Anionic Special Functions(IV) for the HP-41
"Fractional-Integro-Differentiation"Overview
1°) A few Elementary Functions
2°) Kummer's Functions
3°) Error Function
4°) Sine & Hyperbolic Sine Integral
5°) Cosine & Hyperbolic Cosine
Integral
6°) Exponential Integral Ei
7°) Generalized Laguerre's Functions
8°) Bessel & Modified Bessel Functions
of the 1st Kind
9°) Bessel & Modified Bessel Functions
of the 2nd Kind - Non-Integer Order
10°) Spherical Bessel Functions of the 1st Kind
11°) Hermite Functions
12°) Fresnel Integrals
13°) Airy Functions
-The fractional-integro-differentiation of a complex function is quite
well defined.
-For quaternions, octonions and so on, this is more doubtful:
-Even the 1st derivative of a2 is not 2.a
because the multiplication is not commutative in general.
-So, these functions may be regarded as other special functions not
always related to differentiation of special functions... except for complex
arguments !
-I've used M-Code routines which may be replaced by focal programs with
small modifications in the listings below
1°) A few Elementary Functions
Formulae:
• Real Power Dµ ax = [ x ! / Gam(x-µ+1) ] ax-µ assuming x # -1 , -2 , -3 , .....
• Hyperbolic Sine Dµ Sinh a = 2µ-1 sqrt(PI) a1-µ1F~2 ( 1 ; (2-µ)/2 , (3-µ)/2 ; a2/4 )
• Hyperbolic Cosine Dµ Cosh a = (2/a)µ sqrt(PI) 1F~2 ( 1 ; (1-µ)/2 , (2-µ)/2 ; a2/4 )
• Sine Dµ Sin a = 2µ-1 sqrt(PI) a1-µ1F~2 ( 1 ; (2-µ)/2 , (3-µ)/2 ; -a2/4 )
• Cosine Dµ Cos x = (2/a)µ sqrt(PI) 1F~2 ( 1 ; (1-µ)/2 , (2-µ)/2 ; -a2/4 )
• Logarithm Dµ Ln a = a -µ FC(µ)log (a)
where
FC(µ)log (a) = (-1)µ-1 (µ-1)
!
if µ is a positive integer
and
FC(µ)log (a) = [ Ln a - Psi(1-µ) - gamma
] / Gam(1-µ) otherwise
Psi = Digamma Function , gamma = Euler's constant = 0.5772156649... and Gam = Gamma Function.
• Exponential
Dµ Exp a = a -µ 1F~1
( 1 ; 1-µ ; a )
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing these routines )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R4n: temp ( sometimes )
>>> When the programs stop: R01
...... Rnn = the n components of the result
Flags: F09 & F10 ( for "DSHA" "DCHA" "DSINA" "DCOSA" )
Subroutines: A*A1 "1/A" "A^X"
"LNA"
( cf "Anions for the HP41" )
ST*A ST/A AMOVE ASWAP
( cf "Anionic Special Functions(I) for the HP41" )
"HGFB"
( cf "Anionic Special Functions(II) for the HP41" )
GAM 1/G+ PSI
( cf "Gamma, Digamma Functions for the HP41" )
| 01 LBL "DA^X"
02 STO M 03 X<>Y 04 - 05 STO N 06 XEQ "A^X" 07 RCL M 08 1 09 + 10 GAM 11 RCL N 12 1 13 + 14 1/G+ 15 * 16 ST*A 17 X<>Y 18 CLA 19 RTN 20 LBL "DSHA" 21 CF 09 22 CF 10 23 GTO 00 24 LBL "DCHA" 25 SF 09 26 CF 10 27 GTO 00 28 LBL "DSINA" 29 CF 09 30 SF 10 |
31 GTO 00
32 LBL "DCOSA" 33 SF 09 34 SF 10 35 LBL 00 36 STO M 37 12 38 AMOVE 39 14 40 AMOVE 41 A*A1 42 4 43 FS?C 10 44 CHS 45 ST/A 46 CLX 47 SIGN 48 ENTER^ 49 FC?C 09 50 ST+ X 51 RCL M 52 - 53 RCL X 54 1 55 + 56 2 57 ST/ Z 58 / 59 1.002 60 CHS |
61 XEQ "HGFB"
62 X<> Z 63 ST+ X 64 1 65 - 66 STO M 67 41 68 AMOVE 69 X<>Y 70 XEQ "A^X" 71 2 72 RCL M 73 CHS 74 Y^X 75 PI 76 SQRT 77 * 78 ST*A 79 32 80 AMOVE 81 A*A1 82 CLA 83 RTN 84 LBL "DLNA" 85 STO M 86 12 87 AMOVE 88 X<>Y 89 CHS 90 XEQ "A^X" |
91 STO N
92 RCL M 93 FRC 94 X#0? 95 GTO 01 96 LASTX 97 X<=0? 98 GTO 01 99 SIGN 100 CHS 101 RCL M 102 1 103 - 104 Y^X 105 LASTX 106 FACT 107 * 108 ST*A 109 RCL N 110 CLA 111 RTN 112 LBL 01 113 12 114 ASWAP 115 XEQ "LNA" 116 .5772156649 117 1 118 RCL M 119 - 120 PSI |
121 ST+ Y
122 X<>Y 123 ST- 01 124 LASTX 125 1/G+ 126 ST*A 127 A*A1 128 CLA 129 RTN 130 LBL "DE^A" 131 14 132 AMOVE 133 SIGN 134 1 135 RCL Z 136 - 137 1.001 138 CHS 139 XEQ "HGFB" 140 R^ 141 41 142 AMOVE 143 X<>Y 144 CHS 145 XEQ "A^X" 146 32 147 AMOVE 148 A*A1 149 END |
( 315 bytes )
| STACK | INPUTS | OUTPUTS | INPUT | OUTPUT |
| Y | µ | / | / | / |
| X | x | 1.nnn | µ | 1.nnn |
Where 1.nnn is the control number of the result f(a) ; x is required for "DA^X" only
Examples: With a = 1.1
+ 1.2 i + 1.3 j + 1.4 k and µ =
-PI
( quaternion -> 4 STO 00 )
