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#1 2012-10-21 23:53:10
- 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 2012-10-22 00:40:15
- bobbym
- Administrator
Online
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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#3 2012-10-22 00:43:24
- 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 2012-10-22 00:51:43
- 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 2012-10-22 01:07:05
- 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 2012-10-22 01:10:02
- 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 2012-10-22 01:35:47
- 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 2012-10-22 01:53:05
- bobbym
- Administrator
Online
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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#9 2012-10-22 01:56:24
- 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.
#11 2012-10-22 02:07:21
- bobbym
- Administrator
Online
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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#12 2012-10-22 02:11:46
- 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 ei is just a constant. Can you see where to go from here?
#14 2012-10-22 02:41:30
- Sabbir
- Guest
Re: Gamma function
So,will the taylor series give the same result as the integral's.
#15 2012-10-22 02:47:19
- 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 2012-10-22 02:52:18
- bobbym
- Administrator
Online
Re: Gamma function
You can also evaluate the integral numerically.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#18 2012-10-22 03:02:44
- 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 2012-10-22 08:48:48
- bobbym
- Administrator
Online
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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.