Discover published sets by community

If $f$ is analytic and bounded by $1$ in the unit disk, and if $f(0) = 0$, then $|f(z)| \leq |z|$ and $|f'(0)| \leq 1$ for all $z$ in the unit disk. If equality holds for one point, then $f(z) = e^{i\theta}z$ for some real $\theta$.

A function is entire if it is analytic at every point in the complex plane.

A function is holomorphic on an open set if it is analytic at every point in that set.

If a function is entire and bounded in the complex plane, it is constant.

If $f$ is a non-constant analytic function on a domain $D$, then $|f(z)|$ achieves its maximum on the boundary of $D$ and not in the interior.

An isolated singularity $z_0$ of a function $f$ is a pole of order $n$ if $f(z)$ behaves like $1/(z-z_0)^n$ near $z_0$ and $f(z)(z-z_0)^n$ is analytic at $z_0$.

A function is analytic at a point if it is differentiable at that point as well as in some neighborhood around it.

A function is meromorphic on an open set if it is analytic everywhere in that set except for a set of isolated points, which are poles of the function.

Let $f$ be a function analytic in some domain except for a finite number of isolated singularities. The integral of $f$ around a closed contour is $2\pi i$ times the sum of the residues of $f$ inside the contour.

A function $f(z) = u(x, y) + iv(x, y)$ is differentiable at a point in the complex plane if and only if partial derivatives exist and satisfy the equations $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$.

Represents a complex function in the form of a power series that includes terms of negative degree, allowing it to describe functions with singularities.

A point $z_0$ is an essential singularity of a function if the function behaves 'wildly' near $z_0$, with the behavior not being describable as either a pole or a removable singularity.

The coefficient of the $1/(z - z_0)$ term in the Laurent series expansion of a function about an isolated singularity $z_0$.

A real-valued function $u(x, y)$ on an open set in $\mathbb{R}^2$ is called harmonic if it satisfies Laplace's equation: $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$.

A sequence of functions $f_n$ converges uniformly to a function $f$ on a set $E$ if, given any positive number $\epsilon$, there exists an $N$ such that for all $x$ in $E$ and all $n \geq N$, $|f_n(x) - f(x)| < \epsilon$.

Let $f$ and $g$ be analytic inside and on a simple closed contour $C$, where $|g(z)| < |f(z)|$ on $C$. Then $f$ and $f+g$ have the same number of zeros inside $C$.

A transformation of the form $f(z) = \frac{az + b}{cz + d}$, where $ad - bc \neq 0$, which maps the extended complex plane to itself in a conformal manner.

For a meromorphic function $f$ on a domain $D$, and $C$ a positively oriented simple closed contour in $D$ that does not pass through any zeros or poles of $f$, the change in argument of $f(z)$ around $C$ is $2\pi$ times the number of zeros minus the number of poles of $f$ inside $C$.

Under a conformal mapping, the image of a simple, closed contour is a simple, closed contour.

A family of analytic functions on a domain that are locally uniformly bounded is a normal family; that is, from any sequence of functions in the family, a subsequence that converges uniformly on compact subsets can be extracted.

If a function is analytic and non-constant on a domain $D$, it maps open sets to open sets.

Every non-constant polynomial with complex coefficients has at least one complex root. This can be established using complex analysis.

A function that preserves angles and the orientation of small figures. Specifically, it is a function whose derivative exists and is non-zero at every point.