• "DA^X" if x = e f(a) = 1.630330082 + 0.233604201 i + 0.253071217 j + 0.272538234 k ---Execution time = 6s---
• "DSHA" f(a) = 0.093634336 - 0.712248302 i - 0.771602327 j - 0.830956352 k ---Execution time = 25s---
• "DCHA" f(a) = -1.721108064 - 0.629274549 i - 0.681714094 j - 0.734153640 k ---Execution time = 27s---
• "DSINA" f(a) = -0.415438235 - 0.886075931 i - 0.959915592 j - 1.033755253 k ---Execution time = 25s---
• "DCOSA" f(a) = -2.980759335 - 0.217479859 i - 0.235603180 j - 0.253726502 k ---Execution time = 26s---
• "DLNA" f(a) = 3.234305049 - 0.932622695 i - 1.010341253 j - 1.088059811 k ---Execution time = 10s---
with µ = 3 f(a) = -0.123704743 + 0.014013365 i +0.015181146 j + 0.016348926 k ---Execution time = 4s---
• "DE^A" f(a) =
-1.627473729 - 1.341522850 i - 1.453316422 j - 1.565109992 k
---Execution time = 29s---
2°) Kummer's Functions
Formula:
Dµ M(p;q;a) = a -µ Gam(q) 2F~2
( 1 , p ; 1-µ , q ; a )
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DKUMA" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R2n: temp R2n+1 ..... R3n = a
>>> When the programs stop: R01
...... Rnn = the n components of the result
Flags: /
Subroutines: A*A1
( cf "Anions for the HP41" )
ST*A DSA AMOVE ASWAP ( cf "Anionic
Special Functions(I) for the HP41" )
1/G+
( cf "Gamma, Digamma Functions for the HP41" )
X+1 ST<>A
( cf "A few M-Code Routines for the HP-41" )
-Lines 45-46 may be replaced by RCL O RCL N RCL M
-Line 18 may be replaced by PI SIGN +
-Line 02 may be replaced by STO M RDN STO N
X<>Y STO O
| 01 LBL "DKUMA"
02 ST<>A 03 12 04 AMOVE 05 CLX 06 ST*A 07 SIGN 08 STO 01 09 13 10 AMOVE |
11 CLX
12 STO P 13 LBL 01 14 A*A1 15 RCL N 16 RCL O 17 CHS 18 X+1 19 RCL M 20 RCL P |
21 ST+ T
22 ST+ Z 23 + 24 * 25 / 26 ISG P 27 CLX 28 ST*A 29 DSA 30 X#0? |
31 GTO 01
32 21 33 MOVE 34 RCL O 35 CHS 36 XEQ "A^X" 37 RCL O 38 CHS 39 X+1 40 1/G+ |
41 ST*A
42 23 43 ASWAP 44 A*A1 45 RDN 46 ST<>A 47 R^ 48 CLA 49 END |
( 93 bytes / SIZE 3n+1 )
| STACK | INPUTS | OUTPUTS |
| T | / | µ |
| Z | µ | p |
| Y | p | q |
| X | q | 1.nnn |
Where 1.nnn is the control number of the result
Examples: µ = PI p = sqrt(2) q = sqrt (3) a = 0.4 + 0.5 i + 0.6 j + 0.7 k ( quaternion -> 4 STO 00 )
0.4 STO 01
0.5 STO 02
0.6 STO 03
0.7 STO 04
PI 2 SQRT
3 SQRT XEQ "DKUMA"
>>>> 1.004
---Execution time = 38s---
R01 = 0.630946964
R02 = 0.307230135
R03 = 0.368676162
R04 = 0.430122188
-Whence F(a) = 0.630946964 + 0.307230135 i + 0.368676162 j + 0.430122188 k
Note:
-This program does not work if µ = 1 , 2 , 3 , .....
3°) Error Function
Formula:
Dµ Erf (a) = 2µ a1-µ2F~2
[ 1/2 , 1 ; (2-µ)/2 , (3-µ)/2 ; -a2 ]
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DERFA" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R3n: temp
>>> When the programs stop: R01
...... Rnn = the n components of the result
Flags: /
Subroutines: A*A1 "A^X"
( cf "Anions for the HP41" )
ST*A DSA AMOVE ASWAP ( cf "Anionic
Special Functions(I) for the HP41" )
1/G+
( cf "Gamma, Digamma Functions for the HP41" )
-Line 62 may be replaced with E3 /
| 01 LBL "DERFA"
02 STO M 03 .5 04 STO N 05 * 06 STO O 07 12 08 AMOVE 09 SIGN 10 CHS 11 ST*A 12 A*A1 13 12 |
14 ASWAP
15 SIGN 16 RCL M 17 - 18 XEQ "A^X" 19 SIGN 20 RCL O 21 - 22 1/G+ 23 RCL N 24 LASTX 25 + 26 1/G+ |
27 *
28 2 29 RCL M 30 Y^X 31 * 32 ST*A 33 13 34 AMOVE 35 CLX 36 STO P 37 LBL 01 38 A*A1 39 RCL N |
40 ENTER^
41 SIGN 42 RCL O 43 - 44 RCL X 45 RCL N 46 + 47 RCL P 48 ST+ T 49 ST+ Z 50 + 51 * 52 / |
53 ST*A
54 ISG P 55 CLX 56 DSA 57 X#0? 58 GTO 01 59 31 60 AMOVE 61 RCL 00 62 X/E3 63 ISG X 64 CLA 65 END |
( 115 bytes
/ SIZE 3n+1 )
| STACK | INPUT | OUTPUT |
| X | µ | 1.nnn |
Where 1.nnn is the control number of the result
Examples: µ = PI , a = 0.4 + 0.5 i + 0.6 j + 0.7 k ( quaternion -> 4 STO 00 )
0.4 STO 01
0.5 STO 02
0.6 STO 03
0.7 STO 04
PI XEQ "DERFA" >>>> 1.004 ---Execution time = 36s---
R01 = 0.924128302
R02 = 10.81672321
R03 = 12.98006783
R04 = 15.14341247
-Whence, Dµ Erf (a) = 0.924128302 + 10.81672321 i + 12.98006783 j + 15.14341247 k
Note:
-This program does not work if µ = 2 , 3 , 4 , .....
