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...

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

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

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...

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

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

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....

the complex projective plane, usually denoted P2(C) or CP2, is the two-dimensional complex projective space. It is a complex manifold of complex dimension...

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

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...

The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Two special cases...

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...

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...

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...

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...

and the line joining the origin and z, represented as a point in the complex plane, shown as φ {\displaystyle \varphi } in Figure 1. By convention the...

definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function...

the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided. This was known as the complex plane. The complex...

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...

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...

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

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...

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...

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...

mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration...

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...

2π to this angle works as well.) In the complex plane, which is a special interpretation of a Cartesian plane, i is the point located one unit from the...

[clarification needed] With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's...

result, the other hyperbolic functions are meromorphic in the whole complex plane. By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental...