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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
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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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#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
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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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#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
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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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#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
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Re: Gamma function
You can also evaluate the integral numerically.
In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#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
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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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.