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Math: Function: Curve: Gamma function:Definition: What is definition of gamma function? [history]

Mar 18th, 2009 07:38
engatoo engatoo, Roj Merry, John Martin, Knud van Eeden,


----------------------------------------------------------------------
--- Knud van Eeden --- 26 December 2004 - 07:57 pm -------------------
Math: Function: Curve: Gamma function:Definition: What is definition 
of gamma function? [history]
---
The gamma function is defined as the following integral:
              t->infinity
                -
               |
               |    ( x - 1 )    -t
Gamma( x ) =   |   t          . e   . dt
               |
               |
              -
              t=0
where x is real and greater than zero.
---
---
This function is called the
 'gamma function'
or also
 'generalized factorial function'
---
---
Note:
This improper integral will further converge for all values
of x greater than zero.
---
---
Note:
It can be further generalized to the complex domain as:
                infinity
                -
               |
               |    ( z - 1 )    -t
Gamma( z ) =   |   t          . e   . dt
               |
               |
              -
              z=0
where z is a complex number x + y . i, where x greater than zero.
---
---
Note:
The recurrence relation
 Gamma( x + 1 )  =   x       .   Gamma( x )
is a generalization of the factorial relation:
 ( x + 1 )!      = ( x + 1 ) .   x!
---
---
Note:
This gamma function occurs e.g. in the gamma distribution.
---
---
Note:
A variation of this integral led to the Beta function.
---
---
History:
The gamma function arose from work on e.g. interpolation theory and 
integration.
Looked for was a generalization of the integer only factorial function 
x!
The gamma function was the result of this.
Leonhard Euler (in 1729) introduced the beta and gamma function.
He published a solution to this question in 1729 in a letter to 
Christian Goldbach.
It appeared first in Euler's 3 books
'Institutiones Calculi integralis (1768 to 1774)
The name for this integral was introduced by Adrien-Marie Legendre.
Legendre called these 'Eulerian integrals of the first and second 
kind'.
respectively.
The names 'beta' and 'gamma' function were given by
Binet and Gauss respectively.
Carl Friedrich Gauss used this Gamma function e.g. in his calculations
with the hypergeometric series.
---
---
The gamma function comes more handy for its relation to other 
functions
than that it delivers a solution for some problem on its own.
It appears in the simplification of certain infinite and improper
integrals and in the solution of differential equations and
difference equations, which occur in probability theory,
statistics, mathematical physics and engineering mathematics.
---
---
Example:
e.g. to calculate Gamma( 1 ), you replace x with
1 in the formula, and calculate the resulting integral,
by first calculating the integral, and then filling
in the begin and end points.
---
Gamma( 1 ) =
   t->infinity
     -
    |
    |    ( 1 - 1 )    -t
=   |   t          . e   . dt
    |
    |
   -
   t=0
   t->infinity
     -
    |
    |    0    -t
=   |   t  . e   . dt
    |
    |
   -
   t=0
   t->infinity
     -
    |
    |        -t
=   |   1 . e   . dt
    |
    |
   -
   t=0
   t->infinity
     -
    |
    |   -t
=   |  e   . dt
    |
    |
   -
   t=0
      --+ t->infinity
        |
    -t  |
= -e    |
        |
      --+ t=0
          -0
= -0 - (-e  )
= 0 - ( -1 )
= 1
---
So if you know this, what is Gamma( 2 )?
Now using the general recurrence relation
 Gamma( x + 1 )  =   x       .   Gamma( x )
you can immediately say
 Gamma( 1 + 1 )  =   1       .   Gamma( 1 )
 Gamma( 2     )  =   1       .   1
 Gamma( 2     )  =   1
                 =   1!
---
You can check this result by using the formula:
e.g. to calculate Gamma( 2 ) directly, you replace x with
2 in the formula, and calculate the resulting integral,
by first calculating the integral, and then filling
in the begin and end points.
