Prove that $\Gamma'(1)=-\gamma$

Question

Use the product formula for $1/\Gamma(z)$ to prove that $$\Gamma'(1)=-\gamma$$

I know that for Euler constant $\gamma$,

$$\frac{1}{\Gamma(z)} =ze^{\gamma z}\prod _{k=1}^{\infty} (1+\frac{z}{k})e^{-z/k}$$

But I cannot prove it properly. Please show me explicitly. Thank you:)

Take the reciprocals of both sides of your identity, and then take the logarithmic derivative of both sides. Since $\Gamma(1)=1$, this gives you $\Gamma'(1)/\Gamma(1) = \Gamma'(1) = \dots$. — A l'Maeaux, Dec 28 '13 at 10:45

DonAntonio · Accepted Answer · 2013-12-28T11:33:02.183

11

$$\log\Gamma(z)=-\log z-\gamma z-\sum_{k=1}^\infty\left[\log\left(1+\frac zk\right)-\frac zk\right]\implies$$

$$\frac{\Gamma'(z)}{\Gamma(z)}=-\frac1z-\gamma-\sum_{k=1}^\infty\left(\frac1k\frac k{k+z}-\frac1k\right)=-\frac1z-\gamma-\sum_{k=1}^\infty\left(\frac1{k+z}-\frac1k\right)\implies$$

$$\Gamma'(1)=\frac{\Gamma'(1)}{\Gamma(1)}=-1-\gamma-\left(\frac12-1+\frac13-\frac12+\ldots\right)=-1-\gamma-(-1)=-\gamma$$

edited Dec 28 '13 at 11:33

answered Dec 28 '13 at 10:57

DonAntonio

211,718
17
136
287

Prove that $\Gamma'(1)=-\gamma$

1 Answers1

Linked