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Complex plane

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Geometric representation of the complex numbers

This article is about the geometric representation of complex numbers; it is not to be confused withcomplex coordinate plane,complex affine plane, orcomplex projective plane.

The complex plane. The horizontal axis represents thereal numbers $(\mathbb {R} )$ , the vertical axis represents theimaginary numbers $(\mathbb {I} )$ . Thecomplex number $4+4i$ is shown at the point $(4,4).$

{\displaystyle (\mathbb {R} )} — The complex plane. The horizontal axis represents thereal numbers $(\mathbb {R} )$ , the vertical axis represents theimaginary numbers $(\mathbb {I} )$ . Thecomplex number $4+4i$ is shown at the point $(4,4).$

Complex analysis
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Inmathematics, thecomplex plane is theplane formed by thecomplex numbers, with aCartesian coordinate system such that the horizontalx-axis, called thereal axis, is formed by thereal numbers, and the verticaly-axis, called theimaginary axis, is formed by theimaginary numbers.

The complex plane allows for a geometric interpretation of complex numbers. Underaddition, they add likevectors. Themultiplication of two complex numbers can be expressed more easily inpolar coordinates: the magnitude ormodulus of the product is the product of the twoabsolute values, or moduli, and theangle orargument 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 theArgand plane orGauss plane.

Notational conventions

Complex numbers

Incomplex analysis, the complex numbers are customarily represented by the symbolz, which can be separated into its real (x) and imaginary (y) parts:

for example:z = 4 + 5i, wherex andy are real numbers, andi is theimaginary unit. In this customary notation the complex numberz corresponds to the point(x,y) in theCartesian plane; the point(x,y) can also be represented inpolar coordinates with:

${\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\r&={\sqrt {x^{2}+y^{2}}}\\\theta &=\arctan {\frac {y}{x}}.\end{aligned}}$

In the Cartesian plane it may be assumed that therange of thearctangent function takes the values(−π/2, π/2) (inradians), and some care must be taken to define the more complete arctangent function for points(x,y) whenx ≤ 0.^{[note 1]} In the complex plane these polar coordinates take the form

${\begin{aligned}z&=x+iy\\&=|z|\left(\cos \theta +i\sin \theta \right)\\&=|z|e^{i\theta }\end{aligned}}$

where^{[note 2]}

${\begin{aligned}|z|&={\sqrt {x^{2}+y^{2}}}\\\theta &=\arg(z)\\&={\frac {1}{i}}\ln {\frac {z}{|z|}}\\&=-i\ln {\frac {z}{|z|}}.\end{aligned}}$

Here|z| is theabsolute value ormodulus of the complex numberz;θ, theargument ofz, is usually taken on the interval0 ≤θ < 2π; and the last equality (to|z|e^iθ) is taken fromEuler's formula. Without the constraint on the range ofθ, the argument ofz is multi-valued, because thecomplex exponential function is periodic, with period2πi. Thus, ifθ is one value ofarg(z), the other values are given byarg(z) =θ + 2nπ, wheren is any non-zero integer.^[2]

While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of aEuclidean vector space of dimension 2, where theinner product of complex numbersw andz is given by $\Re (w{\overline {z}})$ ; then for a complex numberz its absolute value|z| coincides with its Euclidean norm, and its argumentarg(z) with the angle turning from 1 to z.

The theory ofcontour integration comprises a major part of complex analysis. 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 thepositive direction is counterclockwise. For example, theunit circle is traversed in the positive direction when we start at the pointz = 1, then travel up and to the left through the pointz =i, then down and to the left through−1, then down and to the right through−i, and finally up and to the right toz = 1, where we started.

Almost all of complex analysis is concerned withcomplex 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. Here it is customary to speak of thedomain off(z) as lying in thez-plane, while referring to therange off(z) as a set of points in thew-plane. In symbols we write

${\begin{aligned}z&=x+iy\\f(z)&=w\\&=u+iv\end{aligned}}$

and often think of the functionf as a transformation from thez-plane (with coordinates(x,y)) into thew-plane (with coordinates(u,v)).

Complex plane notation

The complex plane is denoted as $\mathbb {C}$ .

Argand diagram

Geometric representation of the complex-valued pointz =x +yi in the complex plane. The distance along the line from the origin to the pointz =x +yi is themodulus orabsolute value ofz. The angleθ is theargument ofz.

