Complex planeFrom Wikipedia, the free encyclopediaContents1 Complex number 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.4 History in brief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Equality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Elementary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 Conjugation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Addition and subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.3 Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.4 Square root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Polar form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Absolute value and argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.2 Multiplication and division in polar form. . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.1 Eulers formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.2 Natural logarithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.3 Integer and fractional exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6.1 Field structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6.2 Solutions of polynomial equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6.3 Algebraic characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6.4 Characterization as a topological eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Formal construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7.1 Formal development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7.2 Matrix representation of complex numbers. . . . . . . . . . . . . . . . . . . . . . . . . . 151.8 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8.1 Complex exponential and related functions. . . . . . . . . . . . . . . . . . . . . . . . . . 161.8.2 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18iii CONTENTS1.9 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9.1 Control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9.2 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9.3 Fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9.4 Dynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9.5 Electromagnetism and electrical engineering . . . . . . . . . . . . . . . . . . . . . . . . . 191.9.6 Signal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.9.7 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.8 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.9 Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.10 Algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9.11 Analytic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.10History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.11Generalizations and related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.12See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.13Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.14.1 Mathematical references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.14.2 Historical references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.15Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Complex plane 272.1 Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Stereographic projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Cutting the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Multi-valued relationships and branch points . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.2 Restricting the domain of meromorphic functions . . . . . . . . . . . . . . . . . . . . . . 312.3.3 Specifying convergence regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 Gluing the cut plane back together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5 Use of the complex plane in control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6 Other meanings of complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.7 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Complex-valued function 363.1 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38CONTENTS iii3.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Imaginary number 394.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Multiplication of square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Imaginary unit 435.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2 i and i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Proper use. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4.2 Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.4.3 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.4.4 Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.4.5 Other operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.5 Alternative notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.12Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 525.12.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.12.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.12.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Chapter 1Complex numberImReba+bi

a0A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram,representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the imaginary unit which satises i2= 1.12 CHAPTER 1. COMPLEX NUMBERA complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i isthe imaginary unit, that satises the equation i2= 1.[1] In this expression, a is the real part and b is the imaginarypart of the complex number and i is the positive square root of 1.Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane(also called Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part.The complex number a + bi can be identied with the point (a, b) in the complex plane. A complex number whosereal part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a realnumber. In this way, the complex numbers contain the ordinary real numbers while extending them in order to solveproblems that cannot be solved with real numbers alone.As well as their use within mathematics, complex numbers have practical applications in many elds, includingphysics, chemistry, biology, economics, electrical engineering, and statistics. The Italian mathematician GerolamoCardano is the rst known to have introduced complex numbers. He called them ctitious during his attempts tond solutions to cubic equations in the 16th century.[2]1.1 OverviewComplex numbers allow for solutions to certain equations that have no solutions in real numbers. For example, theequation(x + 1)2= 9has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution tothis problem. The idea is to extend the real numbers with the imaginary unit i where i2= 1, so that solutions toequations like the preceding one can be found. In this case the solutions are 1 + 3i and 1 3i, as can be veriedusing the fact that i2= 1:((1 + 3i) + 1)2= (3i)2= (32)(i2) = 9(1) = 9,((1 3i) + 1)2= (3i)2= (3)2(i2) = 9(1) = 9.According to the fundamental theorem of algebra, all polynomial equations with real or complex coecients in asingle variable have a solution in complex numbers.1.1.1 DenitionAcomplex number is a number of the forma + bi, where a and b are real numbers and i is an imaginary unit, satisfyingi2= 1. For example, 3.5 + 2i is a complex number.The real number a is called the real part of the complex number a + bi; the real number b is called the imaginary partof a + bi. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginarypart.[3][4] The real part of a complex number z is denoted by Re(z) or (z); the imaginary part of a complex numberz is denoted by Im(z) or (z). For example,Re(3.5 + 2i) = 3.5Im(3.5 + 2i) = 2.Hence, in terms of its real and imaginary parts, a complex number z is equal to Re(z) + Im(z) i . This expressionis sometimes known as the Cartesian form of z.A real number a can be regarded as a complex number a + 0i whose imaginary part is 0. A purely imaginary numberbi is a complex number 0 + bi whose real part is zero. It is common to write a for a + 0i and bi for 0 + bi. Moreover,when the imaginary part is negative, it is common to write a bi with b > 0 instead of a + (b)i, for example 3 4iinstead of 3 + (4)i.The set of all complex numbers is denoted by , C or C .1.1. OVERVIEW 3ImReyy0xrrz=x+iyz=xiyAn illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y.1.1.2 NotationSome authors[5] write a + ib instead of a + bi. In some disciplines, in particular electromagnetism and electricalengineering, j is used instead of i,[6] since i is frequently used for electric current. In these cases complex numbers4 CHAPTER 1. COMPLEX NUMBERare written as a + bj or a + jb.1.1.3 Complex planeMain article: Complex planeA complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate systemImRe O abz=a+biFigure 1: A complex number plotted as a point (red) and position vector (blue) on an Argand diagram; a+bi is the rectangularexpression of the point.called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001), named after Jean-RobertArgand. The numbers are conventionally plotted using the real part as the horizontal component, and imaginary partas vertical (see Figure 1). These two values used to identify a given complex number are therefore called its Cartesian,rectangular, or algebraic form.A position vector may also be dened in terms of its magnitude and direction relative to the origin. These areemphasized in a complex numbers polar form. Using the polar form of the complex number in calculations may leadto a more intuitive interpretation of mathematical results. Notably, the operations of addition and multiplication takeon a very natural geometric character when complex numbers are viewed as position vectors: addition correspondsto vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e.the angles they make with the x axis). Viewed in this way the multiplication of a complex number by i correspondsto rotating the position vector counterclockwise by a quarter turn (90) about the origin: (a+bi)i = ai+bi2= -b+ai.1.1.4 History in briefMain section: HistoryThe solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots ofnegative numbers when all three roots are real numbers, a situation that cannot be rectied by factoring aided by therational root test if the cubic is irreducible (the so-called casus irreducibilis). This conundrum led Italian mathemati-cian Gerolamo Cardano to conceive of complex numbers in around 1545, though his understanding was rudimentary.1.2. RELATIONS 5Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows thatwith complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbersthus form an algebraically closed eld, where any polynomial equation has a root.Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction,multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.[7] Amore abstract formalism for the complex numbers was further developed by the Irish mathematician William RowanHamilton, who extended this abstraction to the theory of quaternions.1.2 Relations1.2.1 EqualityTwo complex numbers are equal if and only if both their real and imaginary parts are equal. In symbols:z1= z2(Re(z1) = Re(z2) Im(z1) = Im(z2)).1.2.2 OrderingBecause complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linearordering on the set of complex numbers.[8]There is no linear ordering on the complex numbers that is compatible with addition and multiplication. Formally, wesay that the complex numbers cannot have the structure of an ordered eld. This is because any square in an orderedeld is at least 0, but i2= 1.1.3 Elementary operations1.3.1 ConjugationMain article: Complex conjugateThe complex conjugate of the complex number z = x + yi is dened to be x yi. It is denoted z or z*.Formally, for any complex number z: z= Re(z) Im(z) i.Geometrically, z is the reection of z about the real axis.Conjugating twice gives the original complex number: z= z .The real and imaginary parts of a complex number z can be extracted using the conjugate:Re (z) =12(z + z),Im(z) =12i(z z).Moreover, a complex number is real if and only if it equals its conjugate.Conjugation distributes over the standard arithmetic operations:z + w = z + w,z w = z w,6 CHAPTER 1. COMPLEX NUMBERImReyy0xrrz=x+iyz=xiyGeometric representation of z and its conjugate z in the complex planezw = z w,(z/w) = z/ w.The reciprocal of a nonzero complex number z = x + yi is given by1.3. ELEMENTARY OPERATIONS 71z= zz z= zx2+ y2.This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangularcoordinates. Inversive geometry, a branch of geometry studying reections more general than ones about a line, canalso be expressed in terms of complex numbers. In the network analysis of electrical circuits, the complex conjugateis used in nding the equivalent impedance when the maximum power transfer theorem is used.1.3.2 Addition and subtractionbbaaa+bAddition of two complex numbers can be done geometrically by constructing a parallelogram.Complex numbers are added by adding the real and imaginary parts of the summands. That is to say:(a + bi) + (c + di) = (a + c) + (b + d)i.Similarly, subtraction is dened by8 CHAPTER 1. COMPLEX NUMBER(a + bi) (c + di) = (a c) + (b d)i.Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpre-tation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtainedby building a parallelogram three of whose vertices are O, A and B. Equivalently, X is the point such that the triangleswith vertices O, A, B, and X, B, A, are congruent.1.3.3 Multiplication and divisionThe multiplication of two complex numbers is dened by the following formula:(a + bi)(c + di) = (ac bd) + (bc + ad)i.In particular, the square of the imaginary unit is 1:i2= i i = 1.The preceding denition of multiplication of general complex numbers follows naturally from this fundamental prop-erty of the imaginary unit. Indeed, if i is treated as a number so that di means d times i, the above multiplication ruleis identical to the usual rule for multiplying two sums of two terms.(a + bi)(c + di) = ac + bci + adi + bidi (distributive law)= ac+bidi+bci+adi (commutative law of additionthe order of the summandscan be changed)= ac + bdi2+ (bc + ad)i (commutative and distributive laws)= (ac bd) + (bc + ad)i (fundamental property of the imaginary unit).The division of two complex numbers is dened in terms of complex multiplication, which is described above, andreal division. When at least one of c and d is non-zero, we havea + bic + di=(ac + bdc2+ d2)+(bc adc2+ d2)i.Division can be dened in this way because of the following observation:a + bic + di=(a + bi) (c di)(c + di) (c di)=(ac + bdc2+ d2)+(bc adc2+ d2)i.As shown earlier, c di is the complex conjugate of the denominator c + di. At least one of the real part c and theimaginary part d of the denominator must be nonzero for division to be dened.This is called "rationalization" ofthe denominator (although the denominator in the nal expression might be an irrational real number).1.3.4 Square rootSee also: Square roots of negative and complex numbersThe square roots of a + bi (with b 0) are ( + i) , where=a +a2+ b221.4. POLAR FORM 9and= sgn(b)a +a2+ b22,where sgn is the signum function. This can be seen by squaring ( + i) to obtain a + bi.[9][10] Herea2+ b2iscalled the modulus of a + bi, and the square root with non-negative real part is called the principal square root; alsoa2+ b2=z z , where z= a + bi .[11]1.4 Polar formMain article: Polar coordinate systemImRe0rxyFigure 2: The argument and modulus r locate a point on an Argand diagram; r(cos +i sin ) or reiare polar expressions ofthe point.10 CHAPTER 1. COMPLEX NUMBER1.4.1 Absolute value and argumentAn alternative way of dening a point P in the complex plane, other than using the x- and y-coordinates, is to use thedistance of the point from O, the point whose coordinates are (0, 0) (the origin), together with the angle subtendedbetween the positive real axis and the line segment OP in a counterclockwise direction. This idea leads to the polarform of complex numbers.The absolute value (or modulus or magnitude) of a complex number z = x + yi isr = |z| =x2+ y2.If z is a real number (i.e., y = 0), then r = | x |. In general, by Pythagoras theorem, r is the distance of the point Prepresenting the complex number z to the origin. The square of the absolute value is|z|2= z z= x2+ y2.where z is the complex conjugate of z .The argument of z (in many applications referred to as the phase) is the angle of the radius OP with the positivereal axis, and is written as arg(z) . As with the modulus, the argument can be found from the rectangular form x+yi:[12] = arg(z) =arctan(yx) if x > 0arctan(yx) + if x < 0 and y 0arctan(yx) if x < 0 and y< 02if x = 0 and y> 02if x = 0 and y< 0indeterminate if x = 0 and y= 0.The value of is expressed in radians in this article. It can increase by any integer multiple of 2 and still give thesame angle. Hence, the arg function is sometimes considered as multivalued. Normally, as given above, the principalvalue in the interval (,] is chosen. Values in the range [0,2) are obtained by adding 2 if the value is negative.The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the angle 0 is common.The value of equals the result of atan2: = atan2(imaginary, real) .Together, r and give another way of representing complex numbers, the polar form, as the combination of modulusand argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates fromthe polar form is done by the formula called trigonometric formz= r(cos + i sin ).Using Eulers formula this can be written asz= rei.Using the cis function, this is sometimes abbreviated toz= r cis .In angle notation, often used in electronics to represent a phasor with amplitude r and phase , it is written as[13]z= r.1.4. POLAR FORM 110123451 2 3 4 5 62+i3+i5+5iMultiplication of 2 + i (blue triangle) and 3 + i (red triangle). The red triangle is rotated to match the vertex of the blue one andstretched by 5, the length of the hypotenuse of the blue triangle.1.4.2 Multiplication and division in polar formFormulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulasin Cartesian coordinates. Given two complex numbers z1 = r1(cos 1 + i sin 1) and z2 = r2(cos 2 + i sin 2),because of the well-known trigonometric identitiescos(a) cos(b) sin(a) sin(b) = cos(a + b)cos(a) sin(b) + sin(a) cos(b) = sin(a + b)we may derivez1z2= r1r2(cos(1 + 2) + i sin(1 + 2)).In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product.For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i2= 1. The pictureat the right illustrates the multiplication of(2 + i)(3 + i) = 5 + 5i.Since the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or /4 (in radian).On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) andarctan(1/2), respectively. Thus, the formula12 CHAPTER 1. COMPLEX NUMBER4= arctan 12+ arctan 13holds. As the arctan function can be approximated highly eciently, formulas like thisknown as Machin-likeformulasare used for high-precision approximations of .Similarly, division is given byz1z2=r1r2(cos(12) + i sin(12)) .1.5 Exponentiation1.5.1 Eulers formulaEulers formula states that, for any real number x,eix= cos x + i sin xwhere e is the base of the natural logarithm. This can be proved through induction by observing thati0= 1, i1= i, i2= 1, i3= i,i4= 1, i5= i, i6= 1, i7= i,and so on, and by considering the Taylor series expansions of eix, cos(x) and sin(x):eix= 1 + ix +(ix)22!