Geometric representation of z and its conjugate in the complex plane. The distance along the light blue line from the origin to the point z is the modulus or absolute value of z. The angle φ is the argument of z.

In mathematics, the complex plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z In Mathematics, the imaginary part of a Complex number z is the second element of the ordered pair of Real numbers representing z ^[1]

The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768. Jean-Robert Argand ( July 18, 1768 - August 13, 1822) was a non-professional Mathematician. . 1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745. Caspar Wessel ( June 8, 1745 - March 25, 1818) was a Danish-Norwegian Mathematician. . 1818). ^[2] Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 This article is about the zeros of a function which should not be confused with the value at zero. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

The concept of the complex plane allows a geometric interpretation of complex numbers. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Under addition, they add like vectors. Addition is the mathematical process of putting things together The multiplication of two complex numbers can be expressed most 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 Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In particular, multiplication by a complex number of modulus 1 acts as a rotation.

Notational conventions

In complex analysis the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts, like this:

where x and y are real numbers, and i is the imaginary unit. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In this customary notation the complex number z corresponds to the point (x, y) in the Cartesian plane. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

In the Cartesian plane the point (x, y) can also be represented (in polar coordinates) as

In the Cartesian plane it may be assumed that the arctangent takes values from −π to π (in radians), and some care must be taken to define the real arctangent function for points (x, y) when x ≤ 0. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 ^[3] In the complex plane these polar coordinates take the form

where

^[4]

Here |z| is the absolute value or modulus of the complex number z; θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π; and the last equality (to |z|e^iθ) is taken from Euler's formula. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Notice that the argument of z is multi-valued, because the complex exponential function is periodic, with period 2πi. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) Thus, if θ is one value of arg(z), the other values are given by arg(z) = θ + 2nπ, where n is any integer ≠ 0. ^[5]

The theory of contour integration comprises a major part of complex analysis. In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated In this context the direction of travel around a closed curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the positive direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point z = 1, then travel up and to the left through the point z = i, then down and to the left through −1, then down and to the right through −i, and finally up and to the right to z = 1, where we started. In Mathematics, a unit circle is

Almost all of complex analysis is concerned with complex functions – that is, with functions that map some subset of the complex plane into some other (possibly overlapping, or even identical) subset of the complex plane. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Here it is customary to speak of the domain of f(z) as lying in the z-plane, while referring to the range or image of f(z) as a set of points in the w-plane. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the range of a function is the set of all "output" values produced by that function In symbols we write

and often think of the function f as a transformation of the z-plane (with coordinates (x, y)) into the w-plane (with coordinates (u, v)).

Stereographic projections

Main article: Stereographic projection

Sometimes it's useful to think of the complex plane as if it occupied the surface of a sphere. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane Imagine a sphere of unit radius, and put the complex plane right through the middle of it, so the center of the sphere coincides with the origin z = 0 of the complex plane, and the equator on the sphere coincides with the unit circle in the plane.

We can establish a one-to-one correspondence between the points on the surface of the sphere and the points in the complex plane as follows. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Given a point in the plane, draw a straight line connecting it with the north pole on the sphere. That line will intersect the surface of the sphere in exactly one other point. The point z = 0 will be projected onto the south pole of the sphere. Since the interior of the unit circle lies inside the sphere, that entire region (|z| < 1) will be mapped onto the southern hemisphere. The unit circle itself (|z| = 1) will be mapped onto the equator, and the exterior of the unit circle (|z| > 1) will be mapped onto the northern hemisphere. Clearly this procedure is reversible – given any point on the surface of the sphere that is not the north pole, we can draw a straight line connecting that point to the north pole and intersecting the flat plane in exactly one point.

Under this stereographic projection there's just one point – the north pole itself – that is not associated with any point in the complex plane. We perfect the one-to-one correspondence by adding one more point to the complex plane – the so-called point at infinity – and associating it with the north pole on the sphere. This topological space, the complex plane plus the point at infinity, is known as the extended complex plane. In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that And this is why mathematicians speak of a single "point at infinity" when discussing complex analysis. There are two points at infinity (positive, and negative) on the real number line, but there is only one point at infinity (the north pole) in the extended complex plane. ^[6]

Imagine for a moment what will happen to the lines of latitude and longitude when they are projected from the sphere onto the flat plane. The lines of latitude are all parallel to the equator, so they will become perfect circles centered on the origin z = 0. And the lines of longitude will become straight lines passing through the origin (and also through the "point at infinity", since they pass through both the north and south poles on the sphere).

