Exploring Gamma Function: A Grade 11 Student's Perspective

Also, there are many real world applications of the gamma function it relates to the study of queues, it is used in statistical distribution, fluid mechanics, and many others. In summary, the gamma function is a generalization of the factorial function and has a wide range of applications in fields such as probability, fluid mechanics, and statistics. It is defined by an integral and can be used to compute things that are normally thought to be done discretely, such as derivatives.
  • #1
JPC
206
1
Hey , i am in grade 11
not yet studied gamma function, and not sure if it will be in the program

but i have studied it a bit on my own

f(x) = gamma(x)

x = 1 : y = 1
x = 2 : y = 1
x = 3 : y = 2
x = 4 : y = 6
x = 5 : y = 24
x = 6 : y = 120

and i found a pattern :

1 * 1 = 1
2 * 1 = 2
3 * 2 = 6
4 * 6 = 24
5 * 24 = 120

so if we consider the gamma sequence , is it ?

U(n+1) = Un * (n-1)
with n0 = 1

but now , how do u find Gamma of any x (x a any real number) ?
 
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  • #2
You say you have studied the gamma function on your own- didn't you start with the definition? How did you get those numbers? Yes, it is true that for n any positive integer
[tex]\Gamma(n)= (n-1)![/itex]
but that is a consequence of the definition.

The gamma function is defined by
[tex]\Gamma(x)= \int_0^\infty t^{x-1}e^{-t}dt[/itex]

Using that formula, you can show by induction on n (and integration by parts for the induction step) that
[tex]\Gamma(n)= \int_0^\infty t^{n-1}e^{-t}dt= (n-1)![/tex]
for n any positive integer.

The gamma function is defined for all real numbers except negative integers.

Perhaps you are asking how you do that integral for x not a positive integer- the answer is that, in general, you don't! Generally numerical integration is used.

However, it is an interesting exercise for simple values like x= 1/2.
[tex]\Gamma(\frac{1}{2})= \int_0^\infty t^{-\frac{1}{2}}e^{-t} dt[/tex]
Let u= t1/2 so that t= u2 and dt= 2udu. Then the integral is
[tex]\int_0^\infty \frac{1}{u} e^{-u^2}(2 udu)= \int_0^\infty e^{-u^2}du[/itex]

Do you know how to show that that is [itex]\sqrt{\pi}[/itex]?
[tex]\Gamma(\frac{1}{2})= \sqrt{\pi}[/tex]
 
  • #3
hum

sorry haven't done calculus yet
didnt know the gamma function was that complex

but is there a link between the gamma function , and the gamma level adjustments when you edit pictures ?
 
  • #4
  • #5
JPC said:
hum

sorry haven't done calculus yet
didnt know the gamma function was that complex
?

Hi JPC, what you have been studying is (almost) the factorial function, not the gamma function. For postive whole numbers the gamma function is almost the same thing as the gamma function.
 
  • #6
ok

but why have a gamma function ?
whats its use ?
 
  • #7
JPC said:
ok

but why have a gamma function ?
whats its use ?

It's a generalisation of the factorial. The factorial is only defined for integers, and the gamma function is the only function that satisfies some nice "smoothness" criteria (it's continuous, strictly increasing, convex, as well as some more things) which goes through all the points the factorial does. So you might as well define the factorial for non-integers and negative numbers by the gamma function. Then you can calculate the "factorial" of numbers like -1/2.
 
  • #8
Actually, the gama function is a lot more that just a generalization of the factorial. With simple substitutions you can convert to the "Beta" function which is important in the Beta probability distribution.
 
  • #9
Gamma function also let's us compute things that are normally thought to be done decretely like derivatives.
 

1. What is the gamma function?

The gamma function, denoted by the symbol Γ, is a mathematical function that is an extension of the factorial function to real and complex numbers. It is defined as Γ(z) = ∫0 xz-1e-xdx, where z is a complex number.

2. Why is the gamma function important?

The gamma function is important in many areas of mathematics, physics, and statistics. It is used to solve various problems involving factorials, such as calculating probabilities in statistics, performing complex integrations, and solving differential equations. It also has connections to other important mathematical functions, such as the beta function and the zeta function.

3. How is the gamma function related to the factorial function?

The gamma function is an extension of the factorial function to real and complex numbers. This means that when the input to the gamma function is a positive integer, the result is equal to the factorial of that number. For example, Γ(4) = 3! = 6.

4. How is the gamma function calculated?

The gamma function can be calculated using various methods, including numerical approximation, series expansion, and the use of special functions such as the incomplete gamma function. It can also be calculated using a table or graph of values. In general, the gamma function is not easily calculated by hand and is typically calculated using software or calculators.

5. How is the gamma function used in real-world applications?

The gamma function has many real-world applications, including in physics, engineering, and statistics. It is used to solve problems involving growth and decay, such as in radioactive decay and population growth. It is also used in the development of statistical models and in the calculation of probabilities for continuous distributions. Additionally, the gamma function is used in signal processing and image reconstruction in medical imaging.

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