+(ix)33!+(ix)44!+(ix)55!+(ix)66!+(ix)77!+(ix)88!+ = 1 + ix x22! ix33!+x44!+ix55!x66! ix77!+x88!+ =(1 x22!+x44! x66!+x88! )+ i(x x33!+x55! x77!+ )= cos x + i sin x .The rearrangement of terms is justied because each series is absolutely convergent.1.5.2 Natural logarithmEulers formula allows us to observe that, for any complex numberz= r(cos + i sin ).where r is a non-negative real number, one possible value for z's natural logarithm isln(z) = ln(r) + iBecause cos and sin are periodic functions, the natural logarithm may be considered a multi-valued function, with:ln(z) = {ln(r) + ( + 2k)i | k Z}1.6. PROPERTIES 131.5.3 Integer and fractional exponentsWe may use the identityln(ab) = b ln(a)to dene complex exponentiation, which is likewise multi-valued:ln(zn) = ln((r(cos + i sin ))n)= nln(r(cos + i sin ))= {n(ln(r) + ( + k2)i)|k Z}= {nln(r) + ni + nk2i|k Z}.When n is an integer, this simplies to de Moivres formula:zn= (r(cos + i sin ))n= rn(cos n + i sin n).The nth roots of z are given bynz=nr(cos( + 2kn)+ i sin( + 2kn))for any integer k satisfying 0 k n 1. Herenr is the usual (positive) nth root of the positive real number r.While the nth root of a positive real number r is chosen to be the positive real number c satisfying cn= x there isno natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root of zis considered as a multivalued function (in z), as opposed to a usual function f, for which f(z) is a uniquely denednumber. Formulas such asnzn= z(which holds for positive real numbers), do in general not hold for complex numbers.1.6 Properties1.6.1 Field structureThe set C of complex numbers is a eld. Briey, this means that the following facts hold:rst, any two complexnumbers can be added and multiplied to yield another complex number. Second, for any complex number z, itsadditive inverse z is also a complex number; and third, every nonzero complex number has a reciprocal complexnumber. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition andmultiplication for any two complex numbers z1 and z2:z1 + z2= z2 + z1,z1z2= z2z1.These two laws and the other requirements on a eld can be proven by the formulas given above, using the fact thatthe real numbers themselves form a eld.Unlike the reals, C is not an ordered eld, that is to say, it is not possible to dene a relation z1 < z2 that is compatiblewith the addition and multiplication. In fact, in any ordered eld, the square of any element is necessarily positive, soi2= 1 precludes the existence of an ordering on C.When the underlying eld for a mathematical topic or construct is the eld of complex numbers, the topics nameis usually modied to reect that fact. For example:complex analysis, complex matrix, complex polynomial, andcomplex Lie algebra.14 CHAPTER 1. COMPLEX NUMBER1.6.2 Solutions of polynomial equationsGiven any complex numbers (called coecients) a0, , an, the equationanzn+ + a1z + a0= 0has at least one complex solution z, provided that at least one of the higher coecients a1, , an is nonzero. This isthe statement of the fundamental theorem of algebra. Because of this fact, C is called an algebraically closed eld.This property does not hold for the eld of rational numbers Q (the polynomial x2 2 does not have a rational root,since 2 is not a rational number) nor the real numbers R (the polynomial x2+ a does not have a real root for a > 0,since the square of x is positive for any real number x).There are various proofs of this theorem, either by analytic methods such as Liouvilles theorem, or topological onessuch as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degreehas at least one real root.Because of this fact, theorems that hold for any algebraically closed eld, apply to C. For example, any non-emptycomplex square matrix has at least one (complex) eigenvalue.1.6.3 Algebraic characterizationThe eld C has the following three properties: rst, it has characteristic 0. This means that 1 + 1 + + 1 0 forany number of summands (all of which equal one). Second, its transcendence degree over Q, the prime eld of C, isthe cardinality of the continuum. Third, it is algebraically closed (see above). It can be shown that any eld havingthese properties is isomorphic (as a eld) to C. For example, the algebraic closure of Qp also satises these threeproperties, so these two elds are isomorphic. Also, C is isomorphic to the eld of complex Puiseux series. However,specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization isthat C contains many proper subelds that are isomorphic to C.1.6.4 Characterization as a topological eldThe preceding characterization of Cdescribes only the algebraic aspects of C. That is to say, the properties of nearnessand continuity, which matter in areas such as analysis and topology, are not dealt with. The following descriptionof C as a topological eld (that is, a eld that is equipped with a topology, which allows the notion of convergence)does take into account the topological properties. C contains a subset P (namely the set of positive real numbers) ofnonzero elements satisfying the following three conditions:P is closed under addition, multiplication and taking inverses.If x and y are distinct elements of P, then either x y or y x is in P.If S is any nonempty subset of P, then S + P = x + P for some x in C.Moreover, C has a nontrivial involutive automorphism x x* (namely the complex conjugation), such that x x* is inP for any nonzero x in C.Any eld F with these properties can be endowed with a topology by taking the sets B(x, p) = { y | p (y x)(y x)* P } as a base, where x ranges over the eld and p ranges over P. With this topology F is isomorphic as a topologicaleld to C.The only connected locally compact topological elds are R and C. This gives another characterization of C as atopological eld, since C can be distinguished from R because the nonzero complex numbers are connected, whilethe nonzero real numbers are not.1.7. FORMAL CONSTRUCTION 151.7 Formal construction1.7.1 Formal developmentAbove, complex numbers have been dened by introducing i, the imaginary unit, as a symbol. More rigorously, theset C of complex numbers can be dened as the set R2of ordered pairs (a, b) of real numbers. In this notation, theabove formulas for addition and multiplication read(a, b) + (c, d) = (a + c, b + d)(a, b) (c, d) = (ac bd, bc + ad).It is then just a matter of notation to express (a, b) as a + bi.Though this low-level construction does accurately describe the structure of the complex numbers, the followingequivalent denition reveals the algebraic nature of C more immediately. This characterization relies on the notion ofelds and polynomials. A eld is a set endowed with an addition, subtraction, multiplication and division operationsthat behave as is familiar from, say, rational numbers. For example, the distributive law(x + y)z= xz + yzmust hold for any three elements x, y and z of a eld.The set R of real numbers does form a eld.A polynomialp(X) with real coecients is an expression of the formanXn+ + a1X + a0where the a0, ..., an are real numbers. The usual addition and multiplication of polynomials endows the set R[X] ofall such polynomials with a ring structure. This ring is called polynomial ring.The quotient ring R[X]/(X2+ 1) can be shown to be a eld.This extension eld contains two square roots of 1,namely (the cosets of) X and X, respectively. (The cosets of) 1 and X form a basis of R[X]/(X2+ 1) as a realvector space, which means that each element of the extension eld can be uniquely written as a linear combinationin these two elements. Equivalently, elements of the extension eld can be written as ordered pairs (a, b) of realnumbers. Moreover, the above formulas for addition etc. correspond to the ones yielded by this abstract algebraicapproachthe two denitions of the eld C are said to be isomorphic (as elds). Together with the above-mentionedfact that C is algebraically closed, this also shows that C is an algebraic closure of R.1.7.2 Matrix representation of complex numbersComplex numbers a + bi can also be represented by 2 2 matrices that have the following form:(a bb a).Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, andthe sum and product of complex numbers corresponds to the sum and product of such matrices. The geometricdescription of the multiplication of complex numbers can also be phrased in terms of rotation matrices by usingthis correspondence between complex numbers and such matrices. Moreover, the square of the absolute value of acomplex number expressed as a matrix is equal to the determinant of that matrix:|z|2=

a bb a

