**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