Gamma function

Sabbir · 2012-10-21 00:53:10

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).

bobbym · 2012-10-21 01:40:15

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?

zetafunc. · 2012-10-21 01:43:24

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!

zetafunc. · 2012-10-21 01:51:43

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.

Sabbir · 2012-10-21 02:07:05

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

zetafunc. · 2012-10-21 02:10:02

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

Was any part of my explanation unclear?

Sabbir · 2012-10-21 02:35:47

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

bobbym · 2012-10-21 02:53:05

Hi;

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

zetafunc. · 2012-10-21 02:56:24

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.

Sabbir · 2012-10-21 03:00:29

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

bobbym · 2012-10-21 03:07:21

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.

zetafunc. · 2012-10-21 03:11:46

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?

zetafunc. · 2012-10-21 03:14:44

Sorry, I am wrong.

. Ignore my previous post.

Sabbir · 2012-10-21 03:41:30

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

zetafunc. · 2012-10-21 03:47:19

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?

bobbym · 2012-10-21 03:52:18

You can also evaluate the integral numerically.

Sabbir · 2012-10-21 04:00:38

I was just curious about the taylor series

zetafunc. · 2012-10-21 04:02:44

Sabbir wrote:

I was just curious about the taylor series

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

bobbym · 2012-10-21 09:48:48

Hi;

I was just curious about the taylor series

What would you like to know?

Math Is Fun Forum

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

Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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

Re: Gamma function

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