Let $γ\colon[-1,1]\to\mathbb{C}$ , $γ(t)= z_0 + itc$ , $z_0$ fixed and c>0
Prove for x>0 $\lim_{x\to0} \frac{1}{2πi} \int_γ \left(\frac{1}{z-w} - \frac{1} {z-w'}\right)dz = -1$
Where $w=z_0 + x$ , $w'= z_0 - x$
I understand that you have to substitute in w and w' but I can't figure out what to do with $\frac{1}{z-z_0 -x} - \frac{1}{z-z_0 +x}$ What is the next step? Do I Use the $γ(t)$ function?
Any help on this question would be appreciated. Thanks