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Complex plane

The complex plane, also known as the Argand plane, is a two-dimensional geometric representation of the set of complex numbers, where each complex number z = x + iy (with real part x and imaginary part y, and i satisfying i^2 = -1) is identified with the point (x, y) in the Euclidean plane, the horizontal axis serving as the real axis and the vertical axis as the imaginary axis.^[1]^[2] This visualization, first geometrically interpreted by Norwegian surveyor Caspar Wessel in 1797 through vector addition and multiplication, allows complex numbers to be treated as vectors or points, facilitating operations like addition via parallelogram law and multiplication via rotation and scaling.^[3]^[4] Independently developed by Swiss mathematician Jean-Robert Argand in 1806 and popularized by Carl Friedrich Gauss in his 1831 treatise, the complex plane provides a foundational framework for complex analysis, enabling the study of functions of complex variables, conformal mappings, and residues.^[3]^[5] Key features include the modulus |z| = \sqrt{x^2 + y^2} as the distance from the origin, the argument \arg(z) as the angle from the positive real axis, and polar form z = re^{i\theta}, which underpins Euler's formula and applications in physics, engineering, and signal processing.^[1]^[6] The plane extends the real line into a complete field, resolving equations like x^2 + 1 = 0 and supporting the fundamental theorem of algebra, which states every non-constant polynomial has a root in the complexes.^[7]^[8]

Fundamentals

Definition and Notation

The complex plane, denoted by the symbol \mathbb{C}, is the set of all complex numbers, which can be geometrically represented as points in the Euclidean plane \mathbb{R}^2 via the identification z = x + i y, where x = \operatorname{Re}(z) denotes the real part, y = \operatorname{Im}(z) denotes the imaginary part, and i is the imaginary unit satisfying the equation i^2 = -1.^[9]^[10] This correspondence allows complex numbers to be visualized as ordered pairs (x, y) in the plane, with the horizontal axis representing real components and the vertical axis representing imaginary components.^[9] Sometimes referred to as the Gauss plane, it provides a foundational framework for extending real analysis to the complex domain.^[9] Basic arithmetic operations on complex numbers align with vector operations in \mathbb{R}^2. Addition is defined componentwise: for z_1 = x_1 + i y_1 and z_2 = x_2 + i y_2, z_1 + z_2 = (x_1 + x_2) + i (y_1 + y_2), corresponding to the parallelogram law of vector addition.^[10] Multiplication follows the rule z_1 z_2 = (x_1 x_2 - y_1 y_2) + i (x_1 y_2 + x_2 y_1), which can be derived from the distributive property and the relation i^2 = -1, and geometrically represents a scaling and rotation in the plane.^[10] These operations preserve the field structure of \mathbb{C}, making it algebraically complete over the reals.^[11]

Argand Diagram

The Argand diagram provides a visual representation of complex numbers on a two-dimensional Cartesian plane, where the horizontal axis denotes the real part and the vertical axis denotes the imaginary part. A complex number z = x + iy, with x and y real numbers, is plotted as the point (x, y) in this plane.^[2] This graphical framework allows for the intuitive depiction of algebraic operations on complex numbers through geometric transformations.^[12] The concept of the Argand diagram originated with Norwegian mathematician Caspar Wessel, who presented a geometric interpretation of complex numbers as directed line segments in a plane in his 1799 paper "Om Directionens analytiske Betegning" to the Royal Danish Academy of Sciences and Letters.^[3] Independently, Swiss mathematician Jean-Robert Argand described a similar representation in his 1806 essay "Considérations sur la représentation géométrique d'un nombre complexe," which explicitly linked complex numbers to points in the plane.^[13] Although Carl Friedrich Gauss explored equivalent ideas in his unpublished notes around 1796 and later published them in 1831, the diagram is named after Argand due to his clear exposition.^[3] In the Argand diagram, each complex number corresponds to a position vector from the origin to the point (x, y). Addition of complex numbers z_1 = x_1 + i y_1 and z_2 = x_2 + i y_2 is performed as vector addition, resulting in z_1 + z_2 = (x_1 + x_2) + i (y_1 + y_2), which follows the parallelogram law: the resultant vector forms the diagonal of the parallelogram spanned by the two input vectors.^[14] This vector perspective underscores the isomorphism between the complex numbers and the Euclidean plane \mathbb{R}^2.^[15] Multiplication of a complex number by a positive real scalar r > 0 scales the corresponding vector by a factor of r, preserving its direction from the origin.^[16] In contrast, multiplication by the imaginary unit i rotates the vector by 90 degrees counterclockwise around the origin, since i (x + i y) = -y + i x, mapping the point (x, y) to (-y, x).^[17] These operations highlight the diagram's utility in illustrating both magnitude changes and angular transformations inherent to complex arithmetic.^[18] The distance from the origin to the point representing z corresponds to the modulus |z|.^[12]

Polar Representation

In the polar representation, a complex number z = x + i y, where x and y are real numbers, is expressed in terms of its modulus r = |z| and argument \theta = \arg(z) as z = r (\cos \theta + i \sin \theta). This form highlights the geometric interpretation of the complex number as a point in the plane at distance r from the origin and rotated by angle \theta from the positive real axis.^[19] The exponential form z = r e^{i \theta} follows directly from Euler's formula, which states that e^{i \theta} = \cos \theta + i \sin \theta. Leonhard Euler derived this relation in 1748 by expanding the exponential, sine, and cosine functions using their Taylor series around zero:

e^{i \theta} = \sum_{n=0}^{\infty} \frac{(i \theta)^n}{n!}, \quad \cos \theta = \sum_{n=0}^{\infty} (-1)^n \frac{\theta^{2n}}{(2n)!}, \quad \sin \theta = \sum_{n=0}^{\infty} (-1)^n \frac{\theta^{2n+1}}{(2n+1)!}.

