Change of variable in complex integral

Question

When I want to evaluate the following integral $$\int_\Gamma e^{-z^2}dz~~~~~~~~~~~\Gamma:|z|=R,0\leq\arg{z}<\frac{\pi}{4}$$ so I subtitude, letting(choose a single-valued analytic branch) $$z=w^{\frac{1}{8}}, ~dz=\frac{1}{8}w^{\frac{1}{8}-1}dw$$ the integral change to $$\frac{1}{8}\int_{|w|=R^8}\frac{w^{\frac{1}{8}}e^{-w^{\frac{1}{4}}}}{w}dw$$ and according to Cauchy's integral formula $$\frac{1}{8}\int_{|w|=R^8}\frac{w^{\frac{1}{8}}e^{-w^{\frac{1}{4}}}}{w}dw =2\pi i\times\frac{1}{8}w^{\frac{1}{8}}e^{-w^{\frac{1}{4}}}|_{w=0}=0$$ Apparently, the solution is unreasonable. But what is wrong?