As the title says, I am looking at the function:
$$f(z) = z e^{z},$$
and I want to know whether it is differentiable – and if so, to find its derivative (which I can do).
Usually I would either make use of the definition from first principles or the Cauchy–Riemann conditions; however, both seem very messy. I further know that $e^{z}$ is entire in the complex plane but I don't know if that is a sufficent condition to prove its differentiability.
I have seen cases which cite calculating $\frac{\partial f}{\partial \bar{z}}$, but I don't understand the implication or how to achieve this.
Any help would be greatly appreciated. Thanks!
Best Answer
With $\;z=x+iy\;$ :
$$f(z)=(x+iy)(e^x\cos y+ie^x\sin y)=\overbrace{xe^x\cos y-ye^x\sin y}^{=u(x,y)}+\left(\overbrace{xe^x\sin y+ye^x\cos y}^{=v(x,y)}\right)i$$
Now Cauchy-Riemann:
$$u'_x=e^x\left(\cos y+x\cos y-y\sin y\right)=v'_y$$
$$u_y=e^x\left(-x\sin y-\sin y-y\cos y\right)=-v'_x$$