Complex plane
Template:Short description Template:About-distinguish
Template:Complex analysis sidebar
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal Template:Mvar-axis, called the real axis, is formed by the real numbers, and the vertical Template:Mvar-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 Template:Dfn of the product is the product of the two absolute values, or moduli, and the angle or Template:Dfn 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.
Notational conventions
Complex numbers
In complex analysis, the complex numbers are customarily represented by the symbol Template:Mvar, which can be separated into its real (Template:Mvar) and imaginary (Template:Mvar) parts:
<math display="block">z = x + iy</math>
for example: Template:Math, where Template:Mvar and Template:Mvar are real numbers, and Template:Mvar is the imaginary unit. In this customary notation the complex number Template:Mvar corresponds to the point Template:Math in the Cartesian plane; the point Template:Math can also be represented in polar coordinates with:
<math display="block">\begin{align} x &= r\cos\theta \\ y &= r\sin\theta \\ r &= \sqrt{x^2+y^2} \\ \theta &= \arctan\frac{y}{x}. \end{align}</math>
In the Cartesian plane it may be assumed that the range of the arctangent function takes the values Template:Open-open (in radians), and some care must be taken to define the more complete arctangent function for points Template:Math when Template:Math.Template:Refn In the complex plane these polar coordinates take the form
<math display="block">\begin{align} z &= x + iy \\
&= |z|\left(\cos\theta + i\sin\theta\right) \\
&= |z|e^{i\theta}
\end{align}</math>
whereTemplate:Refn
<math display="block">\begin{align} |z| &= \sqrt{x^2+y^2} \\ \theta &= \arg(z) \\
&= \frac{1}{i}\ln\frac{z}{|z|} \\
&= -i\ln\frac{z}{|z|}.
\end{align}</math>
Here Template:Math is the Template:Dfn or Template:Dfn of the complex number Template:Mvar; Template:Mvar, the Template:Dfn of Template:Mvar, is usually taken on the interval Template:Math; and the last equality (to Template:Math) is taken from Euler's formula. Without the constraint on the range of Template:Mvar, the argument of Template:Mvar is multi-valued, because the complex exponential function is periodic, with period Template:Math. Thus, if Template:Mvar is one value of Template:Math, the other values are given by Template:Math, where Template:Mvar is any non-zero integer.Template:Sfn
While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers Template:Mvar and Template:Mvar is given by Template:Nowrap then for a complex number Template:Mvar its absolute value Template:Math coincides with its Euclidean norm, and its argument Template:Math with the angle turning from 1 to Template:Math.
The theory of contour 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 Template:Math. By convention the Template:Em direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point Template:Math, then travel up and to the left through the point Template:Math, then down and to the left through Template:Math, then down and to the right through Template:Math, and finally up and to the right to Template:Math, where we started.
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. Here it is customary to speak of the domain of Template:Math as lying in the Template:Mvar-plane, while referring to the range of Template:Math as a set of points in the Template:Mvar-plane. In symbols we write
<math display="block">\begin{align} z &= x + iy \\ f(z) &= w \\
&= u + iv
\end{align}</math>
and often think of the function Template:Mvar as a transformation from the Template:Mvar-plane (with coordinates Template:Math) into the Template:Mvar-plane (with coordinates Template:Math).
Complex plane notation
The complex plane is denoted as Template:Nowrap
Argand diagram
An Argand diagram is a geometric plot of complex numbers as points Template:Math using the horizontal Template:Mvar-axis as the real axis and the vertical Template:Mvar-axis as the imaginary axis.<ref>Template:Cite web</ref> Though named after Jean-Robert Argand (1768–1822), such plots were first described by Norwegian–Danish land surveyor and mathematician Caspar Wessel (1745–1818).Template:Refn Argand diagrams are frequently used to plot the positions of the zeros 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
It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere 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 a one-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 point Template:Math will be projected onto the south pole of the sphere. Since the interior of the unit circle lies inside the sphere, that entire region (Template:Math) will be mapped onto the southern hemisphere. The unit circle itself (Template:Math) will be mapped onto the equator, and the exterior of the unit circle (Template:Math) 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-called point 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 the extended complex plane. We 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.Template:Sfn
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 Template:Math. 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 origin Template:Math 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 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
<math display="block">w = f(z) = \pm\sqrt{z} = z^{1/2}.</math>
Before we can treat this relationship as a single-valued function, 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
<math display="block">y = g(x) = \sqrt{x} = x^{1/2}</math>
to be the non-negative real number Template:Mvar such that Template:Math. 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 Template:Math varies as the point Template:Mvar moves around the unit circle. We can write <math display="inline">z = re^{i\theta}</math> and take <math display="block">\begin{align} w &= z^{1/2} \\
&= \sqrt{r}\,e^{i\theta/2}, \quad 0\leq\theta\leq 2\pi.
