Why is this? I know what a factorial is, so what does it actually mean to take the factorial of a complex number? Also, are those parts of the complex answer rational or irrational? Do complex factorials give rise to any interesting geometric shapes/curves on the complex plane?
So, basically, factorial gives us the arrangements. Now, the question is why do we need to know the factorial of a negative number?, let's say -5. How can we imagine that there are -5 seats, and we need to arrange it? Something, which doesn't exist shouldn't have an arrangement right? Can someone please throw some light on it?.
Some theorems that suggest that the Gamma Function is the "right" extension of the factorial to the complex plane are the Bohr–Mollerup theorem and the Wielandt theorem.
Possible Duplicate: Prove 0! = 1 0! = 1 from first principles Why does 0! = 1 0! = 1? All I know of factorial is that x! x! is equal to the product of all the numbers that come before it. The product of 0 and anything is 0 0, and seems like it would be reasonable to assume that 0! = 0 0! = 0. I'm perplexed as to why I have to account for this condition in my factorial function (Trying to learn ...
It is a valid question to extend the factorial, a function with natural numbers as argument, to larger domains, like real or complex numbers. The gamma function also showed up several times as certain integrals, so mathematicians gave it a name and of course noted the relationship to factorials. See the graph at the end of this posting.
However, there is a continuous variant of the factorial function called the Gamma function, for which you can take derivatives and evaluate the derivative at integer values.
I was playing with my calculator when I tried $1.5!$. It came out to be $1.32934038817$. Now my question is that isn't factorial for natural numbers only? Like $2!$ is $2\\times1$, but how do we e...
One definition of the factorial that is more general than the usual $$ N! = N\cdot (N-1) \dots 1 $$ is via the gamma function, where $$ \Gamma (N) = (N-1)! = \int_0^ {\infty} x^ {N-1}e^ {-x} dx $$ This definition is not limited to positive integers, and in fact can be taken as the definition of the factorial for non-integers. With this definition, you can quite clearly see that $$ 0! = \Gamma ...
To find the factorial of a number, n n, you need to multiply n n by every number that comes before it. For example, if n= 4 n = 4, then n! = 24 n! = 24 since 4⋅3⋅2⋅1= 24 4 3 2 1 = 24. However, this method is very time consuming and, as n n gets larger, this method also become more difficult, so is there an easier method that I can use to find the factorial of any number?