This is not the only possible stereographic projection of a sphere onto a plane. For instance, the south pole of the sphere might be placed on top of the origin z = 0 in a plane that's tangent to the sphere. The details don't really matter. Any stereographic projection of a sphere onto a plane will produce one "point at infinity", and it will map the lines of latitude and longitude on the sphere into circles and straight lines, respectively, in the plane.

Cutting the plane

When discussing functions of a complex variable it is often convenient to think of a cut in the complex plane. This idea arises naturally in several different contexts.

Multi-valued relationships and branch points

Consider the simple two-valued relationship

Before we can treat this relationship as a single-valued function, the range of the resulting value must be restricted somehow. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function When dealing with the square roots of real numbers this is easily done. For instance, we can just define

to be the non-negative real number y such that y² = x. This idea doesn't work so well in the two-dimensional complex plane. To see why, let's think about the way the value of f(z) varies as the point z moves around the unit circle. We can write

Evidently, as z moves all the way around the circle, w only traces out one-half of the circle. So one continuous motion in the complex plane has transformed the positive square root e⁰ = 1 into the negative square root e^iπ = −1.

This problem arises because the point z = 0 has just one square root, while every other complex number z ≠ 0 has exactly two square roots. On the real number line we could circumvent this problem by erecting a "barrier" at the single point x = 0. A bigger barrier is needed in the complex plane, to prevent any closed contour from completely encircling the branch point z = 0. In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued This is commonly done by introducing a branch cut; in this case the "cut" might extend from the point z = 0 along the positive real axis to the point at infinity, so that the argument of the variable z in the cut plane is restricted to the range 0 ≤ arg(z) < 2π.

We can now give a complete description of w = z^½. To do so we need two copies of the z-plane, each of them cut along the real axis. On one copy we define the square root of 1 to be e⁰ = 1, and on the other we define the square root of 1 to be e^iπ = −1. We call these two copies of the complete cut plane sheets. By making a continuity argument we see that the (now single-valued) function w = z^½ maps the first sheet into the upper half of the w-plane, where 0 ≤ arg(w) < π, while mapping the second sheet into the lower half of the w-plane (where π ≤ arg(w) < 2π). ^[7]

The branch cut in this example doesn't have to lie along the real axis. It doesn't even have to be a straight line. Any continuous curve connecting the origin z = 0 with the point at infinity would work. In some cases the branch cut doesn't even have to pass through the point at infinity. For example, consider the relationship

Here the polynomial z² − 1 vanishes when z = ±1, so g evidently has two branch points. We can "cut" the plane along the real axis, from −1 to 1, and obtain a sheet on which g(z) is a single-valued function. Alternatively, the cut can run from z = 1 along the positive real axis through the point at infinity, then continue "up" the negative real axis to the other branch point, z = −1.

This situation is most easily visualized by using the stereographic projection described above. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis On the sphere one of these cuts runs longitudinally through the southern hemisphere, connecting a point on the equator (z = −1) with another point on the equator (z = 1), and passing through the south pole (the origin, z = 0) on the way. The second version of the cut runs longitudinally through the northern hemisphere and connects the same two equatorial points by passing through the north pole (that is, the point at infinity).

Restricting the domain of meromorphic functions

A meromorphic function is a complex function that is holomorphic and therefore analytic everywhere in its domain except at a finite, or countably infinite, number of points. In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane This article is about both real and complex analytic functions ^[8] The points at which such a function cannot be defined are called the poles of the meromorphic function. In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 Sometimes all these poles lie in a straight line. In that case mathematicians may say that the function is "holomorphic on the cut plane". Here's a simple example.

The gamma function, defined by

where γ is the Euler-Mascheroni constant, and has simple poles at 0, −1, −2, −3, . In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function The Euler–Mascheroni constant (also called the Euler constant) is a Mathematical constant recurring in analysis and Number theory, usually . . because exactly one denominator in the infinite product vanishes when z is zero, or a negative integer. In Mathematics, for a Sequence of numbers a 1 a 2 a 3. the infinite product ^[9] Since all its poles lie on the negative real axis, from z = 0 to the point at infinity, this function might be described as

"holomorphic on the cut plane, the cut extending along the negative real axis, from 0 (inclusive) to the point at infinity. "

Alternatively, Γ(z) might be described as

"holomorphic in the cut plane with −π < arg(z) < π and excluding the point z = 0. "

Notice that this cut is slightly different from the branch cut we've already encountered, because it actually excludes the negative real axis from the cut plane. The branch cut left the real axis connected with the cut plane on one side (0 ≤ θ), but severed it from the cut plane along the other side (θ < 2π).