4°) Sine & Hyperbolic Sine Integral
Formulae:
• Sine Integral Dµ Si a = 2µ-2 PI a1-µ2F~3 ( 1/2 , 1 ; 3/2 , (2-µ)/2 , (3-µ)/2 ; -a2/4 )
• Hyperbolic Sine Integral
Dµ Shi x = 2µ-2 PI a1-µ2F~3
( 1/2 , 1 ; 3/2 , (2-µ)/2 , (3-µ)/2 ; a2/4 )
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DSIA" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R3n: temp
>>> When the programs stop: R01
...... Rnn = the n components of the result
Flag: F01
CF 01 <-> Sine Integral
SF 01 <-> Hyperbolic Sine Integral
Subroutines: A*A1 "A^X"
( cf "Anions for the HP41" )
ST*A ST/A DSA AMOVE ASWAP
( cf "Anionic Special Functions(I) for the HP41" )
1/G+
( cf "Gamma, Digamma Functions for the HP41" )
X+1
( cf "A few M-Code Routines for the HP-41" )
Line 66 may be replaced with E3 /
X+1 ( line 49 ) may be replaced by ISG X
CLX
| 01 LBL "DSIA"
02 CLA 03 STO M 04 12 05 AMOVE 06 4 07 FC? 01 08 CHS 09 ST/A 10 A*A1 11 12 12 ASWAP 13 SIGN 14 RCL M |
15 -
16 XEQ "A^X" 17 2 18 RCL M 19 .5 20 CHS 21 ST* M 22 STO N 23 SIGN 24 + 25 Y^X 26 PI 27 SQRT 28 * |
29 1
30 RCL M 31 + 32 1/G+ 33 * 34 1.5 35 RCL M 36 + 37 1/G+ 38 * 39 ST*A 40 13 41 AMOVE 42 LBL 01 |
43 A*A1
44 RCL N 45 ENTER^ 46 CHS 47 RCL M 48 RCL O 49 X+1 50 STO O 51 ST+ T 52 ST+ Z 53 + 54 ST* Y 55 RCL N 56 - |
57 *
58 / 59 ST*A 60 DSA 61 X#0? 62 GTO 01 63 31 64 AMOVE 65 RCL 00 66 X/E3 67 ISG X 68 CLA 69 END |
(
121 bytes / SIZE 3n+1 )
| STACK | INPUT | OUTPUT |
| X | µ | 1.nnn |
Where 1.nnn is the control number of the result
Examples: µ = PI , a = 1 + 2 i + 3 j + 4 k ( quaternion -> 4 STO 00 )
CF 01 Dµ Si(a) = -6.216067679 + 4.558034704 i + 6.837052053 j + 9.116069403 k ---Execution time = 32s---
SF 01 Dµ Shi(a) = -0.078381628
- 0.067284573 i - 0.100926858 j - 0.134569145 k
---Execution time = 33s---
Note:
-This routine does not work for µ = 2 , 3 , 4 , .....
5°) Cosine & Hyperbolic Cosine Integral
Formulae:
• Cosine Integral
Dµ Ci a = [ (-1)µ-1 (µ-1) ! ] a -µ - 2µ-3 sqrt(PI) a2-µ 2F~3 ( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; -a2/4 ) if µ is a positive integer
Dµ Ci a
= [ Ln a - Psi(1-µ) ] a -µ - 2µ-3
sqrt(PI) a2-µ 2F~3
( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; -a2/4 )
otherwise
• Hyperbolic Cosine Integral
Dµ Chi a = [ (-1)µ-1 (µ-1) ! ] a -µ + 2µ-3 sqrt(PI) a2-µ 2F~3 ( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; a2/4 ) if µ is a positive integer
Dµ
Chi a = [ Ln a - Psi(1-µ) ] a -µ + 2µ-3
sqrt(PI) a2-µ 2F~3
( 1 , 1 ; 2 , (3-µ)/2 , (4-µ)/2 ; a2/4 )
otherwise
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DCIA" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R4n: temp
>>> When the programs stop: R01 ...... Rnn = the n components of the result
Flag: F01
CF 01 <-> Cosine Integral
SF 01 <-> Hyperbolic Cosine Integral
Subroutines: A*A1 A+A
"A^X" "LNA"
( cf "Anions for the HP41" )
ST*A ST/A DSA AMOVE ASWAP
( cf "Anionic Special Functions(I) for the HP41" )
1/G+ PSI
( cf "Gamma, Digamma Functions for the HP41" )
-Line 35 may be replaced by ISG X CLX
-Line 103 ------------------ 1 -
| 01 LBL "DCIA"
02 CLA 03 STO M 04 12 05 AMOVE 06 14 07 AMOVE 08 4 09 FC? 01 10 CHS 11 ST/A 12 A*A1 13 12 14 AMOVE 15 .5 16 STO N 17 CLX 18 ST*A 19 SIGN 20 STO 01 21 13 22 AMOVE 23 LBL 01 24 A*A1 25 SIGN 26 ENTER^ |
27 ST+ X
28 RCL O 29 ST+ Z 30 + 31 / 32 PI 33 INT 34 ENTER^ 35 X+1 36 RCL M 37 ST- Z 38 - 39 RCL N 40 ST* Z 41 * 42 RCL O 43 ST+ Z 44 + 45 * 46 / 47 ST*A 48 ISG O 49 CLX 50 DSA 51 X#0? 52 GTO 01 |
53 32
54 AMOVE 55 41 56 AMOVE 57 2 58 RCL M 59 - 60 XEQ "A^X" 61 A*A1 62 3 63 RCL M 64 - 65 4 66 LASTX 67 - 68 2 69 ST/ Z 70 / 71 1/G+ 72 X<>Y 73 1/G+ 74 * 75 2 76 RCL M 77 3 78 - |
79 Y^X
80 * 81 PI 82 SQRT 83 * 84 FC? 01 85 CHS 86 ST*A 87 13 88 AMOVE 89 41 90 AMOVE 91 RCL M 92 CHS 93 XEQ "A^X" 94 RCL M 95 FRC 96 X#0? 97 GTO 02 98 LASTX 99 X<=0? 100 GTO 02 101 -1 102 LASTX 103 X-1 104 Y^X |
105 LASTX
106 FACT 107 * 108 ST*A 109 GTO 03 110 LBL 02 111 12 112 AMOVE 113 41 114 AMOVE 115 XEQ "LNA" 116 SIGN 117 RCL M 118 - 119 PSI 120 ST- 01 121 LASTX 122 1/G+ 123 ST*A 124 A*A1 125 LBL 03 126 32 127 AMOVE 128 A+A 129 CLA 130 END |
( 219 bytes / SIZE 4n+1 )
| STACK | INPUT | OUTPUT |
| X | µ | 1.nnn |
Where 1.nnn is the control number of the result
Examples: µ = PI , a = 1.1 + 1.2 i + 1.3 j + 1.4 k ( quaternion -> 4 STO 00 )
CF 01 Dµ Ci(a) = 0.770099307 + 0.231600213 i + 0.250900230 j + 0.270200248 k ---Execution time = 43s---
SF 01 Dµ Chi(a) = -0.154706962
+ 0.244319821 i + 0.264679806 j + 0.285039791 k
---Execution time = 48s---
Note:
-This program does not work for µ = 3 , 4 , 5 , .....