Separating the real and imaginary parts in the series for e^{i \theta} yields the trigonometric identity, establishing the connection between exponential and polar forms.^[20]^[19] To convert from Cartesian to polar coordinates, compute r = \sqrt{x^2 + y^2} and \theta = \atantwo(y, x), where \atantwo is the two-argument arctangent function that accounts for the correct quadrant. The reverse conversion uses x = r \cos \theta and y = r \sin \theta. These transformations facilitate analysis in the Argand diagram by aligning with radial and angular measurements.^[21] Multiplication of complex numbers simplifies in polar form: if z_1 = r_1 e^{i \theta_1} and z_2 = r_2 e^{i \theta_2}, then z_1 z_2 = r_1 r_2 e^{i (\theta_1 + \theta_2)}. This operation corresponds to scaling the modulus by the product of the individual moduli and rotating by the sum of the arguments, providing an intuitive geometric view of composition in the complex plane.^[19]

Geometric Properties

Modulus and Argument

The modulus of a complex number z = x + iy, where x and y are real numbers, is defined as |z| = \sqrt{x^2 + y^2}, which represents the Euclidean distance from the origin to the point (x, y) in the complex plane.^[22] This non-negative real value quantifies the length of the vector corresponding to z.^[23] Key properties of the modulus include multiplicativity and the triangle inequality. For complex numbers z_1 and z_2, the modulus satisfies |z_1 z_2| = |z_1| |z_2|, reflecting how multiplication scales distances from the origin.^[24] Additionally, |z_1 + z_2| \leq |z_1| + |z_2|, with equality holding when z_1 and z_2 are non-negative real multiples of each other; this inequality arises from the geometric interpretation of vector addition in the plane.^[25] Geometrically, the set of points where |z| = r for a constant r > 0 forms a circle of radius r centered at the origin.^[26] The argument of a complex number z = x + iy \neq 0 is the angle \theta that the line from the origin to (x, y) makes with the positive real axis, satisfying \tan \theta = y/x with appropriate quadrant adjustment.^[27] Due to the periodicity of angles, the argument is multi-valued: \arg(z) = \theta + 2\pi k for any integer k.^[28] To obtain a single-valued function, the principal argument \Arg(z) is defined as the unique value in the interval (-\pi, \pi].^[29] In the complex plane, the locus where \arg(z) = \theta for a constant \theta consists of all rays emanating from the origin at angle \theta from the positive real axis, excluding the origin itself.^[26]

Complex Conjugation

The complex conjugate of a complex number z = x + iy, where x and y are real numbers, is defined as \bar{z} = x - iy.^[30] This operation replaces the imaginary unit i with -i while keeping the real part unchanged.^[31] The conjugation satisfies several algebraic properties: it is additive, so \overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}, and multiplicative, so \overline{z_1 z_2} = \bar{z_1} \bar{z_2}.^[30] Additionally, applying the operation twice yields the original number, \overline{\bar{z}} = z, and it preserves the modulus, |\bar{z}| = |z|.^[30] Geometrically, in the Argand diagram, complex conjugation acts as a reflection over the real axis.^[32] If z lies in the upper half-plane (where \operatorname{Im}(z) > 0), then \bar{z} lies in the lower half-plane, and vice versa; points on the real axis remain fixed.^[32] This reflection is an isometry of the complex plane that reverses orientation and maps circles and lines to their mirror images across the real axis.^[32] The real and imaginary parts of z can be expressed using conjugation: \operatorname{Re}(z) = \frac{z + \bar{z}}{2} and \operatorname{Im}(z) = \frac{z - \bar{z}}{2i}.^[30] These formulas highlight how conjugation isolates the real and imaginary components symmetrically. When viewing \mathbb{C} as a two-dimensional real vector space isomorphic to \mathbb{R}^2, the expression \operatorname{Re}(z_1 \bar{z_2}) provides the standard Euclidean inner product, corresponding to the dot product of the associated real vectors.^[33] (Note: While Stack Exchange is not a primary source, this identity is standard in linear algebra texts deriving the polarization identity for norms.)

Stereographic Projections

The Riemann sphere, denoted \hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}, compactifies the complex plane by adjoining a point at infinity, yielding a space topologically equivalent to a sphere.^[34] This construction, introduced by Bernhard Riemann, models the extended complex plane as the unit sphere in \mathbb{R}^3, where the equatorial plane identifies with \mathbb{C} via suitable coordinates.^[34] Stereographic projection provides a bijective correspondence between the Riemann sphere minus the north pole and the complex plane. Specifically, projecting from the north pole (0, 0, 1), which represents \infty, onto the equatorial plane Z = 0, maps a point (X, Y, Z) on the unit sphere X^2 + Y^2 + Z^2 = 1 to the complex number z = \frac{X + iY}{1 - Z}.^[34] The inverse stereographic projection maps z = x + iy \in \mathbb{C} to the sphere point \left( \frac{2x}{|z|^2 + 1}, \frac{2y}{|z|^2 + 1}, \frac{|z|^2 - 1}{|z|^2 + 1} \right), with \infty corresponding to the north pole.^[34] This mapping is conformal, preserving angles and orientations, which facilitates the transfer of geometric structures between the plane and sphere.^[35] Moreover, circles on the Riemann sphere project to either circles or straight lines in the complex plane, with lines arising as images of circles passing through the north pole.^[35] On the Riemann sphere, complex conjugation in \mathbb{C} corresponds to reflection through the equatorial plane, an isometry that swaps the upper and lower hemispheres while fixing the equator. This geometric interpretation underscores the symmetry of the extended plane under conjugation.

Riemann Surfaces

Cutting the Plane

In complex analysis, cuts are introduced into the complex plane to resolve the multi-valued nature of certain functions, such as the logarithm or square root, by excluding specific paths where the function values exhibit discontinuities or jumps between branches. These branch cuts serve as artificial barriers that prevent analytic continuation around points where the function would otherwise cycle through multiple values, thereby defining a single-valued branch of the function on the remaining domain. For instance, without a cut, encircling the origin in the complex plane would increment the argument by $2\pi, leading to a change of $2\pi i in the value of the logarithm (or analogous shifts for other multi-valued functions), resulting in a different function value upon return to the starting point.^[36] Common branch cuts take the form of rays or slits emanating from a branch point, often extending to infinity to connect isolated singularities while avoiding other critical points. A standard choice for the principal branch is a ray from the origin along the negative real axis to infinity, which isolates the discontinuity for functions like \log z or \sqrt{z} without intersecting the positive real axis where values are typically real and continuous. More general slits can be drawn as straight lines or curves between branch points, provided they do not enclose regions that would preserve multi-valuedness, ensuring the cut is a simple arc in the plane.^[37] Topologically, introducing a branch cut along such a ray transforms the punctured complex plane into a simply connected domain, where every closed curve can be contracted to a point without crossing the cut, facilitating unique analytic continuation of the function within that region. This modification eliminates non-contractible loops around the branch point, making the domain suitable for defining holomorphic branches and applying theorems like the monodromy theorem. The resulting space supports a consistent choice of function values, with the cut itself becoming a line of discontinuity where the function approaches different limits from either side.^[37] A prominent example is the principal branch of the complex logarithm, \operatorname{Log} z = \ln |z| + i \arg z, defined by placing a branch cut along the ray from 0 to -\infty on the negative real axis. This restricts the argument to the principal value in (-\pi, \pi], ensuring that \operatorname{Im}(\operatorname{Log} z) \in (-\pi, \pi] and yielding a holomorphic function on \mathbb{C} minus the cut. Along the cut, the function jumps by $2\pi i when approached from above versus below, highlighting the cut's role in enforcing single-valuedness.^[37]

Branch Points and Multi-valued Functions

In complex analysis, multi-valued functions arise when a single point in the domain corresponds to multiple values in the range, necessitating careful handling to define analytic branches. Branch points are specific singularities where such functions cannot be rendered single-valued in any punctured neighborhood, as analytic continuation around these points yields a different branch of the function. These points mark locations where the function's behavior fundamentally changes due to the multi-sheeted nature of its inverse image under mappings like exponentiation.^[19] A classic example is the complex logarithm, defined as \log z = \ln |z| + i \arg(z), where \arg(z) is multi-valued, differing by multiples of $2\pi i k for integers k \in \mathbb{Z}. The origin z = 0 and infinity serve as branch points for \log z, as encircling z = 0 once increases the value by $2\pi i, reflecting its infinite-sheeted structure. Similarly, the square root function \sqrt{z} = \sqrt{r} e^{i\theta/2} (in polar form with z = r e^{i\theta}) is multi-valued, with values differing by a sign change upon full rotation around z = 0, and it also branches at infinity. More generally, functions like z^{1/n} for integer n > 1 exhibit branching at z = 0 and infinity, cycling through n distinct values.^[19]^[38] Branch points are classified as algebraic or logarithmic based on the order of branching. Algebraic branch points, such as z = 0 for \sqrt{z}, involve a finite number of sheets—here, two sheets connected at the branch point, forming a 2-sheeted covering. In contrast, logarithmic branch points, like those for \log z, require infinitely many sheets, as the argument winds indefinitely without repetition. The order of an algebraic branch point is finite, corresponding to the minimal integer p such that p traversals return to the original value, as in z^{q/p}.^[19]^[39]^[40] To detect a branch point, consider analytic continuation along a closed path in the complex plane: if encircling a point changes the function's value (e.g., via monodromy), it is a branch point; paths avoiding it preserve the value. For instance, traversing a loop around z = 0 for z^{1/2} swaps the branches, while for \log z, it shifts by $2\pi i. This non-trivial monodromy distinguishes branch points from regular points.^[19]^[41] Riemann surfaces provide a geometric resolution, portraying these functions as single-valued on a multi-sheeted covering of the complex plane, with sheets joined along cuts emanating from branch points. For \sqrt{z}, the two sheets connect at z = 0 and infinity, yielding a compact surface homeomorphic to a sphere; for \log z, infinitely many sheets form a helical structure. The branching order determines the connectivity, enabling global analyticity.^[19]^[38]

Gluing the Cut Plane

To resolve the multi-valuedness arising from branch cuts in the complex plane, Riemann surfaces are constructed by gluing multiple copies of the cut plane along their boundaries, creating a manifold where multi-valued functions become single-valued analytic functions. This gluing process identifies the edges of the cuts in a specific manner, ensuring compatibility of the function values across the seams. For functions with infinite branches, such as the complex logarithm, the construction involves an infinite number of sheets; for algebraic functions defined by polynomial equations, the number of sheets is finite, leading to compact surfaces.^[42] The Riemann surface for the logarithm \log z is formed by stacking infinitely many copies of the plane cut along a ray from the origin, typically the negative real axis, where the argument \arg z is defined continuously on each sheet within an interval of length $2\pi. The upper edge of the cut on one sheet, where \arg z approaches $2\pi k from below for integer k, is glued to the lower edge of the next sheet, where \arg z approaches $2\pi(k+1) from above, with the function values matching via \log z = \ln |z| + i \arg z. This identification results in a helical Riemann surface that spirals infinitely around the branch point at z=0, serving as the universal covering space of the punctured plane \mathbb{C} \setminus \{0\}.^[42] For algebraic functions, such as the square root \sqrt{z}, the construction uses a finite number of sheets—two in this case—each a copy of the plane cut along a branch cut from the branch point at z=0 to infinity. The upper edge of the cut on the first sheet, where \sqrt{z} takes positive values, is glued to the lower edge on the second sheet, and vice versa, with the function switching signs across the seam to ensure analyticity. This gluing yields a compact Riemann surface of genus zero, topologically a sphere, on which \sqrt{z} is single-valued and analytic except at the branch point. More generally, for roots of higher-degree polynomials, the number of sheets equals the degree, and the resulting surface may have positive genus depending on the number and arrangement of branch points.^[42] On these glued Riemann surfaces, analytic continuation of multi-valued functions proceeds along paths that may cross from one sheet to another without encountering discontinuities, as the gluing ensures smooth transitions. For simply connected domains in the base plane, the full Riemann surface acts as a universal cover, allowing paths to lift uniquely while preserving analyticity. Branch points serve as loci where sheets connect, facilitating this global structure.^[42] This construction of Riemann surfaces by gluing cut planes originated in Bernhard Riemann's 1851 doctoral thesis, where he introduced them to rigorously define and integrate hyperelliptic functions arising from algebraic curves.^[43]

Applications in Analysis

Domain Restrictions for Meromorphic Functions

Meromorphic functions are holomorphic in their domain except at isolated poles, where they exhibit singularities of finite order. These functions can be expressed locally as a quotient of two holomorphic functions, with the denominator vanishing at the poles. For multi-valued meromorphic functions, such as $1/\sqrt{z}, branch cuts are essential to restrict the domain and define a single-valued branch that remains meromorphic.^[19]^[44] The principal branch of a multi-valued meromorphic function is typically defined using a standard branch cut, ensuring analyticity except at poles within the slit domain. For instance, the principal branch of \sqrt{z} uses a cut along the negative real axis, making the principal branch of $1/\sqrt{z} holomorphic on \mathbb{C} \setminus ((-\infty, 0] \cup \{0\}), with branch point singularities at z=0 and at infinity. Similarly, for \tan z, which is meromorphic on the entire complex plane with poles at z = (n + 1/2)\pi for integers n, branch cuts can be chosen to avoid these poles when defining related multi-valued extensions.^[19]^[19]^[19] The slit plane, defined as \mathbb{C} minus the branch cuts, serves as the domain where these functions are holomorphic except at poles. For \arcsin z, the principal branch is analytic on \mathbb{C} \setminus ((-\infty, -1] \cup [1, \infty)), with cuts along the real axis from -\infty to -1 and from $1 to \infty, branching at z = \pm 1. This restriction ensures no jumps across the cuts, preserving holomorphy in the slit domain. The choice of cuts for such functions follows conventions that align with the negative real axis for algebraic branches and real-axis exteriors for inverse trigonometric ones, making the functions continuous up to the slits from one side.^[45]^[44]^[45] These domain restrictions impact the computation of residues and contour integrals by defining paths that avoid discontinuities. Cuts ensure contours remain in regions of holomorphy, allowing Cauchy's integral theorem and residue theorem to apply without encircling branch points, thus enabling evaluation of integrals over closed paths homologous to zero while accounting only for poles. For example, in the slit plane for \arcsin z, residues at poles (if present in extensions) can be computed without jump discontinuities interfering.^[19]^[45]

Convergence Regions for Series

In the complex plane, power series expansions of analytic functions converge within open disks centered at the expansion point. A power series \sum_{n=0}^{\infty} a_n (z - c)^n, where a_n and c are complex coefficients, converges absolutely and uniformly to an analytic function inside the disk |z - c| < R, where R is the radius of convergence given by R = \frac{1}{\limsup_{n \to \infty} |a_n|^{1/n}}, and diverges outside this disk.^[46]^[47] This disk represents the largest region around c where the series defines a holomorphic function, with R potentially finite, zero, or infinite depending on the growth of the coefficients.^[48] For functions with isolated singularities, Laurent series extend this representation to annular regions in the complex plane. A Laurent series \sum_{n=-\infty}^{\infty} a_n (z - c)^n converges in an annulus r < |z - c| < R, where the principal part \sum_{n=1}^{\infty} a_{-n} (z - c)^{-n} accounts for the behavior near the singularity at c, and the regular part \sum_{n=0}^{\infty} a_n (z - c)^n behaves like a power series outward.^[49]^[50] Inside the inner radius r and outside the outer radius R, the series diverges, allowing representation of functions holomorphic in punctured disks or between singularities.^[51] Analytic continuation enables extending these series representations beyond individual disks or annuli by overlapping regions. If two power series converge in overlapping disks and agree on the intersection, their analytic continuations define a single holomorphic function on the union, covering larger domains in the complex plane without singularities.^[52] This process can be iterated across chains of overlapping disks to reach maximal domains, sometimes requiring cuts to handle multi-valued extensions beyond convergence boundaries. The exponential function provides a key example of a power series with infinite radius of convergence. The series \sum_{n=0}^{\infty} \frac{z^n}{n!} converges everywhere in the complex plane, defining e^z as an entire function holomorphic on all of \mathbb{C}.^[46]^[53] In contrast, the geometric series \sum_{n=0}^{\infty} z^n converges to \frac{1}{1-z} inside the unit disk |z| < 1, with radius R = 1 determined by the singularity at z = 1, and diverges for |z| \geq 1.^[47]^[54]

