In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal x-axis, called...

in adding more structure, one may view the plane as a 1-dimensional complex manifold, called the complex line. Many fundamental tasks in mathematics...

Poincaré half-plane model. Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to...

regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point ⁠ z...

of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the...

standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their...

These logarithms are equally spaced along a vertical line in the complex plane. A complex-valued function log : U → C {\displaystyle \log \colon U\to \mathbb...

complex ones. The hyperbolic unit j is not a real number but an independent quantity. The collection of all such z is called the split-complex plane....

getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions...

In mathematics, the complex projective plane, usually denoted ⁠ P 2 ( C ) {\displaystyle \mathbb {P} ^{2}(\mathbb {C} )} ⁠ or ⁠ C P 2 , {\displaystyle...

sphere. The extended Euclidean plane can be identified with the extended complex plane, so that equations of complex numbers can be used to describe...

additional examples. In the complex plane, numbers of unit magnitude are called the unit complex numbers. This is the set of complex numbers z such that | z...

represent physical positions, like an affine plane or complex plane. The most basic example is the flat Euclidean plane, an idealization of a flat surface in...

In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases...

generalized to accept complex numbers as arguments. This reveals relations between multiplication of complex numbers, rotations in the complex plane, and trigonometry...

the complex plane, the function e i x {\displaystyle e^{ix}} for real values of x {\displaystyle x} traces out the unit circle in the complex plane. Both...

thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite...

meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the multiplicities of its...

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all...

\infty } can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting...

The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin. This can...

d}}t,\ \qquad \Re (z)>0\,.} The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic...

ex to the complex plane. The exponential function f ( z ) = e z {\displaystyle f(z)=e^{z}} is the unique differentiable function of a complex variable...

geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation x 2 − y 2 = 1. {\displaystyle x^{2}-y^{2}=1...

outside the hyperbolic plane proper. Sometimes the points of the half-plane model are considered to lie in the complex plane with positive imaginary...

A\cos(\omega t+\theta ).} Figure 2 depicts it as a rotating vector in the complex plane. It is sometimes convenient to refer to the entire function as a phasor...

identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complex plane (C), the unit disk is often...

This transform exchanges the part of the complex plane inside the unit circle with the part of the complex plane outside of the unit circle. In particular...

In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are the point's...

multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers T = { z ∈ C...