6°) Exponential Integral Ei
Formulae:
Dµ Ei a = [ (-1)µ-1 (µ-1) ! ] a -µ + a1-µ 2F~2 ( 1 , 1 ; 2 , 2-µ ; a ) if µ is a positive integer
Dµ Ei a = [ Ln a - Psi(1-µ) ] a
-µ + a1-µ 2F~2
( 1 , 1 ; 2 , 2-µ ; a ) otherwise
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DEIA" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R3n: temp
>>> When the programs stop: R01 ...... Rnn = the n components of the result
Flags: /
Subroutines: A*A1 A+A
"A^X"
( cf "Anions for the HP41" )
ST*A DSA AMOVE ASWAP ( cf "Anionic
Special Functions(I) for the HP41" )
1/G+ PSI
( cf "Gamma, Digamma Functions for the HP41" )
-Line 63 may be replaced by 1 -
-Lines 79 to 81 may be replaced by LASTX
| 01 LBL "DEIA"
02 CLA 03 STO M 04 12 05 AMOVE 06 CLX 07 ST*A 08 SIGN 09 STO 01 10 13 11 AMOVE 12 LBL 01 13 A*A1 14 SIGN 15 ENTER^ 16 ST+ X 17 RCL X 18 RCL M |
19 -
20 RCL N 21 ST+ T 22 ST+ Z 23 + 24 * 25 / 26 ST*A 27 ISG N 28 CLX 29 DSA 30 X#0? 31 GTO 01 32 21 33 AMOVE 34 SIGN 35 RCL M 36 - |
37 XEQ "A^X"
38 23 39 ASWAP 40 A*A1 41 2 42 RCL M 43 - 44 1/G+ 45 ST*A 46 13 47 ASWAP 48 12 49 AMOVE 50 RCL M 51 CHS 52 XEQ "A^X" 53 RCL M 54 FRC |
55 X#0?
56 GTO 02 57 LASTX 58 X<=0? 59 GTO 02 60 1 61 CHS 62 LASTX 63 X-1 64 Y^X 65 LASTX 66 FACT 67 * 68 ST*A 69 GTO 03 70 LBL 02 71 12 72 ASWAP |
73 XEQ "LNA"
74 SIGN 75 RCL M 76 - 77 PSI 78 ST- 01 79 1 80 RCL M 81 - 82 1/G+ 83 ST*A 84 A*A1 85 LBL 03 86 32 87 AMOVE 88 A+A 89 CLA 90 END |
( 160 bytes / SIZE 3n+1 )
| STACK | INPUT | OUTPUT |
| X | µ | 1.nnn |
Where 1.nnn is the control number of the result
Example: µ = PI , a = 1.1 + 1.2 i + 1.3 j + 1.4 k ( quaternion -> 4 STO 00 )
Dµ Ei(a) = -0.272860689
+ 0.388792411 i + 0.421191778 j + 0.453591146 k
---Execution time = 51s---
Note:
-"DEIA" does not work for µ = 2 , 3 , 4 , .....
7°) Generalized Laguerre's Functions
Formula:
Dµ Lpq (a)
= [ Gam(p+q+1)/Gam(q+1) ] a -µ2F~2
( 1 , -q ; p+1 , 1-µ ; a )
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DLANA" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R3n: temp
>>> When the programs stop: R01 ...... Rnn = the n components of the result
Flags: /
Subroutines: A*A1 "A^X"
( cf "Anions for the HP41" )
ST*A DSA AMOVE ( cf "Anionic
Special Functions(I) for the HP41" )
1/G+
( cf "Gamma, Digamma Functions for the HP41" )
-Line 52 may be replaced with E3 /
-Line 38 may be replaced by ISG P CLX
-Line 03 may be replaced with STO M RDN STO N
X<>Y STO O
-Lines 02 & 22 may be replaced by 1 +
| 01 LBL "DLANA"
02 X+1 03 ST<>A 04 12 05 AMOVE 06 RCL O 07 CHS 08 XEQ "A^X" 09 RCL M 10 1/G+ 11 LASTX |
12 RCL N
13 + 14 1/G+ 15 / 16 1 17 RCL O 18 - 19 1/G+ 20 * 21 RCL N 22 X+1 |
23 1/G+
24 * 25 ST*A 26 13 27 AMOVE 28 CLX 29 STO P 30 LBL 01 31 A*A1 32 RCL M 33 CHS |
34 RCL O
35 CHS 36 RCL N 37 RCL P 38 X+1 39 STO P 40 ST+ T 41 ST+ Z 42 + 43 * 44 / |
45 ST*A
46 DSA 47 X#0? 48 GTO 01 49 31 50 AMOVE 51 RCL 00 52 X/E3 53 ISG X 54 CLA 55 END |
( 103 bytes / SIZE 3n+1 )
| STACK | INPUTS | OUTPUTS |
| Z | µ | / |
| Y | p | / |
| X | q | 1.nnn |
Where 1.nnn is the control number of the result
Examples: µ = -PI p = sqrt(2) q = sqrt (3) a = 1 + 2 i + 3 j + 4 k ( quaternion -> 4 STO 00 )
1 STO 01
2 STO 02
3 STO 03
4 STO 04
PI CHS 2 SQRT 3 SQRT XEQ "DLANA" >>>> 1.004 ---Execution time = 34s---
R01 = -108.7414887
R02 = -11.05061335
R03 = -16.57592004
R04 = -22.10122675
-Whence, Dµ Lpq (a) = -108.7414887 - 11.05061335 i - 16.57592004 j - 22.10122675 k
Note:
µ must be different from 1 , 2 , 3 , .....
