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A complex function is analytic at a point if it can be represented by a power series converging in a neighborhood of that point. Example: f(z) = e^z is analytic as it can be expressed as a power series for all z.

The Cauchy-Riemann equations are a pair of real partial differential equations which must be satisfied for a function f(z) = u(x, y) + iv(x, y) to be holomorphic. Example: For f(z) = z^2, u(x, y) = x^2 - y^2, v(x, y) = 2xy satisfy the equations.

A contour integral along a path C in the complex plane is the integral of a complex function f(z) over C. Example: The integral of f(z) = z along the unit circle |z|=1 is 0.

The Schwarz lemma states that a holomorphic function f mapping the unit disk to itself with f(0) = 0 satisfies |f(z)| <= |z|, and if equality holds for some non-zero z, f is a rotation. Example: f(z) = az where |a| <= 1 is a function satisfying Schwarz lemma.

An essential singularity of a function is a point where the function behaves in a non-removable and non-pole-like manner. Example: The function f(z) = exp(1/z) has an essential singularity at z = 0.

The residue of a function at a point is the coefficient of the (1/(z - z0)) term in the function's Laurent series around that point. Example: For f(z) = 1/(z^2 + 1), the residue at z = i is 1/(2i).

The Riemann sphere is the extended complex plane which adds a point at infinity, allowing for a compact and thus complete space. Example: The function f(z) = 1/z maps every point in the complex plane to the Riemann sphere except for z = 0.

The argument principle relates the number of zeros and poles of a meromorphic function within a contour to the change in argument of the function around the contour. Example: For f(z) = z/(z^2 + 1), the argument changes by 2π around a contour enclosing the poles at z = i and z = -i.

A complex function f(z) is holomorphic at a point z0 if it is differentiable at z0 and in every neighborhood around z0. Example: f(z) = z^2 is holomorphic everywhere in the complex plane.

A meromorphic function is holomorphic everywhere except at a set of isolated points, which are poles of the function. Example: f(z) = 1/(z - 1) is meromorphic with a pole at z = 1.

The Mandelbrot set is the set of complex numbers c for which the function f(z) = z^2 + c remains bounded when iterated from z = 0. Example: c = 0 is an element of the Mandelbrot set because the corresponding sequence is 0, 0^2 + 0, ..., which is bounded.

A branch point of a multi-valued function is a point where the function is discontinuous when going around a loop. A branch cut is a curve that is introduced to create a single-valued branch of the function. Example: The function f(z) = sqrt(z) has a branch point at z = 0, and the negative real axis is often chosen as a branch cut.

A Laurent series is a representation of a complex function as a power series which includes terms with negative powers of (z - z0). Example: For f(z) = 1/(z - 1), the Laurent series around z = 0 is -sum(n=1 to infinity)(z^n).

The maximum modulus principle states that if f(z) is a non-constant holomorphic function, then |f(z)| cannot have a local maximum that is interior to its domain. Example: If f(z) = z^2 and |f(z)| had a local maximum, it would have to be on the boundary of its domain.