Gamma( 2 ) =
   t->infinity
     -
    |
    |    ( 2 - 1 )    -t
=   |   t          . e   . dt
    |
    |
   -
   t=0
   t->infinity
     -
    |
    |    1    -t
=   |   t  . e   . dt
    |
    |
   -
   t=0
   t->infinity
     -
    |
    |        -t
=   |   t . e   . dt
    |
    |
   -
   t=0
               --+ t->infinity
                 |
        -t    -t |
= -t . e   - e   |
                 |
               --+ t=0
             -0    -0
= 0 - ( 0 . e   - e   )
= 0 - ( -1 )
= 1
---
So if you know this, what is Gamma( 3 )?
Now using the general recurrence relation
 Gamma( x + 1 )  =   x       .   Gamma( x )
you can immediately say
 Gamma( 2 + 1 )  =   2       .   Gamma( 2 )
 Gamma( 3     )  =   2       .   1
 Gamma( 3     )  =   2
                 =   2!
If you check this result by using the formula
e.g. to calculate Gamma( 3 ), you replace x with
3 in the formula, and calculate the resulting integral,
by first calculating the integral, and then filling
in the begin and end points.
---
Gamma( 3 ) =
   t->infinity
     -
    |
    |    ( 3 - 1 )    -t
=   |   t          . e   . dt
    |
    |
   -
   t=0
   t->infinity
     -
    |
    |    2    -t
=   |   t  . e   . dt
    |
    |
   -
   t=0
                                   --+ t->infinity
                                     |
    2    -t            -t        -t  |
= -t  . e   - 2 . t . e   - 2 . e    |
                                     |
                                   --+ t=0
          2    -0            -0        -0
= 0 - ( -0  . e   - 2 . 0 . e   - 2 . e   )
= 0 - ( -0 . 1 - 2 . 0 . 1 - 2 . 1 )
= 0 - ( -2 )
= 2
---
So if you know this, what is Gamma( 4 )?
Now using the general recurrence relation
 Gamma( x + 1 )  =   x       .   Gamma( x )
you can immediately say
 Gamma( 3 + 1 )  =   3       .   Gamma( 3 )
 Gamma( 4     )  =   3       .   2
 Gamma( 4     )  =   6
                 =   3!
If you check this result by using the formula
e.g. to calculate Gamma( 4 ), you replace x with
4 in the formula, and calculate the resulting integral,
by first calculating the integral, and then filling
in the begin and end points.
---
Gamma( 4 ) =
   t->infinity
     -
    |
    |    ( 4 - 1 )    -t
=   |   t          . e   . dt
    |
    |
   -
   t=0
   t->infinity
     -
    |
    |    3    -t
=   |   t  . e   . dt
    |
    |
   -
   t=0
                                                  --+ t->infinity
                                                    |
    3    -t        2    -t            -t        -t  |
= -t  . e   - 3 . t  . e   - 6 . t . e   - 6 . e    |
                                                    |
                                                  --+ t=0
          3    -0        2    -0            -0        -0
= 0 - ( -0  . e   - 3 . 0  . e   - 6 . 0 . e   - 6 . e   )
= 0 - ( - 0 . 1 - 3 . 0 . 1 - 6 . 0 . 1 - 6 . 1 )
= 0 - ( 0 - 0 - 0 - 6 )
= 0 - ( -6 )
= 6
and indeed that is the result you could predict using the
recurrence formula.
---
To continue this idea:
you can immediately say for Gamma( 5 ):
 Gamma( 4 + 1 )  =   4       .   Gamma( 4 )
 Gamma( 5     )  =   4       .   6
 Gamma( 5     )  =   24
                 =   4!
---
To continue this idea:
you can immediately say for Gamma( 6 ):
 Gamma( 5 + 1 )  =   5       .   Gamma( 5 )
 Gamma( 6     )  =   5       .   24
 Gamma( 6     )  =   120
                 =   5!