8°) Bessel & Modified Bessel Functions of
the 1st Kind
Formulae:
• Dµ Jm (a) = 2µ-2m sqrt(PI) a-m-µ Gam(m+1) 2F~3 [ (m+1)/2 , (m+2)/2 ; (m+1-µ)/2 , (m+2-µ)/2 , m+1 ; -a2/4 ]
• Dµ
Im (a) = 2µ-2m sqrt(PI) am-µ
Gam(m+1) 2F~3 [ (m+1)/2 , (m+2)/2
; (m+1-µ)/2 , (m+2-µ)/2 , m+1 ; a2/4 ]
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DJNA" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R3n: temp
>>> When the programs stop: R01 ...... Rnn = the n components of the result
Flag: F01 CF 01 <->
Bessel functions of the 1st kind
SF 01 <-> Modified Bessel Functions of the 1st kind
Subroutines: A*A1 "A^X"
( cf "Anions for the HP41" )
ST*A ST/A DSA AMOVE ASWAP
( cf "Anionic Special Functions(I) for the HP41" )
1/G+
( cf "Gamma, Digamma Functions for the HP41" )
-Line 82 may be replaced with E3 /
-Line 70 may be replaced by ISG X CLX
-Line 29 may be replaced by 1 +
| 01 LBL "DJNA"
02 CLA 03 STO N 04 X<>Y 05 STO M 06 12 07 AMOVE 08 4 09 FC? 01 10 CHS 11 ST/A 12 A*A1 13 12 14 ASWAP 15 RCL N 16 RCL M 17 - |
18 XEQ "A^X"
19 PI 20 SQRT 21 2 22 RCL N 23 ST+ X 24 RCL M 25 - 26 Y^X 27 / 28 RCL N 29 X+1 30 STO N 31 RCL M 32 - 33 2 34 / |
35 STO M
36 1/G+ 37 * 38 RCL M 39 .5 40 + 41 1/G+ 42 * 43 ST*A 44 13 45 AMOVE 46 .5 47 STO P 48 LBL 01 49 A*A1 50 RCL N 51 RCL P |
52 *
53 STO Y 54 LASTX 55 + 56 RCL M 57 RCL O 58 ST+ T 59 ST+ Z 60 + 61 / 62 * 63 RCL M 64 RCL P 65 + 66 RCL N 67 RCL O 68 ST+ Z |
69 ST+ Y
70 X+1 71 STO O 72 * 73 * 74 / 75 ST*A 76 DSA 77 X#0? 78 GTO 01 79 31 80 AMOVE 81 RCL 00 82 X/E3 83 ISG X 84 CLA 85 END |
( 149 bytes / SIZE 3n+1
)
| STACK | INPUTS | OUTPUTS |
| Y | µ | / |
| X | p | 1.nnn |
Where 1.nnn is the control number of the result
Examples: µ = PI p = sqrt(2) a = 1.1 + 1.2 i + 1.3 j + 1.4 k ( quaternion -> 4 STO 00 )
• CF 01 Dµ Jm (a) = -1.115289298 + 0.405457582 i + 0.439245714 j + 0.473033847 k ---Execution time = 30s---
• SF 01 Dµ
Im (a) = -0.213651981 + 0.178633476 i + 0.193519599 j + 0.208405722
k
---Execution time = 30s---
9°) Bessel & Modified Bessel Functions of
the 2nd Kind - Non-Integer Order
Formulae:
• Dµ Ym (a) = 2µ-2m (PI)1/2 a-µ-m csc(m.PI) { -16m Gam(1-m) 2F~3 [ (1-m)/2 , (2-m)/2 ; (1-µ-m)/2 , (2-µ-m)/2 , 1-m ; -a2/4 ]
+ x2m Cos(m.PI) Gam(m+1) 2F~3 [ (m+1)/2 , (m+2)/2 ; (m+1-µ)/2 , (m+2-µ)/2 , m+1 ; -a2/4 ] } where m is not an integer.
• Dµ Km (a) = 2µ-2m-1 (PI)3/2 a-µ-m csc(m.PI) { 16m Gam(1-m) 2F~3 [ (1-m)/2 , (2-m)/2 ; (1-µ-m)/2 , (2-µ-m)/2 , 1-m ; x2/4 ]
- x2m Gam(m+1) 2F~3 [
(m+1)/2 , (m+2)/2 ; (m+1-µ)/2 , (m+2-µ)/2 , m+1 ; a2/4
] } where m is not an integer.