Multi-valued Relationships

In the complex plane, inverse functions often exhibit multi-valued behavior due to the non-simply connected nature of domains excluding branch points, such as the origin. For instance, the equation z = w^2 defines a mapping from the w-plane to the z-plane that serves as a double cover, where each nonzero z corresponds to two values of w = \pm \sqrt{z}, differing by a sign change.^[55] To resolve this, one typically selects a principal branch by introducing a branch cut, such as along the negative real axis, restricting the argument of z to (-\pi, \pi], yielding a single-valued square root function in the cut plane.^[55] This multi-valuedness arises because analytic continuation around the branch point at z=0 swaps the two branches, reflecting the topological obstruction to a global single-valued inverse.^[42] Covering maps provide a framework for understanding such multi-sheeted relationships, where the complex plane or its universal cover projects onto punctured domains in an infinite-to-one manner. The exponential map e^z: \mathbb{C} \to \mathbb{C} \setminus \{0\} exemplifies this, acting as an infinite-sheeted covering of the punctured plane, with each point in the target having infinitely many preimages differing by $2\pi i k for k \in \mathbb{Z}.^[42] The deck transformations, generated by translations z \mapsto z + 2\pi i, form the group of symmetries preserving the fibers, and the inverse, the complex logarithm, is inherently multi-valued, requiring branch choices to define locally.^[42] This structure captures the periodic nature of the exponential, linking it to the cylinder topology of the domain and codomain. Modular functions introduce further multi-valued relationships on subspaces of the complex plane, particularly the upper half-plane \mathcal{H} = \{\tau \in \mathbb{C} \mid \Im(\tau) > 0\}. The j-invariant, defined as j(\tau) = 1728 g_2(\tau)^3 / \Delta(\tau) where g_2 and \Delta are modular forms and the discriminant, is a holomorphic function on \mathcal{H} invariant under the modular group \mathrm{SL}_2(\mathbb{Z}), mapping isomorphically to \mathbb{C} and classifying elliptic curves up to isomorphism.^[56] Its q-expansion j(\tau) = q^{-1} + 744 + 196884 q + \cdots with q = e^{2\pi i \tau} reveals poles at cusps, and extensions like j(N\tau) for level N are multi-valued over \mathbb{C}(j) with degree equal to the index [\Gamma(1) : \Gamma_0(N)], generating fields of modular functions tied to elliptic curve moduli.^[56] These multi-valued mappings are underpinned by the topology of the complex plane, where the fundamental group of the punctured plane \mathbb{C} \setminus \{0\} is isomorphic to \mathbb{Z}, generated by a loop encircling the origin once.^[57] This infinite cyclic group captures the winding number, which induces monodromy on multi-valued functions: analytic continuation along a generator loop increments the logarithm by $2\pi i or swaps branches of the square root, preventing single-valued extensions without cuts or Riemann surface constructions.^[57] Such topological invariants explain the persistent multi-sheeted nature of inverses and coverings around punctures.^[57]

Applications in Other Fields

Control Theory

In control theory, the complex plane, known as the s-plane, is fundamental for analyzing the stability and dynamic behavior of linear time-invariant systems through the Laplace transform. Here, the complex variable s = \sigma + j\omega represents the system's frequencies, with the real part \sigma determining exponential growth (\sigma > 0) or decay (\sigma < 0), and the imaginary part \omega corresponding to oscillatory components. A system is asymptotically stable if all poles of its transfer function lie in the open left half-plane (LHP, where \operatorname{Re}(s) < 0), as this ensures that transient responses decay over time without oscillation divergence.^[58] The Nyquist stability criterion employs the complex plane to evaluate closed-loop stability from the open-loop transfer function G(s). It involves plotting the Nyquist contour, which maps G(j\omega) for \omega ranging from -\infty to \infty along the imaginary axis in the s-plane onto the complex G(j\omega)-plane. The criterion states that the closed-loop system is stable if the number of counterclockwise encirclements of the critical point -1 + j0 by this plot equals the number of open-loop poles in the right half-plane (RHP); for systems with no RHP poles, no net encirclements indicate stability. This graphical method avoids direct computation of closed-loop poles and was originally developed by Harry Nyquist to analyze feedback amplifiers, as presented in his 1932 paper on regeneration theory.^[59] Bode plots extend frequency-domain analysis in the complex plane by separating the transfer function G(j\omega) into magnitude and phase components, plotted against the logarithm of frequency \log \omega. The magnitude plot displays $20 \log_{10} |G(j\omega)| in decibels, while the phase plot shows \arg(G(j\omega)) in degrees; these semi-log graphs reveal asymptotic behaviors near corner frequencies and enable assessment of gain margin (distance from 0 dB at phase crossover) and phase margin (distance from -180° at gain crossover), both critical for robustness against instability. Hendrik Bode introduced this approach during his work on feedback systems at Bell Laboratories in the 1940s, detailed in his 1945 book on network analysis. The root locus technique visualizes pole migration in the s-plane as a design parameter, such as the feedback gain K, varies from 0 to \infty. For a system with open-loop transfer function KG(s), the loci are the paths traced by solutions to $1 + KG(s) = 0 (the characteristic equation), starting at open-loop poles (K=0) and ending at open-loop zeros or infinity (K \to \infty); branches in the LHP indicate stabilizing gains, while crossings into the RHP signal instability thresholds. This method facilitates controller synthesis by highlighting damping and settling time influences from pole positions. Walter R. Evans developed the root locus in the late 1940s, publishing it in 1950 to address synthesis challenges in servomechanisms.^[60] These tools—Nyquist plots, Bode plots, and root loci—emerged in the 1930s and 1940s amid advances in feedback control for telecommunications and servosystems, primarily at Bell Laboratories, revolutionizing stability assessment without solving high-order polynomials directly.^[59]^[60]