\end{align}</math>
Evidently, as Template:Mvar moves all the way around the circle, Template:Mvar only traces out one-half of the circle. So one continuous motion in the complex plane has transformed the positive square root Template:Math into the negative square root Template:Math.
This problem arises because the point Template:Math has just one square root, while every other complex number Template:Math has exactly two square roots. On the real number line we could circumvent this problem by erecting a "barrier" at the single point Template:Math. A bigger barrier is needed in the complex plane, to prevent any closed contour from completely encircling the branch point Template:Math. This is commonly done by introducing a branch cut; in this case the "cut" might extend from the point Template:Math along the positive real axis to the point at infinity, so that the argument of the variable Template:Mvar in the cut plane is restricted to the range Template:Math.
We can now give a complete description of Template:Math. To do so we need two copies of the Template:Mvar-plane, each of them cut along the real axis. On one copy we define the square root of 1 to be Template:Math, and on the other we define the square root of 1 to be Template:Math. We call these two copies of the complete cut plane Template:Dfn. By making a continuity argument we see that the (now single-valued) function Template:Math maps the first sheet into the upper half of the Template:Mvar-plane, where Template:Math, while mapping the second sheet into the lower half of the Template:Mvar-plane (where Template:Math).<ref name="Moretti">Template:Harvnb.</ref>
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 origin Template:Math 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
<math display="block">w = g(z) = \left(z^2 - 1\right)^{1/2}.</math>
Here the polynomial Template:Math vanishes when Template:Math, so Template:Mvar evidently has two branch points. We can "cut" the plane along the real axis, from Template:Math to Template:Math, and obtain a sheet on which Template:Math is a single-valued function. Alternatively, the cut can run from Template:Math along the positive real axis through the point at infinity, then continue "up" the negative real axis to the other branch point, Template:Math.
This situation is most easily visualized by using the stereographic projection described above. On the sphere one of these cuts runs longitudinally through the southern hemisphere, connecting a point on the equator (Template:Math) with another point on the equator (Template:Math), and passing through the south pole (the origin, Template:Math) 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.Template:Refn The points at which such a function cannot be defined are called the poles 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:
The gamma function, defined by
<math display="block">\Gamma (z) = \frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left[\left(1+\frac{z}{n}\right)^{-1}e^{z/n}\right]</math>
where Template:Mvar is the Euler–Mascheroni constant, and has simple poles at Template:Math because exactly one denominator in the infinite product vanishes when Template:Mvar, or a negative integer.Template:Refn Since all its poles lie on the negative real axis, from Template:Math 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, Template:Math might be described as "holomorphic in the cut plane with Template:Math and excluding the point Template:Math."
This cut is slightly different from the Template:Dfn we've already encountered, because it actually Template:Em the negative real axis from the cut plane. The branch cut left the real axis connected with the cut plane on one side Template:Math, but severed it from the cut plane along the other side Template:Math.
Of course, it's not actually necessary to exclude the entire line segment from Template:Math to Template:Math to construct a domain in which Template:Math is holomorphic. All we really have to do is Template:Dfn the plane at a countably infinite set of points Template:Math. But a closed contour in the punctured plane might encircle one or more of the poles of Template:Math, giving a contour integral that is not necessarily zero, by the residue theorem. Cutting the complex plane ensures not only that Template:Math 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 by infinite series, or by continued 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
<math display="block">f(z) = \sum_{n=1}^\infty \left(z^2 + n\right)^{-2}.</math>
Because Template:Math for every complex number Template:Mvar, it's clear that Template:Math is an even function of Template:Mvar, so the analysis can be restricted to one half of the complex plane. And since the series is undefined when
<math display="block">z^2 + n = 0 \quad \iff \quad z = \pm i\sqrt{n},</math>
it makes sense to cut the plane along the entire imaginary axis and establish the convergence of this series where the real part of Template:Mvar is not zero before undertaking the more arduous task of examining Template:Math when Template:Mvar is a pure imaginary number.Template:Refn
In this example the cut is a mere convenience, because the points at which the infinite sum is undefined are isolated, and the Template:Em plane can be replaced with a suitably Template:Em plane. In some contexts the cut is necessary, and not just convenient. Consider the infinite periodic continued fraction
<math display="block">f(z) = 1 + \cfrac{z}{1 + \cfrac{z}{1 + \cfrac{z}{1 + \cfrac{z}{\ddots}}}}.</math>
It can be shown that Template:Math converges to a finite value if Template:Mvar is not a negative real number such that Template:Math. In other words, the convergence region for this continued fraction is the cut plane, where the cut runs along the negative real axis, from −Template:1/4 to the point at infinity.Template:Sfn
Gluing the cut plane back together
We have already seen how the relationship
<math display="block">w = f(z) = \pm\sqrt{z} = z^{1/2}</math>
can be made into a single-valued function by splitting the domain of Template:Mvar into two disconnected sheets. It is also possible to "glue" those two sheets back together to form a single Riemann surface on which Template:Math can be defined as a holomorphic function whose image is the entire Template:Mvar-plane (except for the point Template:Math). Here's how that works.
