Where is ##(z+1)Ln(z)## differentiable?

[Terrell]

Homework Statement

Find the domain in which the complex-variable function $f(z)=(z+1)Ln(z)$ is differentiable. Note: $Ln(z)$ is the principal complex logarithmic function.

Homework Equations

Cuachy-Riemann Equations?

The Attempt at a Solution

The solution I have in mind would be to let $z=x+iy$ then substitute and simplify. Check if it satisfies the Cauchy-Riemann equations, the real and imaginary part of $f$ is continuous and their first-order partial derivative are continuous as well. But, I do not know how to simplify $Arg(z)$ in $Ln(z)=Log_e(z)+iArg(z)$ because $z$ is not fixed.

 

[MathematicalPhysicist]

It should be $\log |z|$ and not $\log z$, i.e. natural logarithm of the modulus of z.

 

[Terrell]

It should be $\log |z|$ and not $\log z$, i.e. natural logarithm of the modulus of z.

Yes, I made a typo, but how do I simplify $Arg(z)$?

 

[MathematicalPhysicist]

Well, $Arg(z)=\arctan y/x$ where $z=x+iy$, this should help you with Cauchy-Riemann.

 

[Terrell]

Well, $Arg(z)=\arctan y/x$ where $z=x+iy$, this should help you with Cauchy-Riemann.

It's what I have on paper but I can't reconcile with what wikipedia have. It has conditions depending on the values of x and y. https://en.wikipedia.org/wiki/Argument_(complex_analysis)

 

[Terrell]

It's what I have on paper but I can't reconcile with what wikipedia have. It has conditions depending on the values of x and y.

After giving it some thought now, it doesn't seem to matter when I start differentiating.