what if we take it one step further?

For all integers(negative integers as well) how would factorials be defined and what would be their values? (I have seen a few pages that say (-1)!=-1, (-3)!=-6, etc. but haven’t seen any satisfactory explanations)
How would the factorial function be extended over all real numbers? For example, $0.5!, \sqrt{2}!,$ or even $\pi !$ . Again, I have seen pages that say 0.5!=0.886... ,etc.

The reason I asked this is because I have seen other functions being understood for values beyond their initial domain.

Thanks for any solutions or explanations!

#Combinatorics

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Comments

See Gamma function.

Pi Han Goh - 3 years, 1 month ago

Thank you

Chandrashekar Giridharan - 3 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$