How can I show that the following functions are holomorphic on their domain of definition and that they satisfy the Cauchy Riemann equation?
1) $\displaystyle f(z) = \frac{1}{z+1}$
2) $\displaystyle f(z) = \frac{e^z}{z}$
I was trying to calculate $f(z) = (x+ iy +1)^{-1}$ and to split it into real and imaginary part to check the Cauchy Riemann equations, but I do not see how to go further. And how can I show that these functions are also holomorphic?
If the satisfy the Cauchy-Riemann equations they are holomorphic.
For the first, $f(x,y)=1/(x+iy+1)=1/((x+1)+iy)\cdot((x+1)-iy)/((x+1)-iy)=1/((x+1)^2+y^2)((x+1)-iy)=u(x,y)+v(x,y)i$, where $u(x,y)=1/((x+1)^2+y^2)(x+1)$ and $v(x,y)=-1/((x+1)^2+y^2)y$.
Now you can check the C-R equations.