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DYNA" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R5n: temp
>>> When the programs stop: R01 ...... Rnn = the n components of the result
Flag: F01 CF 01 <->
Bessel functions of the 2nd kind
SF 01 <-> Modified Bessel Functions of the2nd kind
Subroutines: A*A1 A-A
"A^X"
( cf "Anions for the HP41" )
ST*A ST/A DSA AMOVE ASWAP
( cf "Anionic Special Functions(I) for the HP41" )
1/G+
( cf "Gamma, Digamma Functions for the HP41" )
-Line 124 may be replaced with 2 /
-Line 54 may be replaced by ISG X CLX
-Line 34 may be replaced by PI SIGN +
| 01 LBL "DYNA"
02 STO N 03 X<>Y 04 STO M 05 .5 06 STO O 07 14 08 AMOVE 09 12 10 AMOVE 11 4 12 FC? 01 13 CHS 14 ST/A 15 A*A1 16 12 17 AMOVE 18 CLX 19 ST*A 20 SIGN 21 STO 01 22 GTO 02 23 LBL 00 24 13 25 AMOVE 26 CLX |
27 STO P
28 LBL 01 29 A*A1 30 SIGN 31 RCL N 32 + 33 ENTER^ 34 X+1 35 RCL Y 36 RCL M 37 - 38 RCL O 39 ST* T 40 ST* Z 41 * 42 RCL P 43 ST+ T 44 ST+ Z 45 + 46 STO T 47 / 48 * 49 X<>Y 50 RCL O 51 + 52 RCL N |
53 RCL P
54 X+1 55 ST+ Y 56 STO P 57 * 58 * 59 / 60 ST*A 61 DSA 62 X#0? 63 GTO 01 64 31 65 AMOVE 66 RCL N 67 RCL M 68 - 69 RCL O 70 ST* Y 71 + 72 1/G+ 73 LASTX 74 RCL O 75 + 76 1/G 77 * 78 RTN |
79 LBL 02
80 XEQ 00 81 RCL N 82 1 83 CHS 84 ACOS 85 * 86 FS? 01 87 CLX 88 COS 89 * 90 ST*A 91 23 92 AMOVE 93 12 94 AMOVE 95 41 96 AMOVE 97 RCL N 98 ST+ X 99 XEQ "A^X" 100 A*A1 101 15 102 AMOVE 103 32 104 AMOVE |
105 CLX
106 ST*A 107 SIGN 108 STO 01 109 CHS 110 ST* N 111 XEQ 00 112 16 113 RCL N 114 CHS 115 Y^X 116 * 117 ST*A 118 52 119 AMOVE 120 A-A 121 SIGN 122 CHS 123 PI 124 X/2 125 FC? 01 126 X<>Y 127 PI 128 SQRT 129 * 130 2 |
131 RCL M
132 RCL N 133 ST+ X 134 + 135 Y^X 136 * 137 1 138 CHS 139 ACOS 140 RCL N 141 CHS 142 * 143 SIN 144 / 145 ST*A 146 12 147 AMOVE 148 41 149 AMOVE 150 RCL N 151 RCL M 152 - 153 XEQ "A^X" 154 A*A1 155 CLA 156 END |
( 263 bytes / SIZE 5n+1
)
| STACK | INPUTS | OUTPUTS |
| Y | µ | / |
| X | m | 1.nnn |
Where 1.nnn is the control number of the result and µ is not an integer
Examples: µ = PI m = sqrt(2) a = 1.1 + 1.2 i + 1.3 j + 1.4 k ( quaternion -> 4 STO 00 )
• CF 01 Dµ Ym (a) = -0.710530741 - 0.501187143 i - 0.542952739 j - 0.584718334 k ---Execution time = 72s---
• SF 01 Dµ
Km (a) = 0.019073486 - 0.197769632 i - 0.214250435 j -
0.230731237 k
---Execution time = 70s---
10°) Spherical Bessel Functions of the 1st Kind
Formula:
Dµ jn
(a) = 2µ-2m-1 PI am-µ Gam(m+1)
2F~3 [ (m+1)/2 , (m+2)/2 ; (m+1-µ)/2
, (m+2-µ)/2 , m+3/2 ; -a2/4 ]
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DBS1A" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R3n: temp
>>> When the programs stop: R01 ...... Rnn = the n components of the result
Flags: /
Subroutines: A*A1 "A^X"
( cf "Anions for the HP41" )
ST*A ST/A DSA AMOVE ASWAP
( cf "Anionic Special Functions(I) for the HP41" )
1/G+
( cf "Gamma, Digamma Functions for the HP41" )
-Line 96 may be replaced by E3 /
-Line 91 may be replaced with 1 +
-Lines 62 to 67 may be replaced with SIGN RCL N
+ RCL M - RCL X 1 +
-Line 49 may be replaced by ISG X CLX
-Lines 25 to 27 may be replaced by PI SIGN +
RCL X LASTX +
| 01 LBL "DBS1A"
02 CLA 03 STO N 04 X<>Y 05 STO M 06 12 07 AMOVE 08 A*A1 09 4 10 CHS 11 ST/A 12 12 13 ASWAP 14 RCL N 15 RCL M 16 - 17 XEQ "A^X" 18 13 19 AMOVE 20 .5 |
21 STO P
22 LBL 01 23 A*A1 24 RCL N 25 X+1 26 ENTER^ 27 X+1 28 RCL Y 29 RCL M 30 - 31 RCL P 32 ST* T 33 ST* Z 34 * 35 RCL O 36 ST+ T 37 ST+ Z 38 + 39 / 40 * |
41 RCL N
42 RCL M 43 - 44 RCL N 45 RCL P 46 ST* Z 47 + 48 RCL O 49 X+1 50 STO O 51 ST+ Z 52 ST+ Y 53 * 54 * 55 / 56 ST*A 57 DSA 58 X#0? 59 GTO 01 60 31 |
61 AMOVE
62 RCL N 63 X+1 64 RCL M 65 - 66 ENTER^ 67 X+1 68 2 69 ST/ Z 70 / 71 1/G+ 72 X<>Y 73 1/G+ 74 * 75 RCL N 76 1.5 77 + 78 1/G+ 79 * 80 PI |
81 *
82 2 83 ST/ Y 84 RCL M 85 RCL N 86 ST+ X 87 - 88 Y^X 89 * 90 RCL N 91 X+1 92 1/G+ 93 / 94 ST*A 95 RCL 00 96 X/E3 97 ISG X 98 CLA 99 END |
( 175 bytes / SIZE 3n+1
)
| STACK | INPUTS | OUTPUTS |
| Y | µ | / |
| X | m | 1.nnn |
Where 1.nnn is the control number of the result
Example: µ = PI m = sqrt(2) a = 1.1 + 1.2 i + 1.3 j + 1.4 k ( quaternion -> 4 STO 00 )
Dµ jm (a)
= -0.513818255 + 0.162870796 i + 0.176443362 j + 0.190015928 k
---Execution time = 37s---
11°) Hermite Functions
Formula:
Dµ
