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#1 2012-10-21 00:53:10

Sabbir
Guest

Gamma function

I want to know about gamma function,I have read about it in wikipedia,but I could not understand it very much,so please help(I know about factorial).

#2 2012-10-21 01:40:15

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Gamma function

Hi Sabbir;

Welcome to the forum;

For positive integers

so Gamma(5) = 4! = 24.

The gamma function also defined for other than integers.

But that is all in the pages you visited. Do you have a specific problem that requires the Gamma function?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2012-10-21 01:43:24

zetafunc.
Guest

Re: Gamma function

What do you need help with?

I understand the gamma function as being an extension to the factorial function, defining it for real and complex numbers. It is defined as

or

.

This integral can be annoying to evaluate, so quite often people use a table of values, or use some known properties of distributions to solve. For simplicity, let's try to evaluate Gamma(0.5).

I'll rewrite this as

I'll make a substitution;

So our integral becomes

Do you recognise this integral? It is a disguised form of something called the Gaussian integral. It's defined as

Since it's an even function (it's symmetrical about the y-axis), the area from 0 to infinity is just half of the area from negative infinity to positive infinity. Thus, your integral evaluates to √π. Hope this helps!

#4 2012-10-21 01:51:43

zetafunc.
Guest

Re: Gamma function

You can also evaluate the Gaussian integral using polar co-ordinates (rather than just accepting my definition for it). It's a bit long but I can show you if you'd like.

#5 2012-10-21 02:07:05

Sabbir
Guest

Re: Gamma function

I think if you show me some examples(such as other fractions and complex number)it will be easy for me to understand.

#6 2012-10-21 02:10:02

zetafunc.
Guest

Re: Gamma function

You mean evaluating the gamma function at different values -- 0.75, 0.224, (3+2i), etc.?

Was any part of my explanation unclear?

#7 2012-10-21 02:35:47

Sabbir
Guest

Re: Gamma function

Yes,different values and your explanation was clear but other values will make it easier for me to understand the way of approaching

#8 2012-10-21 02:53:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Gamma function

Hi;

Only certain values yield nice results like that. For say Gamma(10 + 1/3 ) you will have to resort to numerical methods.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#9 2012-10-21 02:56:24

zetafunc.
Guest

Re: Gamma function

There are lots of ways to compute values for the gamma function. For example, this is a useful identity;

Thus, since

then it follows that

so you can use that to find values of the gamma function at 5/2, 7/2, etc.

#10 2012-10-21 03:00:29

Sabbir
Guest

Re: Gamma function

Ok,but it will be helpful to see a complex number example.

#11 2012-10-21 03:07:21

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Gamma function

Those get very complicated. One way is with a Taylor series expanded around infinity.

This is quite close for only two terms of a Taylor series.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#12 2012-10-21 03:11:46

zetafunc.
Guest

Re: Gamma function

Evaluating it at complex numbers is interesting. Let's try to find Gamma(1+i).

but note that

so our integral becomes

I can do that becaues e[sup]i[/sup] is just a constant. Can you see where to go from here?

#13 2012-10-21 03:14:44

zetafunc.
Guest

Re: Gamma function

Sorry, I am wrong.

. Ignore my previous post.

#14 2012-10-21 03:41:30

Sabbir
Guest

Re: Gamma function

So,will the taylor series give the same result as the integral's.

#15 2012-10-21 03:47:19

zetafunc.
Guest

Re: Gamma function

Sabbir wrote:

So,will the taylor series give the same result as the integral's.

They will give you a (very good) approximation to the integral, yes.

May I ask what this is for?

#16 2012-10-21 03:52:18

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Gamma function

You can also evaluate the integral numerically.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

#17 2012-10-21 04:00:38

Sabbir
Guest

Re: Gamma function

I was just curious about the taylor series

#18 2012-10-21 04:02:44

zetafunc.
Guest

Re: Gamma function

Sabbir wrote:

I was just curious about the taylor series

I mean, about the Gamma function -- is it for a particular class?

#19 2012-10-21 09:48:48

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,676

Re: Gamma function

Hi;

I was just curious about the taylor series

What would you like to know?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Offline

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