Of course, it's not actually necessary to exclude the entire line segment from z = 0 to −∞ to construct a domain in which Γ(z) is holomorphic. All we really have to do is puncture the plane at a countably infinite set of points {0, −1, −2, −3, . . . }. But a closed contour in the punctured plane might encircle one or more of the poles of Γ(z), giving a contour integral that is not necessarily zero, by the residue theorem. In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated The residue theorem in Complex analysis is a powerful tool to evaluate Line integrals of Analytic functions over closed curves and can often be used to compute By cutting the complex plane we ensure not only that Γ(z) is holomorphic in this restricted domain – we also ensure that the contour integral of Γ over any closed curve lying in the cut plane is identically equal to zero. And this may be important in some mathematical arguments.

Specifying convergence regions

Many complex functions are defined by infinite series, or by continued fractions. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In analysis, a generalized continued fraction is a generalization of regular continued fractions in canonical form in which the partial numerators and the A fundamental consideration in the analysis of these infinitely long expressions is identifying the portion of the complex plane in which they converge to a finite value. A cut in the plane may facilitate this process, as the following examples show.

Consider the function defined by the infinite series

Since z² = (−z)² for every complex number z, it's clear that f(z) is an even function of z, so the analysis can be restricted to one half of the complex plane. In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive And since the series is undefined when

it makes sense to cut the plane along the entire imaginary axis and establish the convergence of this series where the real part of z is not zero before undertaking the more arduous task of examining f(z) when z is a pure imaginary number. ^[10]

In this example the cut is a mere convenience, because the points at which the infinite sum is undefined are isolated, and the cut plane can be replaced with a suitably punctured plane. In some contexts the cut is necessary, and not just convenient. Consider the infinite periodic continued fraction

It can be shown that f(z) converges to a finite value if and only if z is not a negative real number such that z < −¼. In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators a In other words, the convergence region for this continued fraction is the cut plane, where the cut runs along the negative real axis, from −¼ to the point at infinity. ^[11]

Gluing the cut plane back together

Main article: Riemann surface

We have already seen how the relationship

can be made into a single-valued function by splitting the domain of f into two disconnected sheets. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis It is also possible to "glue" those two sheets back together to form a single Riemann surface on which f(z) = z^½ can be defined as a holomorphic function whose image is the entire w-plane (except for the point w = 0). Here's how that works.

Imagine two copies of the cut complex plane, the cuts extending along the positive real axis from z = 0 to the point at infinity. On one sheet define 0 ≤ arg(z) < 2π, so that 1^½ = e⁰ = 1, by definition. On the second sheet define 2π ≤ arg(z) < 4π, so that 1^½ = e^iπ = −1, again by definition. Now flip the second sheet upside down, so the imaginary axis points in the opposite direction of the imaginary axis on the first sheet, with both real axes pointing in the same direction, and "glue" the two sheets together (so that the edge on the first sheet labeled "θ = 0" is connected to the edge labeled "θ < 4π" on the second sheet, and the edge on the second sheet labeled "θ = 2π" is connected to the edge labeled "θ < 2π" on the first sheet). The result is the Riemann surface domain on which f(z) = z^½ is single-valued and holomorphic (except when z = 0). ^[7]

To understand why f is single-valued in this domain, imagine a circuit around the unit circle, starting with z = 1 on the first sheet. When 0 ≤ θ < 2π we are still on the first sheet. When θ = 2π we have crossed over onto the second sheet, and are obliged to make a second complete circuit around the branch point z = 0 before returning to our starting point, where θ = 4π is equivalent to θ = 0, because of the way we glued the two sheets together. In other words, as the variable z makes two complete turns around the branch point, the image of z in the w-plane traces out just one complete circle.

Formal differentiation shows that

from which we can conclude that the derivative of f exists and is finite everywhere on the Riemann surface, except when z = 0 (that is, f is holomorphic, except when z = 0).