= (a2) ((b)(b)) = a2+ b2.The conjugate z corresponds to the transpose of the matrix.Though this representation of complex numbers with matrices is the most common, many other representations arisefrom matrices other than (0 11 0) that square to the negative of the identity matrix. See the article on 2 2 realmatrices for other representations of complex numbers.16 CHAPTER 1. COMPLEX NUMBER1.8 Complex analysisColor wheel graph of sin(1/ z). Black parts inside refer to numbers having large absolute values.Main article: Complex analysisThe study of functions of a complex variable is known as complex analysis and has enormous practical use in appliedmathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysisor even number theory employ techniques fromcomplex analysis (see prime number theoremfor an example). Unlikereal functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensionalgraphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or byanimating the complex functions dynamic transformation of the complex plane.1.8.1 Complex exponential and related functionsThe notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis.A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalentto the (, )-denition of limits, where the absolute value of real numbers is replaced by the one of complex numbers.From a more abstract point of view, C, endowed with the metric1.8. COMPLEX ANALYSIS 17d(z1, z2) = |z1z2|is a complete metric space, which notably includes the triangle inequality|z1 + z2| |z1| +|z2|for any two complex numbers z1 and z2.Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponentialfunction exp(z), also written ez, is dened as the innite seriesexp(z) := 1 + z +z22 1+z33 2 1+ =n=0znn! .and the series dening the real trigonometric functions sine and cosine, as well as hyperbolic functions such as sinhalso carry over to complex arguments without change. Eulers identity states:exp(i) = cos() + i sin()for any real number , in particularexp(i) = 1Unlike in the situation of real numbers, there is an innitude of complex solutions z of the equationexp(z) = wfor any complex number w 0. It can be shown that any such solution zcalled complex logarithm of asatiseslog(x + iy) = ln |w| + i arg(w),where arg is the argument dened above, and ln the (real) natural logarithm. As arg is a multivalued function, uniqueonly up to a multiple of 2, log is also multivalued. The principal value of log is often taken by restricting theimaginary part to the interval (,].Complex exponentiation zis dened asz= exp( log z).Consequently, they are in general multi-valued. For = 1 / n, for some natural number n, this recovers the non-uniqueness of nth roots mentioned above.Complex numbers, unlike real numbers, do not in general satisfy the unmodied power and logarithm identities, par-ticularly when navely treated as single-valued functions; see failure of power and logarithm identities. For example,they do not satisfyabc= (ab)c.Both sides of the equation are multivalued by the denition of complex exponentiation given here, and the values onthe left are a subset of those on the right.18 CHAPTER 1. COMPLEX NUMBER1.8.2 Holomorphic functionsA function f : C C is called holomorphic if it satises the CauchyRiemann equations. For example, any R-linearmap C C can be written in the formf(z) = az + bzwith complex coecients a and b.This map is holomorphic if and only if b = 0.The second summand bz is real-dierentiable, but does not satisfy the CauchyRiemann equations.Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functionsf and g that agree on an arbitrarily small open subset of C necessarily agree everywhere. Meromorphic functions,functions that can locally be written as f(z)/(z z0)nwith a holomorphic function f, still share some of the featuresof holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0.1.9 ApplicationsComplex numbers have essential concrete applications in a variety of scientic and related areas such as signal pro-cessing, control theory, electromagnetism, uid dynamics, quantum mechanics, cartography, and vibration analysis.Some applications of complex numbers are:1.9.1 Control theoryIn control theory, systems are often transformed from the time domain to the frequency domain using the Laplacetransform.The systems poles and zeros are then analyzed in the complex plane.The root locus, Nyquist plot, andNichols plot techniques all make use of the complex plane.In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e.have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that arein the right half plane, it will be unstable,all in the left half plane, it will be stable,on the imaginary axis, it will have marginal stability.If a system has zeros in the right half plane, it is a nonminimum phase system.1.9.2 Improper integralsIn applied elds, complex numbers are often used to compute certain real-valued improper integrals, by means ofcomplex-valued functions. Several methods exist to do this; see methods of contour integration.1.9.3 Fluid dynamicsIn uid dynamics, complex functions are used to describe potential ow in two dimensions.1.9.4 Dynamic equationsIn dierential equations, it is common to rst nd all complex roots r of the characteristic equation of a lineardierential equation or equation system and then attempt to solve the system in terms of base functions of the formf(t) = ert. Likewise, in dierence equations, the complex roots r of the characteristic equation of the dierenceequation system are used, to attempt to solve the system in terms of base functions of the form f(t) = rt.1.9. APPLICATIONS 191.9.5 Electromagnetism and electrical engineeringMain article: Alternating currentIn electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment ofresistors, capacitors, and inductors can then be unied by introducing imaginary, frequency-dependent resistancesfor the latter two and combining all three in a single complex number called the impedance. This approach is calledphasor calculus.In electrical engineering, the imaginary unit is denoted by j, to avoid confusion with I, which is generally in use todenote electric current, or, more particularly, i, which is generally in use to denote instantaneous electric current.Since the voltage in an AC circuit is oscillating, it can be represented asV (t) = V0ejt= V0 (cos t + j sin t) ,To obtain the measurable quantity, the real part is taken:v(t) = Re(V ) = Re[V0ejt]= V0 cos t.The complex-valued signal V (t) is called the analytic representation of the real-valued, measurable signal v(t) .[14]1.9.6 Signal analysisComplex numbers are used in signal analysis and other elds for a convenient description for periodically varyingsignals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corre-sponding complex functions are considered of which the real parts are the original quantities. For a sine wave of agiven frequency, the absolute value | z | of the corresponding z is the amplitude and the argument arg(z) is the phase.If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodicfunctions are often written as complex valued functions of the formx(t) = Re{X(t)}andX(t) = Aeit= aeieit= aei(t+)where represents the angular frequency and the complex number A encodes the phase and amplitude as explainedabove.This use is also extended into digital signal processing and digital image processing, which utilize digital versions ofFourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, stillimages, and video signals.Another example, relevant to the two side bands of amplitude modulation of AM radio, is:cos(( + )t) + cos (( )t) = Re(ei(+)t+ ei()t)= Re((eit+ eit) eit)= Re(2 cos(t) eit)= 2 cos(t) Re(eit)= 2 cos(t) cos (t).20 CHAPTER 1. COMPLEX NUMBER1.9.7 Quantum mechanicsThe complex number eld is intrinsic to the mathematical formulations of quantummechanics, where complex Hilbertspaces provide the context for one such formulation that is convenient and perhaps most standard. The originalfoundation formulas of quantum mechanicsthe Schrdinger equation and Heisenbergs matrix mechanicsmakeuse of complex numbers.1.9.8 RelativityIn special and general relativity, some formulas for the metric on spacetime become simpler if one takes the timecomponent of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity,but is used in an essential way in quantum eld theory.) Complex numbers are essential to spinors, which are ageneralization of the tensors used in relativity.1.9.9 GeometryFractalsCertain fractals are plotted in the complex plane, e.g. the Mandelbrot set and Julia sets.TrianglesEvery triangle has a unique Steiner inellipsean ellipse inside the triangle and tangent to the midpoints of the threesides of the triangle. The foci of a triangles Steiner inellipse can be found as follows, according to Mardens theo-rem:[15][16] Denote the triangles vertices in the complex plane as a = xA + yAi, b = xB + yBi, and c = xC + yCi. Writethe cubic equation(xa)(xb)(xc)=0 , take its derivative, and equate the (quadratic) derivative to zero. MardensTheorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci ofthe Steiner inellipse.1.9.10 Algebraic number theoryAs mentioned above, any nonconstant polynomial equation (in complex coecients) has a solution in C. A fortiori,the same is true if the equation has rational coecients. The roots of such equations are called algebraic numbers they are a principal object of study in algebraic number theory. Compared to Q, the algebraic closure of Q, whichalso contains all algebraic numbers, C has the advantage of being easily understandable in geometric terms. In thisway, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, morespecically applying the machinery of eld theory to the number eld containing roots of unity, it can be shown thatit is not possible to construct a regular nonagon using only compass and straightedge a purely geometric problem.Another example are Gaussian integers, that is, numbers of the form x + iy, where x and y are integers, which can beused to classify sums of squares.1.9.11 Analytic number theoryMain article: Analytic number theoryAnalytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can beregarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoreticinformation in complex-valued functions. For example, the Riemann zeta function (s) is related to the distributionof prime numbers.1.10. HISTORY 21Construction of a regular pentagon using straightedge and compass.1.10 HistoryThe earliest eeting reference to square roots of negative numbers can perhaps be said to occur in the work of theGreek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considers, apparentlyin error, the volume of an impossible frustum of a pyramid to arrive at the term 81144=3i7 in his calculations,although negative quantities were not conceived of in Hellenistic mathematics and Heron merely replaced it by itspositive ( 14481=37 ).