The Museum of HP Calculators

# Gamma Function for the HP-65

This program is Copyright © 1974 by Hewlett-Packard and is used here by permission. This program was originally published in the HP-65 Pac 2.

This program is supplied without representation or warranty of any kind. Hewlett-Packard Company and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.

## Overview

This program approximates the value of gamma function gamma(x) for 1 <x<70.

gamma(x)  =tx-1e-t dt

## Formulas

gamma(x) = (x - 1)*gamma(x - 1)

For 1<= x <= 2, polynomial approximation can be used.
gamma(x) ~ 1 + b1 (x - 1) + b2 (x- 1)2 + . . . + b8 (x - 1)8

where
b1 =-0.577191652, b2 = 0.988205891
b3 =-0.897056937, b4 = 0.918206857
b5 = -0.756704078, b6 =0.482199394
b7 = -0.193527818, b8 = 0.035868343

Note: This program can be used to find the generalized factorial x! for 0 <= x <= 69.
x! =gamma(x+ 1)

## Instructions

 Step Instructions Input Data/Units Keys Output Data/Units 1 Enter program on card 1 2 Initialize A 3 Enter program on card 2 4 Enter x x A gamma(x) For a new value of x go to 4

## Examples

gamma(5.25) = 35.21
7! = gamma(8) = 5040.00
2.34! = gamma(3.34) = 2.80

## The Program

### Card 1

``` CODE  KEYS
23  LBL
11  A
83  .
00  0
03  3
05  5
08  8
06  6
08  8
03  3
04  4
03  3
33 01  STO 1
83  .
01  1
09  9
03  3
05  5
02  2
07  7
08  8
01  1
08  8
42  CHS
33 02  STO 2
83  .
04  4
08  8
02  2
01  1
09  9
09  9
03  3
09  9
04  4
33 03  STO 3
83  .
07  7
05  5
06  6
07  7
00  0
04  4
00  0
07  7
08  8
42  CHS
33 04  STO 4
83  .
09  9
01  1
08  8
02  2
00  0
06  6
08  8
05  5
07  7
33 05  STO 5
83  .
08  8
09  9
07  7
00  0
05  5
06  6
09  9
03  3
07  7
42  CHS
33 06  STO 6
83  .
09  9
08  8
08  8
02  2
00  0
05  5
08  8
09  9
01  1
33 07  STO 7
83  .
05  5
07  7
07  7
01  1
09  9
01  1
06  6
05  5
02  2
42  CHS
33 08  STO 8
84  R/S
35 01 g NOP
35 01 g NOP
35 01 g NOP
35 01 g NOP
35 01 g NOP
```

### Card 2

``` CODE  KEYS
23  LBL
11  A
41  ENTER
01  1
51  -
00  0
35 24  g x>y
00  0
81  ÷
35 07  g x<>y
01  1
33 08  STO 8
23  LBL
02  2
35 07  g x<>y
35 24  g x>y
22  GTO
01  1
41  ENTER
41  ENTER
41  ENTER
34 01  RCL 1
71  x
34 02  RCL 2
61  +
71  x
34 03  RCL 3
61  +
71  x
34 04  RCL 4
61  +
71  x
34 05  RCL 5
61  +
71  x
34 06  RCL 6
61  +
71  x
34 07  RCL 7
61  +
71  x
83  .
05  5
07  7
07  7
01  1
09  9
01  1
06  6
05  5
02  2
42  CHS
61  +
71  x
01  1
61  +
34 08  RCL 8
71  x
84  R/S
23  LBL
01  1
33  STO
71  x
08  8
35 07  g x<>y
51  -
01  1
22  GTO
02  2
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
35 01  g NOP
```

## Register Use

```R1  b8
R2  b7
R3  b6
R4  b5
R5  b4
R6  b3
R7  b2
R8  used
R9  used
```