AnArgand diagram is a geometricplot of complex numbers as pointsz =x +iy using the horizontalx-axis as the real axis and the verticaly-axis as the imaginary axis.^[3] Though named afterJean-Robert Argand (1768–1822), such plots were first described by Norwegian–Danish land surveyor and mathematicianCaspar Wessel (1745–1818).^{[note 3]} Argand diagrams are frequently used to plot the positions of thezeros and poles of a function in the complex plane.

The lengths of straight lines and curves in the complex plane represent real numbers: the physical length of the line or curve divided by the physical length of the radius of the unit circle. Similarly, angles between any two rays emanating from any point in the complex plane represent real numbers: the radian measure of the physical angle (i.e. the number of radians in the angle). In particular, the argument (phase) of a complex number is a real number, not a physical angle.

Stereographic projections

Main articles:Riemann sphere andStereographic projection

Riemann sphere which maps all points on a sphere except one to all points on the complex plane

It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given asphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is "above" the plane.

We can establish aone-to-one correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows. 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 pointz = 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, minus the north pole. 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 the north pole itself 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-calledpoint at infinity – and identifying it with the north pole on the sphere. This topological space, the complex plane plus the point at infinity, is known as theextended complex plane. We speak of a single "point at infinity" when discussing complex analysis. There are two points at infinity (positive, and negative) on thereal number line, but there is only one point at infinity (the north pole) in the extended complex plane.^[5]

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 originz = 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 yet plausible stereographic situation of the projection of a sphere onto a plane consisting of two or more values. For instance, the north pole of the sphere might be placed on top of the originz = −1 in a plane that is tangent to the circle. 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 acut in the complex plane. This idea arises naturally in several different contexts.

Multi-valued relationships and branch points

Consider the simple two-valued relationship

$w=f(z)=\pm {\sqrt {z}}=z^{1/2}.$

Before we can treat this relationship as a single-valuedfunction, the range of the resulting value must be restricted somehow. When dealing with the square roots of non-negative real numbers this is easily done. For instance, we can just define

$y=g(x)={\sqrt {x}}=x^{1/2}$

to be the non-negative real numbery such thaty² =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 off(z) varies as the pointz moves around the unit circle. We can write ${\textstyle z=re^{i\theta }}$ and take ${\begin{aligned}w&=z^{1/2}\\&={\sqrt {r}}\,e^{i\theta /2},\quad 0\leq \theta \leq 2\pi .\end{aligned}}$

Evidently, asz 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 roote⁰ = 1 into the negative square roote^iπ = −1.

This problem arises because the pointz = 0 has just one square root, while every other complex numberz ≠ 0 has exactly two square roots. On the real number line we could circumvent this problem by erecting a "barrier" at the single pointx = 0. A bigger barrier is needed in the complex plane, to prevent any closed contour from completely encircling thebranch pointz = 0. This is commonly done by introducing abranch cut; in this case the "cut" might extend from the pointz = 0 along the positive real axis to the point at infinity, so that the argument of the variablez in the cut plane is restricted to the range0 ≤ arg(z) < 2π.

We can now give a complete description ofw =z^1/2. To do so we need two copies of thez-plane, each of them cut along the real axis. On one copy we define the square root of 1 to bee⁰ = 1, and on the other we define the square root of 1 to bee^iπ = −1. We call these two copies of the complete cut planesheets. By making a continuity argument we see that the (now single-valued) functionw =z^1/2 maps the first sheet into the upper half of thew-plane, where0 ≤ arg(w) <π, while mapping the second sheet into the lower half of thew-plane (whereπ ≤ arg(w) < 2π).^[6]

The branch cut in this example does not have to lie along the real axis; it does not even have to be a straight line. Any continuous curve connecting the originz = 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

$w=g(z)=\left(z^{2}-1\right)^{1/2}.$

Here the polynomialz² − 1 vanishes whenz = ±1, sog evidently has two branch points. We can "cut" the plane along the real axis, from−1 to1, and obtain a sheet on whichg(z) is a single-valued function. Alternatively, the cut can run fromz = 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 thestereographic projection described above. 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

Ameromorphic function is a complex function that isholomorphic and thereforeanalytic everywhere in its domain except at a finite, orcountably infinite, number of points.^{[note 4]} The points at which such a function cannot be defined are called thepoles of the meromorphic function. Sometimes all of these poles lie in a straight line. In that case mathematicians may say that the function is "holomorphic on the cut plane". By example:

Thegamma function, defined by

$\Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left[\left(1+{\frac {z}{n}}\right)^{-1}e^{z/n}\right]$

whereγ is theEuler–Mascheroni constant, and has simple poles at0, −1, −2, −3, ... because exactly one denominator in theinfinite product vanishes whenz = 0, or a negative integer.^{[note 5]} Since all its poles lie on the negative real axis, fromz = 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 pointz = 0."