[17]The impetus to study complex numbers proper rst arose in the 16th century when algebraic solutions for the roots ofcubic and quartic polynomials were discovered by Italian mathematicians (see Niccol Fontana Tartaglia, GerolamoCardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimesrequired the manipulation of square roots of negative numbers. As an example, Tartaglias formula for a cubicequation of the formx3=px+q [18] gives the solution to the equation x3= x as13((1)1/3+1(1)1/3).At rst glance this looks like nonsense. However formal calculations with complex numbers show that the equationz3= i has solutions i,32+12i and32+12i . Substituting these in turn for 11/3in Tartaglias cubic formula22 CHAPTER 1. COMPLEX NUMBERand simplifying, one gets 0, 1 and 1 as the solutions of x3 x = 0. Of course this particular equation can be solvedat sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, aslater mathematicians showed rigorously, the use of complex numbers is unavoidable.Rafael Bombelli was the rstto explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complexarithmetic trying to resolve these issues.The term imaginary for these quantities was coined by Ren Descartes in 1637, although he was at pains to stresstheir imaginary nature[19][...] sometimes only imaginary, that is one can imagine as many as I said in each equation, butsometimes there exists no quantity that matches that which we imagine.([...] quelquefois seulement imaginaires cest--dire que lon peut toujours en imaginer autant quej'ai dit en chaque quation, mais quil ny a quelquefois aucune quantit qui corresponde celle quonimagine.)A further source of confusion was that the equation 12=11=1 seemed to be capriciously inconsistent withthe algebraic identity ab=ab , which is valid for non-negative real numbers a and b, and which was also usedin complex number calculations with one of a, b positive and the other negative.The incorrect use of this identity(and the related identity1a=1a ) in the case when both a and b are negative even bedeviled Euler. This dicultyeventually led to the convention of using the special symbol i in place of 1 to guard against this mistake. Even so,Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementaryalgebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a naturalway throughout.In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complexexpressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abrahamde Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle topowers of trigonometric functions of that angle could be simply re-expressed by the following well-known formulawhich bears his name, de Moivres formula:(cos + i sin )n= cos n + i sin n.In 1748 Leonhard Euler went further and obtained Eulers formula of complex analysis:cos + i sin = eiby formally manipulating complex power series and observed that this formula could be used to reduce any trigono-metric identity to much simpler exponential identities.The idea of a complex number as a point in the complex plane (above) was rst described by Caspar Wessel in 1799,although it had been anticipated as early as 1685 in Walliss De Algebra tractatus.Wessels memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of thefundamental theorem of algebra. Gauss had earlier published an essentially topological proof of the theorem in1797 but expressed his doubts at the time about the true metaphysics of the square root of 1. It was not until1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largelyestablishing modern notation and terminology. The English mathematician G. H. Hardy remarked that Gauss was therst mathematician to use complex numbers in 'a really condent and scientic way' although mathematicians suchas Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his1831 treatise.[20] Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complexanalysis to a high state of completion, commencing around 1825 in Cauchys case.The common terms used in the theory are chiey due to the founders. Argand called cos +i sin the direction factor,and r=a2+b2the modulus; Cauchy (1828) called cos +i sin the reduced form(l'expression rduite) and apparentlyintroduced the term argument; Gauss used i for1 , introduced the term complex number for a + bi, and called a2+b2the norm. The expression direction coecient, often used for cos +i sin , is due to Hankel (1867), and absolutevalue, for modulus, is due to Weierstrass.Later classical writers on the general theory include Richard Dedekind, Otto Hlder, Felix Klein, Henri Poincar,Hermann Schwarz, Karl Weierstrass and many others.1.11. GENERALIZATIONS AND RELATED NOTIONS 231.11 Generalizations and related notionsThe process of extending the eld R of reals to C is known as CayleyDickson construction. It can be carried furtherto higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4and 8, respectively.However, just as applying the construction to reals loses the property of ordering, more properties familiar from realand complex numbers vanish with increasing dimension. The quaternions are only a skew eld, i.e. for some x,y: xy yx for two quaternions, the multiplication of octonions fails (in addition to not being commutative) to beassociative: for some x, y, z: (xy)z x(yz).Reals, complex numbers, quaternions and octonions are all normed division algebras over R. However, by Hurwitzstheorem they are the only ones. The next step in the CayleyDickson construction, the sedenions, in fact fails to havethis structure.The CayleyDickson construction is closely related to the regular representation of C, thought of as an R-algebra (anR-vector space with a multiplication), with respect to the basis (1, i). This means the following: the R-linear mapC C, z wzfor some xed complex number w can be represented by a 2 2 matrix (once a basis has been chosen). With respectto the basis (1, i), this matrix is(Re(w) Im(w)Im(w) Re(w))i.e., the one mentioned in the section on matrix representation of complex numbers above. While this is a linearrepresentation of C in the 2 2 real matrices, it is not the only one. Any matrixJ=(p qr p), p2+ qr + 1 = 0has the property that its square is the negative of the identity matrix: J2= I. Then{z= aI + bJ: a, b R}is also isomorphic to the eld C, and gives an alternative complex structure on R2. This is generalized by the notionof a linear complex structure.Hypercomplex numbers also generalize R, C, H, and O. For example, this notion contains the split-complex numbers,which are elements of the ring R[x]/(x2 1) (as opposed to R[x]/(x2+ 1)). In this ring, the equation a2= 1 has foursolutions.The eld R is the completion of Q, the eld of rational numbers, with respect to the usual absolute value metric.Other choices of metrics on Q lead to the elds Qp of p-adic numbers (for any prime number p), which are therebyanalogous to R. There are no other nontrivial ways of completing Q than R and Qp, by Ostrowskis theorem.Thealgebraic closure Qp of Qp still carry a norm, but (unlike C) are not complete with respect to it. The completion Cpof Qp turns out to be algebraically closed. This eld is called p-adic complex numbers by analogy.The elds R and Qp and their nite eld extensions, including C, are local elds.1.12 See alsoAlgebraic surfaceCircular motion using complex numbersComplex base systems24 CHAPTER 1. COMPLEX NUMBERComplex geometryComplex square rootDomain coloringEisenstein integerEulers identityGaussian integerMandelbrot setQuaternionRiemann sphere (extended complex plane)Root of unityUnit complex number1.13 Notes[1] Charles P. McKeague (2011), Elementary Algebra, Brooks/Cole, p. 524, ISBN 978-0-8400-6421-9[2] Burton (1995, p. 294)[3] Complex Variables (2nd Edition), M.R. Spiegel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaums Outline Series, McGraw Hill (USA), ISBN 978-0-07-161569-3[4] Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007), Chapter P, College Algebra and Trigonometry (6ed.), Cengage Learning, p. 66, ISBN 0-618-82515-0[5] For example Ahlfors (1979).[6] Brown, James Ward; Churchill, Ruel V. (1996), Complex variables and applications (6th ed.), New York: McGraw-Hill,p. 2, ISBN 0-07-912147-0, In electrical engineering, the letter j is used instead of i.[7] Katz (2004, 9.1.4)[8] http://mathworld.wolfram.com/ComplexNumber.html[9] Abramowitz, Milton; Stegun, Irene A. (1964), Handbook of mathematical functions with formulas, graphs, and mathemat-ical tables, Courier Dover Publications, p. 17, ISBN 0-486-61272-4, Section 3.7.26, p. 17[10] Cooke, Roger (2008), Classical algebra:its nature, origins, and uses, John Wiley and Sons, p. 59, ISBN 0-470-25952-3,Extract: page 59[11] Ahlfors (1979, p. 3)[12] Kasana, H.S. (2005), Chapter 1, Complex Variables: Theory And Applications (2nd ed.), PHI Learning Pvt. Ltd, p. 14,ISBN 81-203-2641-5[13] Nilsson, James William; Riedel, Susan A. (2008), Chapter 9, Electric circuits (8th ed.), Prentice Hall, p. 338, ISBN0-13-198925-1[14] Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN 0-471-92712-0[15] Kalman, Dan (2008a), An Elementary Proof of Mardens Theorem, The American Mathematical Monthly 115: 33038,ISSN 0002-9890[16] Kalman, Dan (2008b), The Most Marvelous Theoremin Mathematics, Journal of Online Mathematics and its Applications[17] Nahin, Paul J. (2007), An Imaginary Tale: The Story of 1, Princeton University Press, ISBN 978-0-691-12798-9,retrieved 20 April 20111.14. REFERENCES 25[18] In modern notation, Tartaglias solution is based on expanding the cube of the sum of two cube roots: (3u+3v)3=33uv(3u+3v)+u+v Withx=3u+3v , p=33uv , q=u+v , u and v can be expressed in terms of p and q as u=q/2+(q/2)2(p/3)3andv=q/2(q/2)2(p/3)3, respectively. Therefore, x=3q/2+(q/2)2(p/3)3+3q/2(q/2)2(p/3)3. When(q/2)2(p/3)3is negative (casus irreducibilis), the second cube root should be regarded as the complex conjugateof the rst one.