Determine where a complex function is analytic

[Fronzbot]

Homework Statement

Determine the region of analyticity of ln|z| + i*arg(z) and justify your answer.

Homework Equations

The Attempt at a Solution

I said that if z=x+iy, the function has a singularity when both x and y are equal to 0 since ln(0) is undefined and arg(z) = arg(x+iy) = -i*log(z+iy/sqrt(x^2+y^2)) which is also undefined.
Thus, I said, the function is analytic everywhere but when |z| = 0.

However, the back of the book claims it is analytic for all z not on the non-positive real axis, but I don't see this. For example, if I pick a value of z = -5 + i*0, ln(-5) is ln(5) + pi*i and arg(z) = pi/ln(10)

The only things I can thing of is that my equation for arg(z) is wrong (I went on wolfram alpha to try and visualize it and I saw that they used it). And the other thing is that arctan(z) would be arctan(y/x) but when y=0, arctan is just equal to zero. My answer makes sense to me, but the books answer does not. What am I missing/doing wrong?

Thank you!

 

[mighty2000]

Your approach is partially correct, but there are a few things that you need to consider in order to fully understand the region of analyticity for this function.

Firstly, let's define our function as f(z) = ln|z| + i*arg(z). As you correctly pointed out, the function has a singularity at z = 0, since ln(0) is undefined and arg(z) is also undefined when z = 0.

Now, let's look at the real and imaginary parts of the function separately. The real part is ln|z|, which is defined for all z ≠ 0. This means that the function is analytic everywhere except at z = 0.

The imaginary part is i*arg(z). Here, we need to be careful with how we define arg(z). In your attempt, you used the formula arg(z) = -i*log(z+iy/sqrt(x^2+y^2)). However, this is not the standard definition of arg(z). The correct definition is arg(z) = arctan(y/x), where x and y are the real and imaginary parts of z, respectively. This definition is valid for all z ≠ 0, except for the non-positive real axis (i.e. the negative real axis and the origin). This means that the function is analytic for all z not on the non-positive real axis.

To summarize, the function f(z) = ln|z| + i*arg(z) is analytic everywhere except at z = 0 and on the non-positive real axis.

I hope this helps to clarify your confusion. Remember to always double check the definitions of the functions you are using, as this can greatly affect your solution. Keep up the good work in your studies!