Imagine two copies of the cut complex plane, the cuts extending along the positive real axis from Template:Math to the point at infinity. On one sheet define Template:Math, so that Template:Math, by definition. On the second sheet define Template:Math, so that Template:Math, 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 "Template:Math" is connected to the edge labeled "Template:Math" on the second sheet, and the edge on the second sheet labeled "Template:Math" is connected to the edge labeled "Template:Math" on the first sheet). The result is the Riemann surface domain on which Template:Math is single-valued and holomorphic (except when Template:Math).<ref name="Moretti"/>
To understand why Template:Mvar is single-valued in this domain, imagine a circuit around the unit circle, starting with Template:Math on the first sheet. When Template:Math we are still on the first sheet. When Template:Math we have crossed over onto the second sheet, and are obliged to make a second complete circuit around the branch point Template:Math before returning to our starting point, where Template:Math is equivalent to Template:Math, because of the way we glued the two sheets together. In other words, as the variable Template:Mvar makes two complete turns around the branch point, the image of Template:Mvar in the Template:Mvar-plane traces out just one complete circle.
Formal differentiation shows that
<math display="block">f(z) = z^{1/2} \quad\Rightarrow\quad f' (z) = \tfrac{1}{2} z^{-1/2}</math>
from which we can conclude that the derivative of Template:Mvar exists and is finite everywhere on the Riemann surface, except when Template:Math (that is, Template:Mvar is holomorphic, except when Template:Math).
How can the Riemann surface for the function
<math display="block">w = g(z) = \left(z^2 - 1\right)^{1/2},</math>
also discussed above, be constructed? Once again we begin with two copies of the Template:Mvar-plane, but this time each one is cut along the real line segment extending from Template:Math to Template:Math – these are the two branch points of Template:Math. 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 Template:Mvar is a single-valued function on this surface by tracing a circuit around a circle of unit radius centered at Template:Math. Commencing at the point Template:Math on the first sheet we turn halfway around the circle before encountering the cut at Template:Math. The cut forces us onto the second sheet, so that when Template:Mvar has traced out one full turn around the branch point Template:Math, Template:Mvar has taken just one-half of a full turn, the sign of Template:Mvar has been reversed (because Template:Math), and our path has taken us to the point Template:Math on the Template:Strong sheet of the surface. Continuing on through another half turn we encounter the other side of the cut, where Template:Math, and finally reach our starting point (Template:Math on the Template:Strong sheet) after making two full turns around the branch point.
The natural way to label Template:Math in this example is to set Template:Math on the first sheet, with Template:Math 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 Template:Em). Imagine this surface embedded in a three-dimensional space, with both sheets parallel to the Template:Mvar-plane. Then there appears to be a vertical hole in the surface, where the two cuts are joined. What if the cut is made from Template:Math down the real axis to the point at infinity, and from Template:Math, 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.
Use in control theory
In control theory, one use of the complex plane is known as the s-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 parameter Template:Mvar of the Laplace transform, hence the name Template:Mvar-plane. Points in the s-plane take the form Template:Math, where 'Template:Mvar' is used instead of the usual 'Template:Mvar' to represent the imaginary component (the variable 'Template:Mvar' is often used to denote electrical current in engineering contexts).
Another related use of the complex plane is with the Nyquist stability criterion. This is a geometric principle which allows the stability of a closed-loop feedback system to be determined by inspecting a Nyquist plot of its open-loop magnitude and phase response as a function of frequency (or loop transfer function) in the complex plane.
The Template:Mvar-plane is a discrete-time version of the Template:Mvar-plane, where [[z-transform|Template:Mvar-transforms]] are used instead of the Laplace transformation.
Quadratic algebras
The complex plane is associated with two distinct quadratic spaces. For a point Template:Math in the complex plane, the squaring function Template:Math and the norm-squared Template:Math are both quadratic forms. The former is frequently neglected in the wake of the latter's use in setting a metric on the complex plane. The complex plane of this article is the quotient ring <math>\R[X]/(X^2 + 1)</math> where the ideal is the quadratic polynomial associated with the imaginary unit. There are two other ideals that yield quotient rings that are two-dimensional real algebras, and hence “complex planes”. These are the quadratic 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 are complex numbers. See also: Template:Section link.
See also
- 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
References
Works cited
- Template:Cite book
- Template:Cite book
- Template:Cite book Reprinted (1973) by Chelsea Publishing Company Template:ISBN.
- Template:Cite book
External links
- Jean-Robert Argand, "Template:Lang", 1806, online and analyzed on BibNum [for English version, click 'Template:Lang']