HP Prime and TI-84 Plus CE: Digamma Function
Calculation
The HP Prime and
TI-84 Plus CE program DIGAM ( ψ )
calculates the digamma function for any positive integers and multiples of 1/2
(such as 3/2, 5/2, 7/2,etc) using the formulas:
ψ(1/2) = -γ – ln 4 =
-1.963510026
ψ(1) = -γ =
-0.5772156649
ψ(n) = Σ(1/x, x, 1,
n-1) - γ for n is an integer
ψ(n) = Σ(2/(2*x+1),
x, 0, n-3/2) - γ – ln 4 for n a multiple of 1/2 for the form p/2, p is odd
where γ =
0.577215664901533 (Euler-Mascheroni constant)
HP Prime Program DIGAM
EXPORT DIGAM(N)
BEGIN
// EWS 2017-12-17
LOCAL K,D,E:=0.577215664901533;
CASE
IF N==1 THEN
RETURN −E; END;
IF N==0.5 THEN
RETURN −E-LN(4); END;
IF FP(N)==0 AND N>1 THEN
RETURN Σ(1/K,K,1,N-1)-E; END;
IF FP(N)==0.5 AND N>1 THEN
RETURN
Σ(2/(2*K+1),K,0,N-3/2)-E-LN(4); END;
DEFAULT RETURN "Error: Invalid
N";
END;
END;
TI-84 Plus CE Program DIGAM
"DIGAMMA
2017-12-17 EWS"
0.577215664901533→E
0→K
Input
"N?",N
If
fPart(N)≠0 and fPart(N)≠0.5
Then
Disp
"INVALID N"
Stop
End
If
N=1
Then
E→D
End
If
fPart(N)=0 and N>1
Then
Σ(1/K,K,1,N-1)-E→D
End
If
N=0.5
Then
E-ln(4)→D
End
If
fPart(N)=0.5 and N>1
Then
Σ(2/(2*K+1),K,0,N-1.5)-E-ln(4)→D
End
Disp
D
Examples
DIGAM(2) =
0.422784335098
DIGAM(3) =
0.922784335098
DIGAM(4) =
1.25611766843
DIGAM(3/2) =
0.03648997398
DIGAM(5/2) =
0.70315664065
DIGAM(7/2) = 1.10315664065
Source:
Keith Oldham, Jan
Myland, & Jerome Spainer. An
Atlas of Functions. 2nd
Edition. Springer: New York. 2009
ISBN 13: 978-0-387-48806-6
Eddie
This blog is property
of Edward Shore, 2017