Quadratic Spaces

The complex plane \mathbb{C}, regarded as a Hermitian space via the standard Hermitian inner product \langle z_1, z_2 \rangle = z_1 \bar{z_2}, admits the associated positive definite form Q(z) = z \bar{z} = |z|^2.^[61] This form satisfies Q(z) > 0 for z \neq 0, ensuring that the only isotropic vector is the zero vector and rendering the space anisotropic.^[61] Viewed as a two-dimensional vector space over the reals, \mathbb{C} carries the standard Euclidean quadratic form Q(z) = |z|^2 = x^2 + y^2. In the broader context of quadratic spaces over the real numbers, non-degenerate forms of a given dimension are classified up to isometry by their signature (number of positive and negative eigenvalues), with the canonical form consisting of a diagonal matrix with +1's and -1's.^[62] For the two-dimensional Euclidean case, the Witt decomposition is trivial, as the positive definite nature precludes hyperbolic planes in the orthogonal decomposition.^[63] This structure finds applications in algebraic geometry, where the complex plane serves as the affine space \mathbb{C}^1 equipped with the Hermitian metric |z|^2, facilitating the study of curves defined by quadratic equations through their intersections and geometric properties under this norm.^[64]

Alternative Interpretations

Other Meanings of Complex Plane

In algebraic geometry, the complex affine plane is formalized as the spectrum of the polynomial ring \mathbb{C}[x,y], denoted \operatorname{Spec}(\mathbb{C}[x,y]), where points correspond to maximal ideals (x-a, y-b) for a,b \in \mathbb{C}, and the structure sheaf assigns coordinate rings to open sets, enabling the study of algebraic varieties as geometric objects over the complex numbers.^[65] This construction provides a foundation for schemes, generalizing classical affine varieties and allowing intersection theory and cohomology to be developed in a categorical framework.^[66] The projective plane \mathbb{P}^2(\mathbb{C}), obtained by projectivizing the affine plane via homogeneous coordinates [x:y:z], compactifies the space by incorporating points at infinity, which is essential for resolving singularities in curves and analyzing global properties like the degree-genus formula for plane curves.^[67] In quantum mechanics, the complex plane represents the state space of a qubit through stereographic projection of the Bloch sphere, mapping pure states |\psi\rangle = \alpha |0\rangle + \beta |1\rangle (with |\alpha|^2 + |\beta|^2 = 1) to points z = \beta / \alpha \in \mathbb{C} \cup \{\infty\}, where the north pole projects to infinity and facilitates visualization of quantum superpositions and measurements on the Riemann sphere.^[68] This projection preserves the geometry of SU(2) rotations as Möbius transformations on the complex plane, aiding in the analysis of quantum gates and entanglement in qubit systems.^[68] In formulations of complex spacetime within relativistic quantum mechanics, the structure incorporates complex coordinates to unify space and time dimensions, potentially extending 2D Minkowski spacetime into a complex plane where Lorentz invariance manifests through analytic continuations, though empirical verification remains theoretical.^[69] In computer science, particularly lattice-based cryptography, the Gaussian integers \mathbb{Z} = \{a + bi \mid a,b \in \mathbb{Z}\} embed as a square lattice in the complex plane, providing the ring structure for polynomial extensions like \mathbb{Z}/(x^n - 1) in NTRU variants such as CNTRU, where security relies on the hardness of shortest vector problems in these ideal lattices.^[70] These lattices enable efficient key generation and encryption by leveraging the Euclidean algorithm in \mathbb{Z}, with applications in post-quantum schemes resistant to quantum attacks due to the geometric density and approximation factors in the complex plane.^[70]