This cut is slightly different from thebranch cut we've already encountered, because it actuallyexcludes 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 fromz = 0 to−∞ to construct a domain in whichΓ(z) is holomorphic. All we really have to do ispuncture 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 ofΓ(z), giving acontour integral that is not necessarily zero, by theresidue theorem. Cutting the complex plane ensures not only thatΓ(z) is holomorphic in this restricted domain – but also that the contour integral of the gamma function over any closed curve lying in the cut plane is identically equal to zero.

Specifying convergence regions

Many complex functions are defined byinfinite series, or bycontinued fractions. 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

$f(z)=\sum _{n=1}^{\infty }\left(z^{2}+n\right)^{-2}.$

Becausez² = (−z)² for every complex numberz, it's clear thatf(z) is aneven function ofz, so the analysis can be restricted to one half of the complex plane. And since the series is undefined when

$z^{2}+n=0\quad \iff \quad z=\pm i{\sqrt {n}},$

it makes sense to cut the plane along the entire imaginary axis and establish the convergence of this series where the real part ofz is not zero before undertaking the more arduous task of examiningf(z) whenz is a pure imaginary number.^{[note 6]}

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

$f(z)=1+{\cfrac {z}{1+{\cfrac {z}{1+{\cfrac {z}{1+{\cfrac {z}{\ddots }}}}}}}}.$

Itcan be shown thatf(z) converges to a finite value ifz is not a negative real number such thatz < −1⁄4. In other words, the convergence region for this continued fraction is the cut plane, where the cut runs along the negative real axis, from −1⁄4 to the point at infinity.^[8]

Gluing the cut plane back together

Main article:Riemann surface

We havealready seen how the relationship

$w=f(z)=\pm {\sqrt {z}}=z^{1/2}$

can be made into a single-valued function by splitting the domain off into two disconnected sheets. It is also possible to "glue" those two sheets back together to form a singleRiemann surface on whichf(z) =z^1/2 can be defined as a holomorphic function whose image is the entirew-plane (except for the pointw = 0). Here's how that works.

Imagine two copies of the cut complex plane, the cuts extending along the positive real axis fromz = 0 to the point at infinity. On one sheet define0 ≤ arg(z) < 2π, so that1^1/2 =e⁰ = 1, by definition. On the second sheet define2π ≤ arg(z) < 4π, so that1^1/2 =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 whichf(z) =z^1/2 is single-valued and holomorphic (except whenz = 0).^[6]

To understand whyf is single-valued in this domain, imagine a circuit around the unit circle, starting withz = 1 on the first sheet. When0 ≤θ < 2π we are still on the first sheet. Whenθ = 2π we have crossed over onto the second sheet, and are obliged to make a second complete circuit around the branch pointz = 0 before returning to our starting point, whereθ = 4π is equivalent toθ = 0, because of the way we glued the two sheets together. In other words, as the variablez makes two complete turns around the branch point, the image ofz in thew-plane traces out just one complete circle.

Formal differentiation shows that

$f(z)=z^{1/2}\quad \Rightarrow \quad f'(z)={\tfrac {1}{2}}z^{-1/2}$

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

How can the Riemann surface for the function

$w=g(z)=\left(z^{2}-1\right)^{1/2},$

also discussedabove, be constructed? Once again we begin with two copies of thez-plane, but this time each one is cut along the real line segment extending fromz = −1 toz = 1 – these are the two branch points ofg(z). 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 thatg is a single-valued function on this surface by tracing a circuit around a circle of unit radius centered atz = 1. Commencing at the pointz = 2 on the first sheet we turn halfway around the circle before encountering the cut atz = 0. The cut forces us onto the second sheet, so that whenz has traced out one full turn around the branch pointz = 1,w has taken just one-half of a full turn, the sign ofw has been reversed (becausee^iπ = −1), and our path has taken us to the pointz = 2 on thesecond sheet of the surface. Continuing on through another half turn we encounter the other side of the cut, wherez = 0, and finally reach our starting point (z = 2 on thefirst sheet) after making two full turns around the branch point.

