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Why does Mathematica sometimes give Gamma, the funtion and sometimes !, the factorial? What is the differences?
A Gamma function (Gamma[x]) is a continuous function that extends the factorial function (x!) to real and complex numbers. It is defined as Γ(x) = (x-1)!, where x is a positive real number. On the other hand, a Factorial function (Factorial[n]) is a discrete function that only accepts positive integers as input and returns n! as the output.
The Gamma function has a smooth, curved graph that increases rapidly as x increases. The Factorial function, on the other hand, has a steeper, more linear graph that increases at a slower rate. However, the two functions have the same value at integer inputs.
No, the Gamma and Factorial functions have different domains and ranges, and therefore cannot be used interchangeably. The Gamma function can handle real and complex numbers, while the Factorial function can only handle positive integers.
Yes, both functions have special properties that make them useful in various mathematical applications. For example, the Gamma function satisfies the recurrence relation Γ(x+1) = xΓ(x), while the Factorial function satisfies the identity n! = n * (n-1)!.
You can simply use the built-in functions Gamma[x] and Factorial[n] in Mathematica and input the desired value for x or n. Mathematica will return the corresponding values for the Gamma and Factorial functions, allowing for easy comparison. Additionally, you can use the Plot function to visualize the graphs of the two functions for a range of inputs and compare their trends.