Question

Show that w=z+e^z is analytic and hence find dw/dz

Answer 1

Let's rewrite the function

$w(x+iy)=x+iy+e^{x+iy}=x+iy+e^x(\cos y+i\sin y)$

$u=Re(w)=x+e^x\cos y$

$v=Im(w)=y+e^x\sin y$

$\frac{\partial u}{\partial x}=1+e^x\cos y=\frac{\partial v}{\partial y}$

$\frac{\partial u}{\partial y}=-e^x\sin y=-\frac{\partial v}{\partial x}$

Therefore the function is analytic  and

$w'=1+e^z$