The natural way to labelθ = arg(z) in this example is to set−π <θ ≤π on the first sheet, withπ <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 isupside down). Imagine this surface embedded in a three-dimensional space, with both sheets parallel to thexy-plane. Then there appears to be a vertical hole in the surface, where the two cuts are joined. What if the cut is made fromz = −1 down the real axis to the point at infinity, and fromz = 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 areorientable two-dimensional surfaces ofgenus one.

Use in control theory

See also:Complex number § Control theory

Incontrol theory, one use of the complex plane is known as thes-plane. 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 parameters of theLaplace transform, hence the names-plane. Points in the s-plane take the forms =σ +jω, where 'j' is used instead of the usual 'i' to represent the imaginary component (the variable 'i' is often used to denoteelectrical current in engineering contexts).

Another related use of the complex plane is with theNyquist stability criterion. This is a geometric principle which allows the stability of a closed-loop feedback system to be determined by inspecting aNyquist plot of its open-loop magnitude and phase response as a function of frequency (or looptransfer function) in the complex plane.

Thez-plane is adiscrete-time version of thes-plane, wherez-transforms are used instead of the Laplace transformation.

Quadratic algebras

The complex plane is associated with two distinctquadratic spaces. For a pointz =x +iy in the complex plane, thesquaring functionz² and the norm-squaredx² + y² are bothquadratic forms. The former is frequently neglected in the wake of the latter's use in setting ametric on the complex plane. The complex plane of this article is thequotient ring $\mathbb {R} [X]/(X^{2}+1)$ where theideal is the quadratic polynomial associated with theimaginary unit. There are two other ideals that yield quotient rings that are two-dimensional real algebras, and hence “complex planes”. These are thequadratic algebras over the real number field.

Another meaning of "complex plane"

The preceding sections of this article deal with the complex plane in terms of a geometric representation 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 is an additional possibility.

Two-dimensional complex vector space, a "complex plane" in the sense that it is a two-dimensional vector space whose coordinates arecomplex numbers. See also:Complex affine space § Two dimensions.

See also

Mandelbrot fractal, imaged on a complex plane

Complex coordinate space
Complex geometry
Complex line
Constellation diagram
Riemann sphere, which is an extended complex plane
s-plane
In-phase and quadrature components
Real line

Notes

^A detailed definition of the complex argument in terms of thecomplete arctangent can be found at the description of theatan2 function.
^All the familiar properties of the complex exponential function, the trigonometric functions, and the complex logarithm can be deduced directly from thepower series fore^z. In particular, the principal value oflogr, where|r| = 1, can be calculated without reference to any geometrical or trigonometric construction.^[1]
^ Wessel's memoir was presented to the Danish Academy in 1797; Argand's paper was published in 1806.^[4]
^See alsoProof that holomorphic functions are analytic.
^The infinite product forΓ(z) isuniformly convergent on any bounded region where none of its denominators vanish; therefore it defines a meromorphic function on the complex plane.^[7]
^WhenRe(z) > 0 this sum converges uniformly on any bounded domain by comparison withζ(2), whereζ(s) is theRiemann zeta function.

References

^Whittaker & Watson 1927,Appendix.
^Whittaker & Watson 1927, p. 10.
^Weisstein, Eric W. (8 February 2024)."Argand Diagram".MathWorld. Retrieved17 February 2024.
^Whittaker & Watson 1927, p. 9.
^Flanigan 1983, p. 305.
^^a ^bMoretti 1964, pp. 113–119.
^Whittaker & Watson 1927, pp. 235–236.
^Wall 1948, p. 39.

Works cited

Flanigan, Francis J. (1983).Complex Variables: Harmonic and Analytic Functions. Dover.ISBN 0-486-61388-7.
Moretti, Gino (1964).Functions of a Complex Variable. Prentice-Hall.
Wall, H. S. (1948).Analytic Theory of Continued Fractions. D. Van Nostrand Company. Reprinted (1973) by Chelsea Publishing CompanyISBN 0-8284-0207-8.
Whittaker, E. T.;Watson, G. N. (1927).A Course in Modern Analysis (Fourth ed.). Cambridge University Press.

External links

Wikimedia Commons has media related toComplex plane.

Jean-Robert Argand, "Essai sur une manière de représenter des quantités imaginaires dans les constructions géométriques", 1806, online and analyzed onBibNum[for English version, click 'à télécharger']

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