Hm (a) = [ 2m+µ (PI) a-µ /
Gam((1-m)/2) ] 2F~2 [ 1 , -m/2 ; (1-µ)/2
, (2-µ)/2 ; a2 ]
- [ 2m+µ (PI) a1-µ / Gam((-m)/2) ] 2F~2
[ 1 , (1-m)/2 ; 1-µ/2 , (3-µ)/2 ; a2 ]
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DHMTA" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R4n: temp
>>> When the programs stop: R01 ...... Rnn = the n components of the result
Flag: F10
Subroutines: A*A1 A-A
"A^X"
( cf "Anions for the HP41" )
ST*A ST/A DSA AMOVE ASWAP
( cf "Anionic Special Functions(I) for the HP41" )
1/G+
( cf "Gamma, Digamma Functions for the HP41" )
-Line 66 may be replaced by 1 -
-Lines 82-72-67-58-55 may be replaced with 2 /
-Lines 28 to 30 may be replaced by PI SIGN +
RCL X LASTX +
| 01 LBL "DHMTA"
02 CHS 03 STO N 04 X<>Y 05 CHS 06 STO M 07 12 08 AMOVE 09 14 10 AMOVE 11 A*A1 12 12 13 ASWAP 14 SF 10 15 LBL 01 16 RCL M 17 XEQ "A^X" 18 13 19 AMOVE 20 CLX |
21 STO O
22 .5 23 STO P 24 LBL 02 25 A*A1 26 RCL N 27 RCL M 28 X+1 29 ENTER^ 30 X+1 31 RCL P 32 ST* T 33 ST* Z 34 * 35 RCL O 36 ST+ T 37 ST+ Z 38 + 39 * 40 / |
41 ST*A
42 ISG O 43 CLX 44 DSA 45 X#0? 46 GTO 02 47 31 48 AMOVE 49 FC?C 10 50 RTN 51 SIGN 52 ST+ M 53 ST+ N 54 RCL M 55 X/2 56 1/G+ 57 RCL N 58 X/2 59 1/G+ 60 * |
61 ST*A
62 14 63 ASWAP 64 XEQ 01 65 RCL N 66 X-1 67 X/2 68 1/G+ 69 2 70 RCL M 71 + 72 X/2 73 1/G+ 74 * 75 ST*A 76 42 77 AMOVE 78 A-A 79 STO O 80 SIGN |
81 RCL M
82 + 83 X/2 84 1/G+ 85 2 86 RCL M 87 RCL N 88 + 89 2 90 - 91 Y^X 92 / 93 PI 94 * 95 CHS 96 ST*A 97 RCL O 98 CLA 99 END |
( 179 bytes / SIZE 4n+1 )
| STACK | INPUTS | OUTPUTS |
| Y | µ | / |
| X | m | 1.nnn |
Where 1.nnn is the control number of the result
Example: µ = -PI m = sqrt(2) a = 1.1 + 1.2 i + 1.3 j + 1.4 k ( quaternion -> 4 STO 00 )
Dµ Hm (a) = 2.770233592
- 1.881749624 i - 2.038562092 j - 2.195374561 k
---Execution time = 100s---
12°) Fresnel Integrals
Formulae:
• Fresnel Cosine Integral Dµ C(a) = 22µ-3/2 PI3/2 a1-µ3F~4 [ 1/4 , 3/4 , 1 ; (2-µ)/4 , (3-µ)/4 , (4-µ)/4 , (5-µ)/4 ; -(PI)2 a4/16 ]
• Fresnel Sine Integral Dµ
S(a) = 22µ-11/2 PI5/2 a3-µ3F~4
[ 3/4 , 1 , 5/4 ; (4-µ)/4 , (5-µ)/4 , (6-µ)/4 , (6-µ)/4
; -(PI)2 a4/16 ]
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DCSA" )
• R01 ...... • Rnn = the n components of the anion
Rn+1 .......... R3n: temp
>>> When the programs stop: R01 ...... Rnn = the n components of the result
Flag: F01 CF 01 <->
Fresnel Cosine Integral Dµ C(a)
SF 01 <-> Fresnel Sine Integral
Dµ S(a)
Subroutines: A*A1 A-A
"A^X"
( cf "Anions for the HP41" )
ST*A ST/A DSA AMOVE ASWAP
( cf "Anionic Special Functions(I) for the HP41" )
1/G+
( cf "Gamma, Digamma Functions for the HP41" )
Lines 61 to 65 may be replaced by RCL X PI SIGN
+ ENTER^ X<> L ST+ L
| 01 LBL "DCSA"
02 CLA 03 STO M 04 12 05 AMOVE 06 13 07 AMOVE 08 A*A1 09 12 10 AMOVE 11 A*A1 12 PI 13 4 14 STO N 15 / 16 X^2 17 CHS 18 ST*A 19 12 20 AMOVE 21 31 22 AMOVE 23 3 24 FC? 01 25 SIGN 26 RCL M 27 - 28 XEQ "A^X" |
29 13
30 AMOVE 31 LBL 01 32 A*A1 33 SIGN 34 PI 35 INT 36 2 37 FS? 01 38 X^2 39 RCL M 40 - 41 RCL N 42 FS? 01 43 ST+ T 44 ST/ T 45 ST/ Z 46 / 47 RCL O 48 ST+ T 49 ST+ Z 50 + 51 / 52 * 53 PI 54 INT 55 2 56 FC? 01 |
57 CLX
58 + 59 RCL M 60 - 61 ENTER^ 62 X+1 63 ENTER^ 64 X+1 65 SIGN 66 CLX 67 RCL N 68 ST/ Z 69 ST/ Y 70 ST/ L 71 CLX 72 RCL O 73 ST+ Z 74 ST+ Y 75 ST+ L 76 X<> L 77 * 78 * 79 / 80 ST*A 81 ISG O 82 CLX 83 DSA 84 X#0? |
85 GTO 01
86 31 87 AMOVE 88 PI 89 X^2 90 LASTX 91 3 92 Y^X 93 FS? 01 94 * 95 SQRT 96 2 97 RCL M 98 2 99 FC? 01 100 CLX 101 - 102 ST+ X 103 1.5 104 - 105 Y^X 106 * 107 2 108 FS? 01 109 X^2 110 RCL M 111 - 112 RCL N |
113 /
114 1/G+ 115 ST* Y 116 X<> L 117 RCL N 118 1/X 119 + 120 1/G+ 121 ST* Y 122 X<> L 123 RCL N 124 1/X 125 + 126 1/G+ 127 ST* Y 128 X<> L 129 RCL N 130 1/X 131 + 132 1/G+ 133 * 134 ST*A 135 RCL 00 136 E3 137 / 138 ISG X 139 CLA 140 END |
( 227
bytes / SIZE 3n+1 )
| STACK | INPUT | OUTPUT |
| X | µ | 1.nnn |
Where 1.nnn is the control number of the result
Example: µ = -PI , a = 1.1 + 1.2 i + 1.3 j + 1.4 k ( quaternion -> 4 STO 00 )
• CF 01 Dµ C (a) = 0.418951593 - 0.420163597 i - 0.455177231 j - 0.490190864 k ---Execution time = 62s---
• SF 01 Dµ
S (a) = -0.233950973 + 0.363367171 i + 0.393647768 j + 0.423928366
k
---Execution time = 63s---
Notes:
-With Dµ C (a) , µ must be different
from 2 , 3 , 4 , ....