[19] Descartes, Ren (1954) [1637], La Gomtrie | The Geometry of Ren Descartes with a facsimile of the rst edition, DoverPublications, ISBN 0-486-60068-8, retrieved 20 April 2011[20] Hardy, G. H.; Wright, E. M. (2000) [1938], An Introduction to the Theory of Numbers, OUP Oxford, p. 189 (fourthedition), ISBN 0-19-921986-91.14 References1.14.1 Mathematical referencesAhlfors, Lars (1979), Complex analysis (3rd ed.), McGraw-Hill, ISBN 978-0-07-000657-7Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN 0-387-90328-3Joshi, Kapil D. (1989), Foundations of Discrete Mathematics, New York: John Wiley & Sons, ISBN 978-0-470-21152-6Pedoe, Dan (1988), Geometry: A comprehensive course, Dover, ISBN 0-486-65812-0Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), Section 5.5 Complex Arithmetic, Nu-merical Recipes: The Art of Scientic Computing (3rd ed.), New York: Cambridge University Press, ISBN978-0-521-88068-8Solomentsev, E.D. (2001), Complex number, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-41.14.2 Historical referencesBurton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-009465-9Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2Nahin, Paul J. (1998), An Imaginary Tale: The Story of1 (hardcover edition ed.), Princeton UniversityPress, ISBN 0-691-02795-1A gentle introduction to the history of complex numbers and the beginnings of complex analysis.H.D. Ebbinghaus; H. Hermes; F. Hirzebruch; M. Koecher; K. Mainzer; J. Neukirch; A. Prestel; R. Remmert(1991), Numbers (hardcover ed.), Springer, ISBN 0-387-97497-0An advanced perspective on the historical development of the concept of number.1.15 Further readingThe Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf,2005; ISBN 0-679-45443-8. Chapters 47 in particular deal extensively (and enthusiastically) with complexnumbers.Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations andthe structures of modern algebra.26 CHAPTER 1. COMPLEX NUMBERVisual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN 0-19-853447-7 (hardcover, 1997).History of complex numbers and complex analysis with compelling and useful visual interpretations.Conway, John B., Functions of One Complex Variable I (Graduate Texts in Mathematics), Springer; 2 edition(12 September 2005). ISBN 0-387-90328-3.1.16 External linksHazewinkel, Michiel, ed. (2001), Complex number, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Introduction to Complex Numbers from Khan AcademyImaginary Numbers on In Our Time at the BBC.Eulers work on Complex Roots of Polynomials at Convergence. MAAMathematical Sciences Digital Library.John and Bettys Journey Through Complex NumbersThe Origin of Complex Numbers by John H. Mathews and Russell W. HowellDimensions: a math lm. Chapter 5 presents an introduction to complex arithmetic and stereographic projec-tion. Chapter 6 discusses transformations of the complex plane, Julia sets, and the Mandelbrot set.Chapter 2Complex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established bythe real axis and the orthogonal imaginary axis. It can be thought of as a modied Cartesian plane, with the realpart of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacementalong the y-axis.[1]The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they addlike vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinatesthemagnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argumentof the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number ofmodulus 1 acts as a rotation.The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are namedafter Jean-Robert Argand (17681822), although they were rst described by Danish land surveyor and mathemati-cian Caspar Wessel (17451818).[2] Argand diagrams are frequently used to plot the positions of the poles and zeroesof a function in the complex plane.2.1 Notational conventionsIn complex analysis, the complex numbers are customarily represented by the symbol z, which can be separated intoits real (x) and imaginary (y) parts:z= x + iyfor example: z = 4 + 5i, where x and y are real numbers, and i is the imaginary unit. In this customary notation thecomplex number z corresponds to the point (x, y) in the Cartesian plane.In the Cartesian plane the point (x, y) can also be represented in polar coordinates as(x, y) = (r cos , r sin ) (r, ) =(x2+ y2, arctan yx).In the Cartesian plane it may be assumed that the arctangent takes values from /2 to /2 (in radians), and somecare must be taken to dene the real arctangent function for points (x, y) when x 0.[3] In the complex plane thesepolar coordinates take the formz= x + iy= |z| (cos + i sin ) = |z|eiwhere|z| =x2+ y2; = arg(z) =1i lnz|z|= i lnz|z|.[4]2728 CHAPTER 2. COMPLEX PLANEImReyy0xrrz=x+iyz=xiyGeometric representation of z and its conjugate z in the complex plane. The distance along the light blue line from the origin to thepoint z is the modulus or absolute value of z. The angle is the argument of z.Here |z| is the absolute value or modulus of the complex number z; , the argument of z, is usually taken on the interval0 < 2; and the last equality (to |z|ei) is taken fromEulers formula. Notice that the argument of z is multi-valued,because the complex exponential function is periodic, with period 2i. Thus, if is one value of arg(z), the other2.2. STEREOGRAPHIC PROJECTIONS 29values are given by arg(z) = + 2n, where n is any integer 0.[5] While seldom used explicitly, the geometric viewof the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where theinner product of complex numbers w and z is given by (wz) ; then for a complex number z its absolute value |z|coincides with its Euclidean norm, and its argument arg(z) with the angle turning from 1 to z.The theory of contour integration comprises a major part of complex analysis. In this context the direction of travelaround a closed curve is important reversing the direction in which the curve is traversed multiplies the value of theintegral by 1. By convention the positive direction is counterclockwise. For example, the unit circle is traversed inthe positive direction when we start at the point z = 1, then travel up and to the left through the point z = i, then downand to the left through 1, then down and to the right through i, and nally up and to the right to z = 1, where westarted.Almost all of complex analysis is concerned with complex functions that is, with functions that map some subsetof the complex plane into some other (possibly overlapping, or even identical) subset of the complex plane. Here itis customary to speak of the domain of f(z) as lying in the z-plane, while referring to the range or image of f(z) as aset of points in the w-plane. In symbols we writez= x + iy; f(z) = w = u + ivand often think of the function f as a transformation of the z-plane (with coordinates (x, y)) into the w-plane (withcoordinates (u, v)).2.2 Stereographic projectionsMain article: Stereographic projectionIt can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius,Riemann sphere which maps all but one point on a sphere to all points on the complex planeplace its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit30 CHAPTER 2. COMPLEX PLANEcircle 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 poleand the points in the complex plane as follows. Given a point in the plane, draw a straight line connecting it with thenorth pole on the sphere. That line will intersect the surface of the sphere in exactly one other point. The point z =0 will be projected onto the south pole of the sphere. Since the interior of the unit circle lies inside the sphere, thatentire region (|z| < 1) will be mapped onto the southern hemisphere.The unit circle itself (|z| = 1) will be mappedonto the equator, and the exterior of the unit circle (|z| > 1) will be mapped onto the northern hemisphere, minus thenorth pole. Clearly this procedure is reversible given any point on the surface of the sphere that is not the northpole, we can draw a straight line connecting that point to the north pole and intersecting the at plane in exactly onepoint.Under this stereographic projection the north pole itself is not associated with any point in the complex plane. Weperfect the one-to-one correspondence by adding one more point to the complex plane the so-called point at inn-ityand identifying it with the north pole on the sphere. This topological space, the complex plane plus the point atinnity, is known as the extended complex plane. We speak of a single point at innity when discussing complexanalysis. There are two points at innity (positive, and negative) on the real number line, but there is only one pointat innity (the north pole) in the extended complex plane.