MHB Deduce analyticity of each function

[Dustinsfl]

What does it mean to deduce analyticity?

Given the function: \(f(z) = z^2 + 5iz + 3 - i\)

1. The C-R equations are satisfied
2. It is a polynomial so it is infinitely differentiable
3. Since it is in \(C^{\infty}\), we know it has a Taylor series about some point \(z_0\).

Is that deducing it analyticity or is it something else?

Additionally, for the function \(f(z) = \sin(2z)\), again, we know that the C-R equations are satisfied and the transcendental sine is \(C^{\infty}\). So let \(z_0\in IOC\) where IOC is interval of convergence. Then a T.S exist about \(z_0\). Would this be deducing \(\sin(2z)\) is analytic.

 

[mathbalarka]

The 3 conditions you mention are essentially equivalent : if $f(z)$ satisfy at least one of them, all of the others are automatically satisfied and $f(z)$ is called complex analytic.

In this case, the function is polynomial and is infinitely differentiable at all point in $\Bbb C$, hence is analytic.