This article is about the geometric representation of complex numbers. For the two-dimensional projective space with complex-number coordinates, see complex projective plane.
Mathematical analysis → Complex analysis
Complex analysis
Complex numbers
Real number
Imaginary number
Complex plane
Complex conjugate
Unit complex number
Complex functions
Complex-valued function
Analytic function
Holomorphic function
Cauchy–Riemann equations
Formal power series
Basic theory
Zeros and poles
Cauchy's integral theorem
Local primitive
Cauchy's integral formula
Winding number
Laurent series
Isolated singularity
Residue theorem
Conformal map
Schwarz lemma
Harmonic function
Laplace's equation
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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 the real axis, is formed by the real numbers, and the vertical y-axis, called the imaginary axis, is formed by the imaginary numbers.
The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
The complex plane is sometimes called the Argand plane or Gauss plane.
In mathematics, the complexplane 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 complexplane. This allows a geometric interpretation of the complex numbers and their...
applicable to all complex numbers; see § Complexplane for the extension of exp x {\displaystyle \exp x} to the complexplane. Using the power series...
These logarithms are equally spaced along a vertical line in the complexplane. A complex-valued function log : U → C {\displaystyle \log \colon U\to \mathbb...
everywhere within some neighbourhood of z0 in the complexplane. Given a complex-valued function f of a single complex variable, the derivative of f at a point...
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...
getting a complex analytic function whose domain is the whole complexplane with a finite number of curve arcs removed. Many basic and special complex functions...
additional examples. In the complexplane, numbers of unit magnitude are called the unit complex numbers. This is the set of complex numbers z such that | z...
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 complexplane. The complex...
definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complexplane". A function...
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complexplane is a function that is holomorphic on all...
thought of as deformed versions of the complexplane: locally near every point they look like patches of the complexplane, but the global topology can be quite...
the complexplane, the function e i x {\displaystyle e^{ix}} for real values of x {\displaystyle x} traces out the unit circle in the complexplane. When...
identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complexplane (C), the unit disk is often...
mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complexplane. Contour integration...
cosine functions to the whole complexplane, and the domain of the other trigonometric functions to the complexplane with some isolated points removed...
meromorphic functions. For example, if a function is meromorphic on the whole complexplane plus the point at infinity, then the sum of the multiplicities of its...
function to a meromorphic function that is holomorphic in the whole complexplane except zero and the negative integers, where the function has simple...
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...
2π to this angle works as well.) In the complexplane, which is a special interpretation of a Cartesian plane, i is the point located one unit from the...
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...
\infty } can be added to the complexplane as a topological space giving the one-point compactification of the complexplane. When this is done, the resulting...
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