[6]Imagine for a moment what will happen to the lines of latitude and longitude when they are projected from the sphereonto the at plane. The lines of latitude are all parallel to the equator, so they will become perfect circles centeredon the origin z = 0. And the lines of longitude will become straight lines passing through the origin (and also throughthe point at innity, 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 consistingof two or more values.For instance, the north pole of the sphere might be placed on top of the origin z = 1 in aplane thats tangent to the circle. The details don't really matter. Any stereographic projection of a sphere onto aplane will produce one point at innity, and it will map the lines of latitude and longitude on the sphere into circlesand straight lines, respectively, in the plane.2.3 Cutting the planeWhen discussing functions of a complex variable it is often convenient to think of a cut in the complex plane. Thisidea arises naturally in several dierent contexts.2.3.1 Multi-valued relationships and branch pointsConsider the simple two-valued relationshipw = f(z) = z= z1/2.Before we can treat this relationship as a single-valued function, the range of the resulting value must be restrictedsomehow. When dealing with the square roots of non-negative real numbers this is easily done. For instance, we canjust deney= g(x) =x= x1/2to be the non-negative real number y such that y2= x. This idea doesn't work so well in the two-dimensional complexplane. To see why, lets think about the way the value of f(z) varies as the point z moves around the unit circle. Wecan writez= reiand take w = z1/2=r ei/2(0 2).Evidently, as z moves all the way around the circle, w only traces out one-half of the circle. So one continuous motionin the complex plane has transformed the positive square root e0= 1 into the negative square root ei= 1.2.3. CUTTING THE PLANE 31This problem arises because the point z = 0 has just one square root, while every other complex number z 0 hasexactly two square roots. On the real number line we could circumvent this problem by erecting a barrier at thesingle point x = 0. A bigger barrier is needed in the complex plane, to prevent any closed contour from completelyencircling the branch point z = 0. This is commonly done by introducing a branch cut; in this case the cut mightextend from the point z = 0 along the positive real axis to the point at innity, so that the argument of the variable zin the cut plane is restricted to the range 0 arg(z) < 2.We can now give a complete description of w = z. To do so we need two copies of the z-plane, each of them cutalong the real axis. On one copy we dene the square root of 1 to be e0= 1, and on the other we dene the squareroot of 1 to be ei= 1. We call these two copies of the complete cut plane sheets. By making a continuity argumentwe see that the (now single-valued) function w = z maps the rst sheet into the upper half of the w-plane, where 0 arg(w) < , while mapping the second sheet into the lower half of the w-plane (where arg(w) < 2).[7]The branch cut in this example doesn't have to lie along the real axis. It doesn't even have to be a straight line. Anycontinuous curve connecting the origin z = 0 with the point at innity would work. In some cases the branch cutdoesn't even have to pass through the point at innity. For example, consider the relationshipw = g(z) =(z21)1/2.Here the polynomial z2 1 vanishes when z = 1, so g evidently has two branch points. We can cut the plane alongthe real axis, from 1 to 1, and obtain a sheet on which g(z) is a single-valued function. Alternatively, the cut can runfrom z = 1 along the positive real axis through the point at innity, then continue up the negative real axis to theother branch point, z = 1.This situation is most easily visualized by using the stereographic projection described above. On the sphere oneof these cuts runs longitudinally through the southern hemisphere, connecting a point on the equator (z = 1) withanother point on the equator (z = 1), and passing through the south pole (the origin, z = 0) on the way. The secondversion of the cut runs longitudinally through the northern hemisphere and connects the same two equatorial pointsby passing through the north pole (that is, the point at innity).2.3.2 Restricting the domain of meromorphic functionsA meromorphic function is a complex function that is holomorphic and therefore analytic everywhere in its domainexcept at a nite, or countably innite, number of points.[8] The points at which such a function cannot be denedare called the poles of the meromorphic function. Sometimes all these poles lie in a straight line. In that casemathematicians may say that the function is holomorphic on the cut plane. Heres a simple example.The gamma function, dened by(z) =ezzn=1[(1 +zn)1ez/n]where is the EulerMascheroni constant, and has simple poles at 0, 1, 2, 3, ... because exactly one denominatorin the innite product vanishes when z is zero, or a negative integer.[9] Since all its poles lie on the negative real axis,from z = 0 to the point at innity, this function might be described as holomorphic on the cut plane, the cut extendingalong the negative real axis, from 0 (inclusive) to the point at innity.Alternatively, (z) might be described as holomorphic in the cut plane with < arg(z) < and excluding the pointz = 0.Notice that this cut is slightly dierent from the branch cut we've already encountered, because it actually excludesthe negative real axis from the cut plane. The branch cut left the real axis connected with the cut plane on one side(0 ), but severed it from the cut plane along the other side ( < 2).Of course, its not actually necessary to exclude the entire line segment from z = 0 to to construct a domain inwhich (z) is holomorphic. All we really have to do is puncture the plane at a countably innite set of points {0,1, 2, 3, ...}. But a closed contour in the punctured plane might encircle one or more of the poles of (z), givinga contour integral that is not necessarily zero, by the residue theorem. By cutting the complex plane we ensure notonly that (z) is holomorphic in this restricted domain we also ensure that the contour integral of over any closedcurve lying in the cut plane is identically equal to zero.32 CHAPTER 2. COMPLEX PLANE2.3.3 Specifying convergence regionsMany complex functions are dened by innite series, or by continued fractions. A fundamental consideration in theanalysis of these innitely long expressions is identifying the portion of the complex plane in which they converge toa nite value. A cut in the plane may facilitate this process, as the following examples show.Consider the function dened by the innite seriesf(z) =n=1(z2+ n)2.Since z2= (z)2for every complex number z, its clear that f(z) is an even function of z, so the analysis can berestricted to one half of the complex plane. And since the series is undened whenz2+ n = 0 z= in,it makes sense to cut the plane along the entire imaginary axis and establish the convergence of this series where thereal part of z is not zero before undertaking the more arduous task of examining f(z) when z is a pure imaginarynumber.[10]In this example the cut is a mere convenience, because the points at which the innite sum is undened are isolated,and the cut plane can be replaced with a suitably punctured plane. In some contexts the cut is necessary, and not justconvenient. Consider the innite periodic continued fractionf(z) = 1 +z1 +z1 +z1 +z....It can be shown that f(z) converges to a nite value if and only if z is not a negative real number such that z < . Inother words, the convergence region for this continued fraction is the cut plane, where the cut runs along the negativereal axis, from to the point at innity.[11]2.4 Gluing the cut plane back togetherMain article: Riemann surfaceWe have already seen how the relationshipw = f(z) = z= z1/2can be made into a single-valued function by splitting the domain of f into two disconnected sheets. It is also possibleto glue those two sheets back together to form a single Riemann surface on which f(z) = z1/2 can be dened as aholomorphic function whose image is the entire w-plane (except for the point w = 0). Heres how that works.Imagine two copies of the cut complex plane, the cuts extending along the positive real axis from z = 0 to the pointat innity. On one sheet dene 0 arg(z) < 2, so that 11/2 = e0= 1, by denition. On the second sheet dene 2 arg(z) < 4, so that 11/2 = ei= 1, again by denition. Now ip the second sheet upside down, so the imaginaryaxis points in the opposite direction of the imaginary axis on the rst sheet, with both real axes pointing in the samedirection, and glue the two sheets together (so that the edge on the rst sheet labeled " = 0 is connected to theedge labeled " < 4" on the second sheet, and the edge on the second sheet labeled " = 2" is connected to the edgelabeled " < 2" on the rst sheet). The result is the Riemann surface domain on which f(z) = z1/2 is single-valuedand holomorphic (except when z = 0).