References

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[PDF] 1 The Complex Plane
1 The Complex Plane. A complex number z is given by a pair of real numbers x and y and is written in the form z = x+iy, where i satisfies i2 = −1.
[2]
The Complex Plane - Stanford CCRMA
A complex plane (or Argand diagram) is any 2D graph in which the horizontal axis is the real part and the vertical axis is the imaginary part of a complex ...
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[PDF] I.3. The Complex Plane
Sep 11, 2023 · Caspar Wessel (June 8, 1745–March 25, 1818) was a Norwegian mathematician who first gave a geometric interpretation to the representation of.
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[PDF] A Short History of Complex Numbers - URI Math Department
Wessel's approach used what we today call vectors. He uses the geometric addition of vectors (parallelogram law) and defined multiplication of vectors in terms ...
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[PDF] Teaching the Complex Numbers: What History and Philosophy of ...
Jan 1, 2013 · 1 Gauss published a memoir about the geometric interpretation of complex numbers in 1832, which launched its wide acceptance in the mathematical ...
[6]
2.1 The Complex Plane - BOOKS
A complex number z is an ordered pair of real numbers x and y which are distinguished from each other by adjoining the symbol i to the second number.
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Lesson on the Complex Plane - University of Illinois
Mar 1, 2015 · This lessons explains how, ℂ , the Field of Complex Numbers is ideally suited to describe the geometry of the Cartesian plane, just as the field ...
[8]
[PDF] 17.1 Complex Numbers - Penn Math
DEFINITION 17.1 Complex Number. A number of the form z = x + iy where x and y are real numbers and i is a number such that i² = -1 is called a complex number. ...
[9]
Complex Plane -- from Wolfram MathWorld
is the real line. The complex plane is sometimes called the Argand plane or Gauss plane, and a plot of complex numbers in the plane is sometimes called an ...
[10]
Complex Number Primer - The Definition - Pauls Online Math Notes
Given two complex numbers z1=a+bi z 1 = a + b i and z2=c+di z 2 = c + d i we define addition and multiplication as follows, z1+z2=(a+c)+(b+d)iz1z2=(ac−bd)+(ad+ ...
[11]
Complex Number -- from Wolfram MathWorld
Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions.
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The Complex Plane - BOOKS
A number that lies on the real axis has zero for its imaginary part. Such a number is called real or, for emphasis, pure real. A number that lies on the ...
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[PDF] The Square Root of i - UNL Digital Commons
Jean Robert Argand rediscovered Wessel's work in 1806 with his publication of the Argand diagram. In this geometrical representation of complex numbers ...
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[PDF] Complex numbers - UCSB
Because complex numbers obey the same addition rule that applies to vectors in a plane, they can be added graphically by the parallelogram rule (Fig. 2.2a) ...
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[PDF] Complex numbers - University of Iowa Physics
The real part of the complex number is plotted on the horizontal axis and the imaginary part of the vertical axis. The point (x,y) represents the complex number ...
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[PDF] Math 147 — Complex Analysis - UCI Mathematics
Nov 26, 2024 · When drawn with axes, the complex plane is known as the Argand diagram and we refer, respectively, to the real and imaginary axes. z w z + w.
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Complex numbers: multiplication
Stated more briefly, multiplication by i gives a 90° counterclockwise rotation about 0. You can analyze what multiplication by –i does in the same way. You'll ...Missing: Argand | Show results with:Argand
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[PDF] COMPLEX EIGENVALUES Math 21b, O. Knill
If z = x + iy is written as a vector x y. , then multiplication with an other complex number w is a dilation-rotation: a scaling by |w| and a rotation by arg(w) ...Missing: diagram | Show results with:diagram
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[PDF] Ahlfors, Complex Analysis
Complex Analysis has successfully maintained its place as the standard elementary text on functions of one complex variable. There is, never-.
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[PDF] e, π and i - How Euler Did It - Mathematical Association of America
In this column we will look at the origins of the Euler identity, see what Euler contributed, and consider whether it is correctly named. Phase 1: 1702 to 1729.
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Complex Analysis - THEODORE GAMELIN - Google Books
2 Polar Representation. 5. 3 Stereographic Projection. 11. 4 The Square and Square Root Functions. 15. 5 The Exponential Function. 19. 6 The Logarithm Function.
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[PDF] Introduction to Complex Numbers. Summary - UC Davis Mathematics
Jan 14, 2007 · Just use the definition of + (when used to denote the addition of complex numbers) and the familiar properties of the addition of real numbers.
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Complex Number Primer - Pauls Online Math Notes
The modulus of a complex number is always a real number and in fact it will never be negative since square roots always return a positive number or zero.
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[PDF] Complex Numbers
Geometrically, the product of z1 and z2 has length equal to the product of the lengths of z1 and z2 and argument equal to the sum of the arguments of z1 and z2.<|control11|><|separator|>
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[PDF] Complex conjugation, Modulus, and Inequalities
The vector addition corresponds to a triangle in the complex plane whose three sides have length |ζ|, |w| and |ζ + w|. Hence, the triangle inequality is simply ...
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Graphics for complex analysis -- Douglas N. Arnold
Jan 29, 1997 · ... constant modulus, and rays of constant argument to rays of constant argument. ... circle in the plane is another line or circle. This ...
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[PDF] Chapter 6 Complex Numbers
Figure 7.6 shows circles along the real and imaginary axes under the mapping. Lines in the complex plane are also mapped to circles on the unit sphere. The ...
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4. Polar and Exponential Forms - Pauls Online Math Notes
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[PDF] The argument of a complex number
With this definition, we identify θ as the so-called principal value of the argument, which we denote by Arg z (note the capital A). On the other hand, in many ...
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[PDF] Complex numbers - Purdue Math
Jan 27, 2021 · For a complex number z = x + iy the number z = x − iy is called the conjugate. Operation of conjugation respects addition and multiplication: z ...
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Complex Numbers
A complex number is a number of the form a + bi, where a and b are real numbers. The imaginary number i satisfies i² = -1.
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2.3 Complex Conjugate and Norm - BOOKS
Geometrically, you should be able to see that the complex conjugate of any complex number is found by reflecting in the real axis, as shown in Figure 2.4.
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2 + 2\text{Re}(z\bar w)$ for any $z, w \in \mathbb{C}$. - Mathematics ...
Nov 12, 2016 · Just calculate |z+w|2=(z+w)(¯z+w)=(z+w)(ˉz+¯w)=zˉz+w¯w+z¯w+ˉzw=|z|2+|w|2+z¯w+ˉzw. Now, as z¯w and ˉzw are conjugate, their sum is twice ...
[34]
[PDF] Extended Complex Plane and Riemann Sphere
z. Figure 1: The plane and the sphere intersected together. Now we want to find the formula for the stereographic projection 𝑧 = 𝑥 + 𝑖𝑦 of the point 𝑍 ...<|control11|><|separator|>
