For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.
Complex analysis is a branch of mathematical analysis that investigates functions of one or several complex variables. Typical topics include Cauchy's integral formula, singularities, poles, holomorphic and meromorphic functions, Laurent and Taylor series, maximum modulus principle, isolated zeros principle, etc.
Independent variable and dependent variable of complex functions are both complex numbers, and may be separated into real (denoted by $\Re$) and imaginary parts (denoted by $\Im$) : $z = x +iy$ and $w = f(z) =u(x,y)+iv(x,y)$ where $x, y \in \mathbb{R}$, $u(x,y)$ and $v(x,y)$ being real-valued functions.
A fundamental result of complex analysis is the Cauchy-Riemann equations, which give the conditions for a function to have a complex derivative, which is a complex generalization of the (real) derivative. If the complex derivative is defined at each point in an open connected subset of $\mathbb C$, the function is called analytic, or holomorphic. If the complex derivative is defined at each point in $\mathbb C$, the function is called entire.
