Being analytic means that a function is differentiable at every point in its domain. In other words, the function has a derivative at every point and the derivative is continuous.
To prove that ln (z) is analytic, we need to show that it is differentiable at every point in its domain. This can be done using the Cauchy-Riemann equations, which state that a function is differentiable if its partial derivatives exist and satisfy certain conditions.
The domain of ln (z) is all complex numbers except for 0 and negative real numbers. This is because ln (z) is undefined for these values.
No, ln (z) cannot be extended to be analytic at 0. This is because the function has a singularity at 0, meaning it is not defined at this point. However, it can be extended to be analytic on the complex plane by excluding the point 0.
The natural logarithm function, denoted as ln (x), is the inverse of the exponential function e^x. This means that ln (x) gives the power to which e must be raised to get x. Similarly, ln (z) gives the complex power to which e must be raised to get z, and it shares many of the same properties as the natural logarithm function.