- #1
Frank Li
- 8
- 0
Γ(n) = ∫x→∞ tn-1 e-t dt?
I was watching about factorials on Youtube channel by the Numberphile, a topic named "0! = 1". Inside that video, they mentioned about this function, and I would like to look deeper into this topic.jedishrfu said:Can you be more explicit? Like where did you find it? What it's used for?
It looks like the gamma function.
https://en.m.wikipedia.org/wiki/Gamma_function
jedishrfu said:
Yes, I remember that video.
jedishrfu said:There are some other videos on YouTube that get into more detail about the function and it's uses
Basically though, it came about as mathematicians try to extend factorials to work with rational numbers and beyond which is a common theme in math. Find a pattern and keep extending it outward i.e. Generalizing it more and then prove that it works in the new contexts.
Creativity in action.
The Gamma function is a mathematical function that is used to extend the concept of factorial to non-integer values. It is denoted by the Greek letter gamma (Γ) and is defined as Γ(z) = ∫0∞ xz-1e-x dx, where z is a complex number. It has applications in various fields such as statistics, physics, and engineering.
The Gamma function is an extension of the factorial function, which is only defined for positive integers. In fact, for positive integer values of z, Γ(z) is equal to (z-1)!. However, the Gamma function allows for the calculation of factorials for non-integer values, making it a more versatile and useful tool in mathematics.
The Gamma function has numerous applications in fields such as physics, engineering, and statistics. For example, it is used in quantum mechanics to calculate probabilities of energy levels in atoms and molecules. In engineering, it is used to model the behavior of materials under stress. In statistics, it is used in the calculation of various probability distributions such as the chi-square distribution.
One way to understand the Gamma function is to think of it as a smoother version of the factorial function. While the factorial function jumps from one integer to the next, the Gamma function allows for the calculation of values in between. It can also be thought of as a continuous version of the factorial function, as it provides a way to calculate factorials for non-integer values.
Yes, the Gamma function has several special properties that make it a powerful tool in mathematics. Some of these include the reflection formula, recurrence relation, and duplication formula. It also has connections to other mathematical functions such as the beta function and the zeta function. Additionally, it has an infinite number of zeros, known as the Gamma function zeros, which have applications in number theory and complex analysis.