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Question: Psi(rational)

Question:Psi(rational)

vv 13650 Maple
convert simplify psi
September 03 2020
1

 

During the computation of a series, I needed the value Psi(1/12)  of the digamma function at a rational point. 

simplify, convert/elementary or other usual convertions do not help.

But Maple knows the formula for Psi(r) when r is rational:

 

> 

FunctionAdvisor(special_values, Psi)[24][1]: convert(%,`global`);

Psi(n+p/q) = q*(Sum(1/(k*q+p), k = 0 .. n-1))+2*(Sum(cos(2*Pi*p*k/q)*ln(sin(Pi*k/q)), k = 1 .. floor((1/2)*q+1/2)-1))-(1/2)*Pi*cot(Pi*p/q)-ln(2*q)-gamma

 (1)
> 

(simplify@value)(eval(%, [n=0, p=1, q=12]));

Psi(1/12) = ((2*3^(1/2)-6)*ln(2+3^(1/2))+(-Pi-2*gamma-6*ln(2)-3*ln(3))*3^(1/2)-Pi+2*gamma+6*ln(2)+3*ln(3))/(2*3^(1/2)-2)

 (2)
> 

evalf(%);

-12.44790533 = -12.44790533

 (3)

The question is: why does not Maple use this formula when asked? Or, is there a convertion which I was missing?

 
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