[35]
[PDF] Complex Analysis and Conformal Mapping
Apr 18, 2024 · The resulting space is identified with a two-dimensional sphere via stereographic projection π: S → C, [1, 23], which is one-to-one (and ...
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Branch Cuts and Branches | Complex Variables with Applications
One way to get a single valued function out of a multiple valued function is to introduce branch cuts in the complex plane.
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[PDF] Branch Points and Branch Cuts (18.04, MIT). - MIT Mathematics
Oct 11, 1999 · In the next section we will consider examples of branch points, branch cuts and Riemann surfaces for other multiple valued functions.
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Multivalued Function -- from Wolfram MathWorld
Branch Cut, Branch Point, Complex Function, Function, Riemann Surface, Single-Valued Function. Explore with Wolfram|Alpha. WolframAlpha. More things to try ...<|control11|><|separator|>
[39]
Algebraic Branch Point -- from Wolfram MathWorld
An algebraic branch point is a singular boundary point of one sheet of a multivalued function about which a finite number p of distinct sheets hang together.
[40]
Logarithmic Branch Point -- from Wolfram MathWorld
A branch point whose neighborhood of values wrap around an infinite number of times as their complex arguments are varied.
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Branch Point -- from Wolfram MathWorld
A branch point of an analytic function is a point in the complex plane whose complex argument can be mapped from a single point in the domain to multiple ...
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[PDF] A concise course in complex analysis and Riemann surfaces ...
... Riemann sphere then one needs to use here that stereographic projection is a conformal map. We shall see later in this chapter that the holomorphic maps f ...
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[PDF] FOUNDATIONS OF A GENERAL THEORY OF FUNCTIONS OF A ...
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[PDF] Branch Cuts for Complex Elementary Functions (or - FreeBSD
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DLMF: §4.23 Inverse Trigonometric Functions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions
### Summary of Principal Branches and Branch Cuts for Inverse Trigonometric Functions
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[PDF] 9. Power series Definition 9.1. Let z 0 be a complex number. An ...
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[PDF] 1 Basics of Series and Complex Numbers
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[PDF] Power Series - UC Davis Math
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[PDF] Complex Analysis III: Laurent Series and Singularities
Feb 26, 2014 · A “Laurent series” for a function is a generalization of the Taylor series, and like a Taylor series, it is essentially a power series.
[50]
[PDF] 19. LAurent Series If a holomorphic function is defined on an open ...
This is holomorphic on the annulus U = 1z ∈ C|1 < |z| < 2l. Therefore it has a Laurent expansion centred at zero which converges on the annulus. To find the ...
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[PDF] 18.04 S18 Topic 7: Taylor and Laurent series - MIT OpenCourseWare
When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series ...Missing: plane | Show results with:plane
[52]
[PDF] 1. Review: Complex numbers, Analytic functions - OSU Math
If the radius of convergence were larger, f would be analytic in a larger domain since convergent power series are, we have seen it, analytic. Corollary 10.19.
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[PDF] Exponentials, Trig Functions, and Complex Power Series
A series ∑ ck of complex numbers ck = ak + ibk (ak and bk real) is said to converge if the corresponding series of real and imaginary parts, ∑ ak and ∑ bk, both ...
[54]
[PDF] A First Course in Complex Analysis - CSI Math
... converges absolutely for |z| < 1 to the function. 1/(1 − z). The convergence is uniform on any set of the form {z ∈ C : |z| ≤ r } for any r < 1. Proof ...
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[PDF] Complex functions, single and multivalued.
If the function has two branch points like p(z − a)(z − b), a common choice is to take the cut along the line between z = a and z = b, and gives two sheets ...
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[PDF] (Elliptic Modular Curves) JS Milne
This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts. BibTeX ...
[57]
None
Below is a merged summary of the fundamental group of the punctured plane or circle and its relation to multi-valuedness in complex functions, based on the provided segments from Hatcher’s *Algebraic Topology* (https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) and related content. To retain all information in a dense and organized manner, I will use a combination of narrative text and a table in CSV format for key details. The response consolidates overlapping information while preserving unique insights from each segment.
[58]
[PDF] Understanding Poles and Zeros 1 System Poles and Zeros - MIT
the complex s-plane, whose axes represent the real and imaginary parts of the complex variable s. Such plots are known as pole-zero plots. It is usual to ...
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[PDF] Regeneration Theory - By H. NYQUIST
Regeneration Theory. By H. NYQUIST. Regeneration or feed-back is of considerable importance in many appli- cations of vacuum tubes. The most obvious example ...
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https://ieeexplore.ieee.org/document/5060121
[61]
Hermitian Form -- from Wolfram MathWorld
A Hermitian form on a vector space V over the complex field C is a function f:V×V->C such that for all u,v,w in V and all a,b in R, 1. f(au+bv,w)=af
[62]
[PDF] Further linear algebra. Chapter V. Bilinear and quadratic forms.
Definition 5.1 Let q be a quadratic form on vector space V over C, and ... Clearly, in the complex case it is the integer r that appears in the canonical form.
[63]
Quadratic $D$-forms with applications to hermitian forms - arXiv
Jun 15, 2019 · ... hermitian forms. ... Some generalized notions of isotropy, metabolicity and isometry are introduced and used to find a Witt decomposition for ...
[64]
isotropic quadratic space - PlanetMath
Mar 22, 2013 · ... complex numbers are algebraically ... totally isotropic quadratic space. Defines, totally isotropic ...
[65]
[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5
Oct 17, 2005 · C = Spec C[x, y]. (This discussion will apply with C replaced by any algebraically closed field.) Sadly, C[x, y] is not a Principal Ideal ...
[66]
[PDF] an introduction to affine schemes - UChicago Math
Jul 26, 2009 · The goal of this paper is to present the basic concepts of algebraic geometry, in particular affine schemes and sheaf theory, in such a way that.
[67]
[PDF] Algebraic Curves in the Projective Plane - MIT OpenCourseWare
One can talk about projective algebraic curves, and this helps us understand the be- haviour of ordinary algebraic curves (as they go out to infinity).
[68]
[PDF] arXiv:quant-ph/0201014v2 16 Sep 2002
Stereographic projection of the Bloch sphere onto the extended complex plane. Then, P maps |ψi = (a, b)T to a complex number z = P(|ψi) =.
[69]
On the existence of complex spacetime in relativistic quantum ...
The aim of this paper is to prove the existence of such a complex spacetime in our physical world and to show that what the current relativistic quantum ...Missing: 2D | Show results with:2D
[70]
Generalized NTRU Algorithms on Algebraic Rings - MDPI
To validate the generality of the G-NTRU framework, we introduce the CNTRU (Complex Number NTRU), a variant of the NTRU over the ring of complex numbers.Missing: plane | Show results with:plane
[71]
Fano Plane -- from Wolfram MathWorld
The Fano plane is the configuration consisting of the two-dimensional finite projective plane the Galois field of order 2 GF(2).