I try to plot the digamma function of (1/2 + 1/x) but I'm not sure that I'm plotting the good one.

I've tried:

special.polygamma(0, (1/2 + 1/x))

and

special.polygamma(1, (1/2 + 1/x))

but I don't have the same result as with mathcad.

I've tried to code it like that:

def F(x): return mpmath.diff(lambda x: gamma(1/2 + 1/x),1)/gamma(1/2 + 1/x)

But It returns zero division error even when x is in ]0,1]

Any idea?