I'm a student of electrical engineering, preparing for the theoretical exam covering complex analysis. I'm confused about the analyticity of a complex function, so I'm asking for clarification because our professor inadequately explained this concept.
What I understand from my lecture notes: suppose there's a complex-valued function $f(z)$. Such function is analytic if the following conditions are satisfied:
- $f$ is differentiable at $z_0$, therefore $f'(z_0)$ exists
- $f$ is differential at every point of some $\epsilon$-spherical neighborhood of $z_0$
- $f$ can be expanded as a Taylor series in the vicinity of $z_0$
At that point, Cauchy-Riemann equations are nowhere mentioned. However, I've found online that C-R equations are a necessary condition for a complex-valued function being holomorphic (or analytic, although terms are used interchangeably). And here's where I'm starting to get lost.
If C-R equations are a necessary condition for holomorphicity, but they are not sufficient conditions for complex differentiability, then how can they ensure given function $f$ being holomorphic, if according to what I've read, such function must be differentiable at given point $z_0$ in order to be considered holomorphic? Can, therefore, a function be holomorphic (or analytic) if C-R conditions are not satisfied?
What I've read thus far
- Are the Cauchy-Riemann equations a necessary and sufficient condition for a function to be analytic?
- Analyticity of a function in $x$ and $y$, without employing the Cauchy-Riemann eqns
- Complex analytic function and Cauchy-Riemann conditions question
- Reference request for undergraduate complex analysis.
- Prove a function is holomorphic