Newsgroups: sci.math
From: mcohen@[deleted] (Martin Cohen)
Subject: Psi function at rational values
Date: Thu, 27 Apr 1995 22:18:33 GMT

A short while post I posted a query about computing the
psi function (Gamma'(x)/Gamma(x)) at rational values.
I have since found these sources:

"An Atlas of Functions" by Jerome Spanier and Keith Oldham,
section 44:4.

Higher Transcendental Functions, Vol I, Erdelyi and others,
p. 18, sec 1.7.3. This givies enough info so you can
derive it for yourself.

Here is Gauss' formula for psi(m/n) for 0 < m < n from
"An Atlas of Functions" by Jerome Spanier and Keith Oldham,
section 44:4. "g" is Euler's constant (gamma), FLOOR(a, b) is
integer part of a/b. This is my input to Derive.

P(m,n):=-g-pi/2*COT(m*pi/n)+2*SUM(COS(2*j*m*pi/n)*LN(2*SIN(j*pi/n)),
j,1,FLOOR(n-1,2))-LN(n)-LN(n-2*FLOOR(n-1,2))

Here is some sample output from Derive.

P(1,2)

-2*LN(2)-g

P(1,3)

-3*LN(3)/2-g-SQRT(3)*pi/6

P(1,6)

-3*LN(3)/2-2*LN(2)-g-SQRT(3)*pi/2

P(5,6)

-3*LN(3)/2-2*LN(2)-g+SQRT(3)*pi/2

P(1,8)

SQRT(2)*LN(3-2*SQRT(2))/2-4*LN(2)-g-SQRT(2)*pi/2-pi/2

P(1,9)

-2*COS(pi/9)*LN(COS(pi/18))+2*SIN(pi/18)*LN(SIN(2*pi/9))+2*COS(2*pi/9)*LN(SIN(~
pi/9))-5*LN(3)/2-pi*COT(pi/9)/2-g

P(1,10)

3*SQRT(5)*LN(3/2-SQRT(5)/2)/4-5*LN(5)/4+LN(4)/4-5*LN(2)/2-g-pi*SQRT(SQRT(5)/2+~
5/4)

P(1,11)

-2*LN(2*COS(5*pi/22))*SIN(pi/22)-2*LN(2*COS(3*pi/22))*SIN(5*pi/22)-2*LN(2*COS(~
pi/22))*COS(pi/11)+2*LN(2*SIN(2*pi/11))*SIN(3*pi/22)+2*LN(2*SIN(pi/11))*COS(2*~
pi/11)-LN(11)-pi*COT(pi/11)/2-g

Then I noticed this:

P(1,5)-P(2,10)

2*LN(2)

P(1,2)-P(2,4)

2*LN(2)

P(1,2)

-2*LN(2)-g

P(2,4)

-4*LN(2)-g

I realized that the book said that at least one of m and n
in P(n, m) had to be odd, so, to be safe, I defined this:

Q(m,n):=P(m/GCD(m,n),n/GCD(m,n))

Q(2,10)

SQRT(5)*LN(3/2-SQRT(5)/2)/4-5*LN(5)/4-LN(4)/4+LN(2)/2-g-pi*SQRT(SQRT(5)/10+1/4)

P(2,10)

SQRT(5)*LN(3/2-SQRT(5)/2)/4-5*LN(5)/4-3*LN(4)/4-LN(2)/2-g-pi*SQRT(SQRT(5)/10+1~
/4)

Q(2,10)-Q(1,5)

0

So that's ok.

Just for fun:

P(1,1)

"+-"inf

As it should be.

Q(1,12)

SQRT(3)*LN(2-SQRT(3))-3*LN(3)/2-3*LN(2)-g-pi*(SQRT(3)/2+1)

Q(1,15)

-2*SIN(pi/30)*LN(COS(7*pi/30))-2*COS(pi/15)*LN(COS(pi/30))+2*SIN(7*pi/30)*LN(S~
IN(2*pi/15))+2*LN(2*SIN(pi/15))*COS(2*pi/15)+SQRT(5)*LN(3/2-SQRT(5)/2)/4-5*LN(~
5)/4-LN(4)/4-3*LN(3)/2-SQRT(5)*LN(2)*SIN(7*pi/30)+LN(2)*SIN(7*pi/30)+LN(2)/2-p~
i*COT(pi/15)/2-g

Q(1,16)

-SQRT(2-SQRT(2))*LN(SQRT(SQRT(2)/4+1/2)/(1-SQRT(1/2-SQRT(2)/4)))-SQRT(SQRT(2)+~
2)*LN(SQRT(1/2-SQRT(2)/4)/(1-SQRT(SQRT(2)/4+1/2)))+SQRT(2)*LN(3-2*SQRT(2))/2-5~
*LN(2)-g-pi*SQRT(SQRT(2)/2+1)-SQRT(2)*pi/2-pi/2

Q(1,17)

-2*LN(2*COS(7*pi/34))*SIN(3*pi/34)-2*LN(2*COS(5*pi/34))*SIN(7*pi/34)-2*LN(2*CO~
S(3*pi/34))*COS(3*pi/17)-2*LN(2*COS(pi/34))*COS(pi/17)+2*LN(2*SIN(4*pi/17))*SI~
N(pi/34)+2*LN(2*SIN(3*pi/17))*SIN(5*pi/34)+2*LN(2*SIN(2*pi/17))*COS(4*pi/17)+2~
*LN(2*SIN(pi/17))*COS(2*pi/17)-LN(17)-pi*COT(pi/17)/2-g

P(1,18)

-2*SIN(pi/18)*LN(COT(2*pi/9))-2*COS(2*pi/9)*LN(COT(pi/9))-2*COS(pi/9)*LN(COT(p~
i/18))-5*LN(3)/2-2*LN(2)-pi*COT(pi/18)/2-g

P(1,20)

-SQRT(5/2-SQRT(5)/2)*LN((SQRT(5)+1)/(4*(1-SQRT(5/8-SQRT(5)/8))))-SQRT(SQRT(5)/~
2+5/2)*LN((1-SQRT(5))/(4*(SQRT(SQRT(5)/8+5/8)-1)))+3*SQRT(5)*LN(3/2-SQRT(5)/2)~
/4-5*LN(5)/4+LN(4)/4-7*LN(2)/2-g-pi*SQRT(SQRT(5)/2+5/4)-SQRT(5)*pi/2-pi/2
-- 
Marty Cohen (mcohen@[deleted]) - Not the guy in Philly
  This is my opinion [deletia]
          Use this material of your own free will

[Affiliation deleted at author's request -- djr]