---
and so on
---
Thus in general
 Gamma( n     )  =   (n-1)!
in the case you choose n to be a positive integer.
so you see clearly this designed link between the
gamma function and the faculty function
---
---
Book: see also:
---
[book: Abramowitz, Milton / Stegun, Irene A. - handbook of 
mathematical functions - Dover - p. 255 'Gamma (factorial) function' 
(with lots of formulas)]
---
[book: Ball Rouse, W. W. - A Short Account of the History of 
Mathematics - New York: Dover Publications - p. 396 'The beta function 
and gamma function were invented by Euler']
---
[book: Bronshtein, I. N. / Semendyayev, K. A. - handbook of 
mathematics - p. 91 'Factorials and the gamma function']
---
[book: Childers, Donald G. - probability and random processes (using 
MatLab with applications to continuous and discrete time systems) - 
publisher: McGraw-Hill - ISBN 0-256-13361-1 - p. 31 'Factorials and 
the gamma function']
---
[book: Daintith, John / Nelson, R. David - the Penguin dictionary of 
mathematics - ISBN 0-14-051119-9 - p. 142 'Gamma function']
---
[book: Dvorak, Stanislav / Musset, Anthony - BASIC in action - 1984 - 
Butterworths, Engeland - 1984 - ISBN 0-4080-01395-8 - subprograms 
iteration polynom and Stirling formula faculty x!, p. 51]
---
[book: Eves, Howard - an introduction to the history of mathematics - 
p. 434 'The beta and gamma functions of advanced calculus are 
accredited to Euler, although they were adumbrated by Wallis']
---
[book: encyclop<ae>dia Britannica (volume 28) - encyclop<ae>dia 
Britannica, Chicago - 1993 - 0-85229-571-5 - p. 496 (History: invented 
by Leonhard Euler, in his trial to generalize the integer only faculty 
function n! to all real (or complex) numbers)]
---
[book: Hastings, Kevin J. - Probability and statistics - ISBN 0-201-
59278-9 - p. 137 'Gamma function']
---
[book: Kline, Morris - mathematical thought from ancient to modern 
times / volume 2 - ISBN: 0-19-506136-5 - p. 422 'Further special 
functions (discusses extensive history of the gamma function)']
---
[book: Knuth, Donald Ervin - the art of computer programming, 
fundamental algorithms (volume 1) - Addison-Wesley - ISBN 0-201-03809-
9 - p. 46 'Permutations and factorials: gamma function (discusses the 
extensive history gamma function)]
---
[book: Press, William H. / Teukolsky, Saul A. / Flannery, Brian P. / 
Vettering, William T. - numerical recipes in C / the art of scientific 
computing]
---
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[book: Spanier, Jerome / Oldham, Keith B. - an atlas of functions - 
ISBN 0-89116-573-8 - p. 27:14 'Gamma']
---
[book: Spiegel, Murray R. - advanced mathematics for engineers & 
scientists - McGraw-Hill (Schaum) - 1971 - ISBN 07-060216-6 - p. 103 
(proof <GAMMA>(n+1) = n * <GAMMA>(n) using partial integration]
---
[book: Struik, Dirk J. - a conscise history of mathematics - p. 
168 'First appearence of the Beta and Gamma integrals']
---
[book: Weisstein, Eric W. - CRC concise encyclopedia of mathematics - 
ISBN 0-8493-9640-9 - p. 696 'Gamma function' (lot of formulas)]
---
[book: Wells, David - the Penguin Dictionary of Curious and 
Interesting numbers, p. 21]
---
[book: Wylie, Clarence R. - advanced engineering mathematics - McGraw-
Hill - p. 616 'The gamma and beta functions' (showing a graph of the 
gamma function)]
---
[book: Zwillinger, Daniel - CRC standard mathematical tables and 
formula - p. 494 'Gamma function']
---
---
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