-With Dµ S (a) , µ ----------------------
4 , 5 , 6 , ....
13°) Airy Functions
Formulae:
Dµ Ai(a) = 3µ-4/3 a -µ { 32/3 Gam(1/3) 2F~3 [ 1/3 , 1 ; (1-µ)/3 , (2-µ)/3 , (3-µ)/3 ; a3/9 ] - a Gam(2/3) 2F~3 [ 2/3 , 1 ; (4-µ)/3 , (2-µ)/3 , (3-µ)/3 ; a3/9 ] }
Dµ Bi(a) = 3µ-5/6
a -µ { 32/3 Gam(1/3) 2F~3
[ 1/3 , 1 ; (1-µ)/3 , (2-µ)/3 , (3-µ)/3 ; a3/9
] + a Gam(2/3) 2F~3 [ 2/3 , 1 ; (4-µ)/3
, (2-µ)/3 , (3-µ)/3 ; a3/9 ] }
"DAIRYA" calculates Dµ Ai(a)
& Dµ Bi(a) simultaneously and stores the
results in R01 thru R2n
Data Registers: • R00 = n > 1 ( Registers R00 thru Rnn are to be initialized before executing "DAIRYA" )
• R01 ...... • Rnn = the n components of the anion
R2n+1 .......... R4n: temp
>>> When the programs stop: R01
...... Rnn = the n components of Dµ Ai(a)
Rn+1 ..... R2n = ------------------- Dµ
Bi(a)
Flag: F10
Subroutines: A*A1 A-A
A+A "A^X"
( cf "Anions for the HP41" )
ST*A ST/A DSA AMOVE ASWAP
( cf "Anionic Special Functions(I) for the HP41" )
1/G+
( cf "Gamma, Digamma Functions for the HP41" )
-Lines 125 to 127 may be replaced by 4 3 /
RCL M X<>Y
-Lines 73-74 may be replaced by SIGN ST* X
ST- L
-Lines 47 to 50 may be replaced by CLX FS? 10 SIGN
RCL M -
-Lines 43-33-27-21 may be replaced by 3 /
| 01 LBL "DAIRYA"
02 STO M 03 3 04 STO N 05 12 06 AMOVE 07 A*A1 08 A*A1 09 9 10 ST/A 11 12 12 ASWAP 13 14 14 AMOVE 15 SIGN 16 ENTER^ 17 SF 10 18 LBL 01 19 FS? 10 20 ST+ X 21 X/3 22 1/G+ 23 / 24 2 25 RCL M 26 - 27 X/3 |
28 1/G+
29 * 30 3 31 RCL M 32 - 33 X/3 34 1/G+ 35 * 36 1 37 RCL N 38 FC? 10 39 CLX 40 + 41 RCL M 42 - 43 X/3 44 1/G+ 45 * 46 STO O 47 RCL M 48 CHS 49 FS? 10 50 X+1 51 XEQ "A^X" 52 CLX 53 X<> O 54 ST*A |
55 13
56 AMOVE 57 LBL 02 58 A*A1 59 SIGN 60 FS? 10 61 ST+ X 62 1 63 RCL N 64 FC? 10 65 CLX 66 + 67 RCL M 68 - 69 RCL N 70 LASTX 71 - 72 ENTER^ 73 X-1 74 SIGN 75 CLX 76 RCL N 77 ST/ T 78 ST/ Z 79 ST/ Y 80 ST/ L 81 CLX |
82 RCL O
83 ST+ T 84 ST+ Z 85 ST+ Y 86 ST+ L 87 X<> L 88 * 89 * 90 / 91 ST*A 92 ISG O 93 CLX 94 DSA 95 X#0? 96 GTO 02 97 31 98 AMOVE 99 FC?C 10 100 RTN 101 14 102 ASWAP 103 RCL N 104 2 105 RCL Y 106 / 107 Y^X 108 1 |
109 XEQ 01
110 42 111 AMOVE 112 A+A 113 3 114 RCL M 115 5 116 6 117 / 118 - 119 Y^X 120 ST*A 121 13 122 ASWAP 123 A-A 124 3 125 RCL M 126 4 127 X/3 128 - 129 Y^X 130 ST*A 131 32 132 AMOVE 133 R^ 134 CLA 135 END |
( 231 bytes / SIZE
4n+1 )
| STACK | INPUT | OUTPUT |
| X | µ | 1.nnn |
Where 1.nnn is the control number of Dµ Ai(a) [ Dµ Bi(a) is in registers Rn+1 thru R2n ]
Example: µ = PI , a = 1.1 + 1.2 i + 1.3 j + 1.4 k ( quaternion -> 4 STO 00 )
• R01 thru R04 -> Dµ Ai (a) = -0.264935707 + 0.693505589 i + 0.751297721 j + 0.809089853 k ---Execution time = 86s---
• R05 thru R08 -> Dµ
Bi (a) = -2.400100467 - 0.665074361 i - 0.709663892 j - 0.764253423
k
Note:
"DAIRYA" does not work if µ = 1 , 2 , 3 , ....
Reference: