1. Let's start by stating the problem: Understanding what an analytic function is in complex analysis. 2. An analytic function is a complex function that is locally given by a convergent power series. In simpler terms, a function $f(z)$ is analytic at a point $z_0$ if it can be expressed as $$f(z) = \sum_{n=0}^\infty a_n (z - z_0)^n$$ for all $z$ near $z_0$, where $a_n$ are complex coefficients. 3. Important rules and properties: - Analytic functions are infinitely differentiable. - They satisfy the Cauchy-Riemann equations: $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$ where $f(z) = u(x,y) + iv(x,y)$ with $z = x + iy$. 4. To check if a function is analytic, verify if it satisfies the Cauchy-Riemann equations and if the partial derivatives are continuous. 5. Example: Consider $f(z) = z^2 = (x + iy)^2 = x^2 - y^2 + 2ixy$. - Here, $u(x,y) = x^2 - y^2$ and $v(x,y) = 2xy$. - Compute partial derivatives: $$\frac{\partial u}{\partial x} = 2x, \quad \frac{\partial u}{\partial y} = -2y$$ $$\frac{\partial v}{\partial x} = 2y, \quad \frac{\partial v}{\partial y} = 2x$$ - Check Cauchy-Riemann: $$2x = 2x \quad \text{and} \quad -2y = -2y$$ - Both hold true everywhere, so $f(z) = z^2$ is analytic everywhere. 6. Summary: Analytic functions are complex functions that can be represented by power series locally, satisfy the Cauchy-Riemann equations, and are infinitely differentiable. This foundational concept is crucial in complex analysis and has many applications in physics and engineering.