What is the Gamma Function and How Can Grade 11 Students Understand It?

[JPC]

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) ?

 

[HallsofIvy]

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
\Gamma(n)= (n-1)![/itex]<br /> but that is a consequence of the definition.<br /> <br /> The gamma function is <b>defined</b> by<br /> \Gamma(x)= \int_0^\infty t^{x-1}e^{-t}dt[/itex]&lt;br /&gt; &lt;br /&gt; Using that formula, you can show by induction on n (and integration by parts for the induction step) that&lt;br /&gt; \Gamma(n)= \int_0^\infty t^{n-1}e^{-t}dt= (n-1)!&lt;br /&gt; for n any positive integer.&lt;br /&gt; &lt;br /&gt; The gamma function is defined for all real numbers except negative integers.&lt;br /&gt; &lt;br /&gt; Perhaps you are asking how you do that integral for x not a positive integer- the answer is that, in general, you don&amp;#039;t! Generally numerical integration is used.&lt;br /&gt; &lt;br /&gt; However, it is an interesting exercise for simple values like x= 1/2.&lt;br /&gt; \Gamma(\frac{1}{2})= \int_0^\infty t^{-\frac{1}{2}}e^{-t} dt&lt;br /&gt; Let u= t&lt;sup&gt;1/2&lt;/sup&gt; so that t= u&lt;sup&gt;2&lt;/sup&gt; and dt= 2udu. Then the integral is&lt;br /&gt; \int_0^\infty \frac{1}{u} e^{-u^2}(2 udu)= \int_0^\infty e^{-u^2}du[/itex]&amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; Do you know how to show that that is \sqrt{\pi}? &amp;lt;br /&amp;gt; \Gamma(\frac{1}{2})= \sqrt{\pi}

 

[JPC]

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 ?

 

[Eighty]

but is there a link between the gamma function , and the gamma level adjustments when you edit pictures ?

Nope. Check the very last sentence (as of right now) of http://en.wikipedia.org/wiki/Gamma_correction .

 

[uart]

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.

 

[JPC]

ok

but why have a gamma function ?
whats its use ?

 

[Eighty]

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.

 

[HallsofIvy]

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.

 

[Gib Z]

Gamma function also let's us compute things that are normally thought to be done decretely like derivatives.