How can the Riemann surface for the function

also discussed above, be constructed? Once again we begin with two copies of the z-plane, but this time each one is cut along the real line segment extending from z = −1 to z = 1 – these are the two branch points of g(z). In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis We flip one of these upside down, so the two imaginary axes point in opposite directions, and glue the corresponding edges of the two cut sheets together. We can verify that g is a single-valued function on this surface by tracing a circuit around a circle of unit radius centered at z = 1. Commencing at the point z = 2 on the first sheet we turn halfway around the circle before encountering the cut at z = 0. The cut forces us onto the second sheet, so that when z has traced out one full turn around the branch point z = 1, w has taken just one-half of a full turn, the sign of w has been reversed (since e^iπ = −1), and our path has taken us to the point z = 2 on the second sheet of the surface. Continuing on through another half turn we encounter the other side of the cut, where z = 0, and finally reach our starting point (z = 2 on the first sheet) after making two full turns around the branch point.

The natural way to label θ = arg(z) in this example is to set −π < θ ≤ π on the first sheet, with π < θ ≤ 3π on the second. The imaginary axes on the two sheets point in opposite directions so that the counterclockwise sense of positive rotation is preserved as a closed contour moves from one sheet to the other (remember, the second sheet is upside down). Imagine this surface embedded in a three-dimensional space, with both sheets parallel to the xy-plane. Then there appears to be a vertical hole in the surface, where the two cuts are joined together. What if the cut is made from z = −1 down the real axis to the point at infinity, and from z = 1, up the real axis until the cut meets itself? Again a Riemann surface can be constructed, but this time the "hole" is horizontal. Topologically speaking, both versions of this Riemann surface are equivalent – they are orientable two-dimensional surfaces of genus one. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Mathematics, genus has a few different but closely related meanings Topology Orientable surface

Use of the complex plane in control theory

In control theory, one use of the complex plane is known as the 's-plane'. Control theory is an interdisciplinary branch of Engineering and Mathematics, that deals with the behavior of Dynamical systems The desired output It is used to visualise the roots of the equation describing a system's behaviour (the characteristic equation) graphically. The equation is normally expressed as a polynomial in the parameter 's' of the Laplace transform, hence the name 's' plane. In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic

Another related use of the complex plane is with the Nyquist stability criterion. The Nyquist stability criterion, named after Harry Nyquist, provides a simple test for Stability of a Closed-loop Control system by examining This is a geometric principle which allows the stability of a control system to be determined by inspecting a Nyquist plot of its frequency-phase response (or transfer function) in the complex plane. A Nyquist plot is used in automatic control and Signal processing for assessing the stability of a system with Feedback. A transfer function is a mathematical representation in terms of spatial or temporal frequency of the relation between the input and output of a ( linear time-invariant)

The 'z-plane' is a discrete-time version of the s-plane, where z-transforms are used instead of the Laplace transformation. In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real

Other meanings of "complex plane"

The preceding sections of this article deal with the complex plane as the geometric analogue of the complex numbers. Although this usage of the term "complex plane" has a long and mathematically rich history, it is by no means the only mathematical concept that can be characterized as "the complex plane". There are at least three additional possibilities.

1. 1+1-dimensional Minkowski space, also known as the split-complex plane, is a "complex plane" in the sense that the algebraic split-complex numbers can be separated into two real components that are easily associated with the point (x, y) in the Cartesian plane. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real
2. The set of dual numbers over the reals can also be placed into one-to-one correspondence with the points (x, y) of the Cartesian plane, and represent another example of a "complex plane". A variety of dualities in mathematics are listed at Duality (mathematics.
3. The vector space C×C, the Cartesian product of the complex numbers with themselves, is also a "complex plane" in the sense that it is a two-dimensional vector space whose coordinates are complex numbers. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

See also

• Constellation diagram
• Laplace transform
• Riemann sphere
• Riemann surface
• S plane
• Z-transform

External links

• Eric W. Weisstein, Argand Diagram at MathWorld. A constellation diagram is a representation of a signal modulated by a digital Modulation scheme such as Quadrature amplitude modulation or Phase-shift keying In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional The S plane is a mathematical domain where instead of viewing processes in the time domain modelled with time-based functions they are viewed as equations in the frequency domain In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

Notes

References

• Francis J. Flanigan, Complex Variables: Harmonic and Analytic Functions, Dover, 1983 ISBN 0-486-61388-7.
• Gino Moretti, Functions of a Complex Variable, Prentice-Hall, Inc. , 1964.
• H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc. , 1948; reprinted (1973) by Chelsea Publishing Company ISBN 0-8284-0207-8.
• E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, fourth edition, Cambridge University Press, 1927. Edmund Taylor Whittaker ( 24 October[[ 873]] - 24 March[[ 956]] was a mathematician who contributed widely to Applied mathematics, Mathematical physics (George Neville Watson ( 31 January 1886 – 2 February 1965) was an English mathematician a noted master in the application of Complex