16 Complex Analysis
Complex analysis is the study of complex-differentiable functions of a complex variable, where a complex-differentiable function is a function whose derivative is the same from every direction.
Here, again, we identifty the ideas that are fundamental to the study of complex analysis.
Complex differentiability. Complex differentiability is a property: the derivative is the same from every direction. Locally the function is a uniform scaling and a rotation.
Cauchy’s theorem. Cauchy’s theorem is a theorem: the integral of a holomorphic function around a closed loop in a simply connected region is zero. A simply connected region is a region with no holes.
Analyticity. Analyticity is a property: a holomorphic function is infinitely differentiable and equals a power series in a neighborhood of each point. A power series is an infinite sum of increasing integer powers.
definition [d]
A holomorphic function on an open set \(U\) in the complex plane is a function \(f: U \to \mathbb{C}\) that is complex-differentiable at every point of \(U\).
The complex derivative at \(z_{0}\) is
\[ f'(z_{0}) = \lim_{z \to z_{0}} \dfrac{f(z)-f(z_{0})}{z-z_{0}} \]
where
- \(U\) is an open set in the complex plane
- \(f\) is a complex-valued function
- \(z_{0}\) is a point of \(U\)
Note:
- The limit must exist and be the same along every path to \(z_{0}\).
16.1 Examples
16.3 References
- Needham, T. Visual Complex Analysis. Oxford University Press. — complex differentiability as the same derivative in every direction, Cauchy’s theorem, and analyticity.
- Waleffe, F. Vector and Complex Calculus. — path independence of integrals of holomorphic functions.
- Howell, K. B. notes on complex analysis. — the definition of a holomorphic function by the complex derivative.