. , , ,

,,,

- - — ,

1.

2.

2.1 -

2.2

3. -

4.

5.


, , , .

. .

:

.

- . Γ(z).

, - .

- , .


1.

-:

1. - n

(n) = (n - 1)! = 12... (n - 1). (1)

2. x>0 - .

. (2)

3. - :

(3)

1.

- (6).

:

6 , (1):

(6) =(6-1)! = 5! = 120

: 120.

2.

- (0,5).

:

(2):


.

.

: .

3.

- (1,5).

:

(3):

y = 1.5 + 2 = 3.5.

.

: .


2.

2.1 -

G(a) =(2.1)

.

1. -

=ty, t > 0 ,

G(a) =

, t 1+t ,


t 0 , :

,:

(2.2)

(2,1) ,

(2.3)


2. -

(2.4)

- , . n=1 (2.4)

2.2

- -. .

- x>0 ( z) :

.


, . g=5 n=6, , eps . , : Re z > 0.

- x>0 Gam(z+1)=z*Gam(z) Gam(z+1). , -, .

-, , ( ), -, - x, .

LnGam() - :

Ck ( 1).

k C
1 2.5066282746310005
2 1.0000000000190015
3 76.18009172947146
4 -86.50532032941677
5 24.01409824083091
6 -1.231739572450155
7 0.1208650973866179e-2
8 -0.5395239384953e-5

1. Ck

- . .


3 -

- 3, 4, 5, 6.

:

-   X ;

-   RS ;

-   GN ;

-   Y ;

-   RES -;

-   GAM , -.

3 GAMMA

4 GAMMA_ WHOLE


5 - - GAMMA_LN


6 - - GAMMA_POINT


4.

;

(SETQ CN '(2.5066282746310005 1.0000000000190015 76.18009172947146 -86.50532032941677 24.01409824083091

-1.231739572450155 0.1208650973866179e-2 -0.5395239384953e-5))

;

(DEFUN GAMMA_LN (X)

(SETQ SER (CADR CN))

(SETQ L (CDDR CN))

(SETQ Y X)

(DO

((J 2))

((>= J 8))

(SETQ Y (+ Y 1))

(SETQ CO (CAR L))

(SETQ SER (+ SER (/ CO Y)))

(SETQ L (CDR L))

(SETQ J (+ J 1))

)

(SETQ Y (+ X 5.5))

(SETQ Y (- Y (* (+ X 0.5) (LOG Y))))

(SETQ Y (+ (* -1 Y) (LOG (* (CAR CN) (/ SER X)))))

)

; -

;

(DEFUN GAMMA (X)

(EXP (GAMMA_LN X))

)

;

(DEFUN GAMMA_WHOLE (X)

(SETQ X (- X 1))

(DO

((RES 1) (RS 1))

((EQL X 0) RS)

(SETQ RS (* RES RS))

(SETQ X (- X 1))

(SETQ RES (+ RES 1))

)

)

;

(DEFUN GAMMA_POINT (X)

(IF (> X 0)

(PROGN

(SETQ Y (+ X 2))

(SETQ GAM (* (SQRT (* 2 (/ PI Y))) (EXP (+ (* Y (LOG Y)) (- (/ (- 1 (/ 1 (* 30 Y Y))) (* 12 Y)) Y)))))

(SETQ RES (/ GAM (* X (+ X 1))))

)

;

(PROGN

(SETQ J 0)

(SETQ Y X)

(DO

(())

((>= Y 0))

(SETQ J (+ J 1))

(SETQ Y (+ Y 1))

)

(SETQ GAM (GAMMA_POINT Y))

(DO

((I 0))

((>= I (- J 1)))

(SETQ GAM (/ GAM (+ X I)))

(SETQ I (+ I 1))

)

(SETQ RES GAM)

)

)

RES)

;

(SETQ FUNC 0)

(SETQ INPUT_STREAM (OPEN " D:\GAMMA.TXT" :DIRECTION :INPUT))

(SETQ FUNC (READ INPUT_STREAM))

(CLOSE INPUT_STREAM)

; -

(SETQ OUTPUT_STREAM (OPEN "D:\RESULT.TXT" :DIRECTION :OUTPUT))

(PRINT 'RESULT_OF_GAMMA_FUNCTION OUTPUT_STREAM)

;

(PRINT (MAPCAR 'GAMMA FUNC) OUTPUT_STREAM)

;

(PRINT (MAPCAR 'GAMMA_WHOLE FUNC) OUTPUT_STREAM)

;

(PRINT (MAPCAR 'GAMMA_POINT FUNC) OUTPUT_STREAM)

(TERPRI OUTPUT_STREAM)

(CLOSE OUTPUT_STREAM)

;END


5

1.

7 . -

8 . -

2.

9 . -


10 . -

3.

11 . -

12 . -


, . , , .

. - , - , - . .


1.            , .. [] / .., ... .: , 2007. 708 .

2.    - - [ ] : http://www.cyberguru.ru/cpp-sources/algorithms/vytchislenie-gamma-funktsii-i-beta-funktsii.html

3.    - [ ] : http://ru.wikipedia.org/wiki/_

4.            , .. : . [] / .., 3- .:-, 2006. C. 412.

5.            , .. . [] / .., ... .: , 2006. C. 346.

6.            , .. [] / .., .., ... : , 2002. 160 .

7.            , .. Lisp. [ ] / .., .. . .: , 2003. . 79.

8.            . [] / ., .. .: , 1990. 460 .

1. 2. 2.1 - 2.2 3. - 4.

 

 

 

! , , , .
. , :