[7]2.5. USE OF THE COMPLEX PLANE IN CONTROL THEORY 33To understand why f is single-valued in this domain, imagine a circuit around the unit circle, starting with z = 1 onthe rst sheet. When 0 < 2 we are still on the rst sheet. When = 2 we have crossed over onto the secondsheet, and are obliged to make a second complete circuit around the branch point z = 0 before returning to our startingpoint, where = 4 is equivalent to = 0, because of the way we glued the two sheets together. In other words, asthe variable z makes two complete turns around the branch point, the image of z in the w-plane traces out just onecomplete circle.Formal dierentiation shows thatf(z) = z1/2 f(z) =12z1/2from which we can conclude that the derivative of f exists and is nite everywhere on the Riemann surface, exceptwhen z = 0 (that is, f is holomorphic, except when z = 0).How can the Riemann surface for the functionw = g(z) =(z21)1/2,also discussed above, be constructed? Once again we begin with two copies of the z-plane, but this time each one iscut along the real line segment extending from z = 1 to z = 1 these are the two branch points of g(z). We ip oneof these upside down, so the two imaginary axes point in opposite directions, and glue the corresponding edges of thetwo cut sheets together. We can verify that g is a single-valued function on this surface by tracing a circuit around acircle of unit radius centered at z = 1. Commencing at the point z = 2 on the rst sheet we turn halfway around thecircle before encountering the cut at z = 0. The cut forces us onto the second sheet, so that when z has traced out onefull turn around the branch point z = 1, w has taken just one-half of a full turn, the sign of w has been reversed (sinceei= 1), and our path has taken us to the point z = 2 on the second sheet of the surface. Continuing on throughanother half turn we encounter the other side of the cut, where z = 0, and nally reach our starting point (z = 2 on therst sheet) after making two full turns around the branch point.The natural way to label = arg(z) in this example is to set < on the rst sheet, with < 3 on thesecond. The imaginary axes on the two sheets point in opposite directions so that the counterclockwise sense ofpositive 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 the xy-plane.Then there appears to be a vertical hole in the surface, where the two cuts are joined together. What if the cut is madefrom z = 1 down the real axis to the point at innity, and from z = 1, up the real axis until the cut meets itself? Againa Riemann surface can be constructed, but this time the hole is horizontal. Topologically speaking, both versionsof this Riemann surface are equivalent they are orientable two-dimensional surfaces of genus one.2.5 Use of the complex plane in control theoryIn control theory, one use of the complex plane is known as the 's-plane'. It is used to visualise the roots of theequation describing a systems behaviour (the characteristic equation) graphically. The equation is normally expressedas a polynomial in the parameter 's of the Laplace transform, hence the name 's plane. Points in the s-plane take theform s = + j , where 'j' is used instead of the usual 'i' to represent the imaginary component.Another related use of the complex plane is with the Nyquist stability criterion. This is a geometric principle whichallows the stability of a closed-loop feedback system to be determined by inspecting a Nyquist plot of its open-loopmagnitude and phase response as a function of frequency (or loop transfer function) in the complex plane.The 'z-plane' is a discrete-time version of the s-plane, where z-transforms are used instead of the Laplace transfor-mation.2.6 Other meanings of complex planeThe preceding sections of this article deal with the complex plane as the geometric analogue of the complex numbers.Although this usage of the term complex plane has a long and mathematically rich history, it is by no means the34 CHAPTER 2. COMPLEX PLANEonly mathematical concept that can be characterized as the complex plane. There are at least three additionalpossibilities.1. 1+1-dimensional Minkowski space, also known as the split-complex plane, is a complex plane in the sensethat the algebraic split-complex numbers can be separated into two real components that are easily associatedwith the point (x, y) in the Cartesian plane.2. The set of dual numbers over the reals can also be placed into one-to-one correspondence with the points (x,y) of the Cartesian plane, and represent another example of a complex plane.3. The vector space CC, the Cartesian product of the complex numbers with themselves, is also a complexplane in the sense that it is a two-dimensional vector space whose coordinates are complex numbers.2.7 TerminologyWhile the terminology complex plane is historically accepted, the object could be more appropriately named com-plex line as it is a 1-dimensional complex vector space.2.8 See alsoMandelbrot fractal, imagined on a complex planeConstellation diagramRiemann sphereS planeIn-phase and quadrature components2.9. NOTES 352.9 Notes[1] Although this is the most common mathematical meaning of the phrase complex plane, it is not the only one possible.Alternatives include the split-complex plane and the dual numbers, as introduced by quotient rings.[2] Wessels memoir was presented to the Danish Academy in 1797; Argands paper was published in 1806. (Whittaker &Watson, 1927, p. 9)[3] A detailed denition of the complex argument in terms of the real arctangent can be found here.[4] It can be shown (Whittaker & Watson, 1927, Appendix) that all the familiar properties of the complex exponential function,the trigonometric functions, and the complex logarithm can be deduced directly from the power series for ez. In partic-ular, the principal value of logr, where |r| = 1, can be calculated without reference to any geometrical or trigonometricconstruction.[5] (Whittaker & Watson, 1927, p. 10)[6] (Flanigan, 1983, p. 305)[7] (Moretti, 1964, pp. 113119)[8] See also Proof that holomorphic functions are analytic.[9] It can be shown that the innite product for (z) is uniformly convergent on any bounded region where none of its de-nominators vanish; therefore it denes a meromorphic function on the complex plane. (Whittaker & Watson, 1927, pp.235236)[10] When Re(z) > 0 this sum converges uniformly on any bounded domain by comparison with (2), where (s) is the Riemannzeta function.[11] (Wall, 1948, p. 39)2.10 ReferencesFlanigan, 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) byChelsea Publishing Company ISBN 0-8284-0207-8.Whittaker, E. T.; Watson, G. N. (1927). A Course in Modern Analysis (Fourth ed.). Cambridge UniversityPress.2.11 External linksWeisstein, Eric W., Argand Diagram, MathWorld.Chapter 3Complex-valued functionAn exponential function Anof a discrete (integer) variable n, similar to geometric progressionIn mathematics, a complex-valued function (sometimes referred to as complex function) is a function whose valuesare complex numbers. In other words, it is a function that assigns a complex number to each member of its domain.This domain does not necessarily have any structure related to complex numbers. Most important uses of such363.1. COMPLEX ANALYSIS 37functions in complex analysis and in functional analysis are explicated below.A vector space and a commutative algebra of functions over complex numbers can be dened in the same way as forreal-valued functions. Also, any complex-valued function f on an arbitrary set X can be considered as an orderedpair of two real-valued functions: (Ref, Imf) or, alternatively, as a real-valued function on X {0, 1} (the disjointunion of two copies of X) such that for any x:Eulers formula features a complex-valued function of a real (or angular) variable Re f(x) = F(x, 0)Imf(x) = F(x, 1)Some properties of complex-valued functions (such as measurability and continuity) are nothing more than correspondingproperties of real-valued functions.3.1 Complex analysisMain article: Complex analysis38 CHAPTER 3. COMPLEX-VALUED FUNCTIONComplex analysis considers holomorphic functions on complex manifolds, such as Riemann surfaces. The property ofanalytic continuation makes them very dissimilar from smooth functions, for example. Namely, if a function denedin a neighborhood can be continued to a wider domain, then this continuation is unique.As real functions, any holomorphic function is innitely smooth and analytic. But there is much less freedom inconstruction of a holomorphic function than in one of a smooth function.3.2 Functional analysisComplex-valued L2spaces on sets with a measure have a particular importance because they are Hilbert spaces. Theyoften appear in functional analysis (for example, in relation with Fourier transform) and operator theory. A majoruser of such spaces is quantum mechanics, as wave functions.The sets on which the complex-valued L2is constructed have the potential to be more exotic than their real-valuedanalog. For example, complex-valued function spaces are used in some branches of p-adic analysis for algebraicreasons: complex numbers form an algebraically closed eld (which facilitates operator theory), whereas neither realnumbers nor p-adic numbers are not.Also, complex-valued continuous functions are an important example in the theory of C*-algebras: see Gelfandrepresentation.3.3 See alsoFunction of a complex variable, the dual concept3.4 External linksWeisstein, Eric W., Complex Function, MathWorld.Chapter 4Imaginary numberImaginary Number and Imaginary numbers redirect here. For the 2013 EP by The Maine, see Imaginary Num-bers (EP).An imaginary number is a complex number that can be written as a real number multiplied by the imaginary uniti,[note 1] which is dened by its property i2= 1.[1] The square of an imaginary number bi is b2. For example, 5i isan imaginary number, and its square is 25. Except for 0 (which is both real and imaginary[2]), imaginary numbersproduce negative real numbers when squared.An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where thereal numbers a and b are called, respectively, the real part and the imaginary part of the complex number.[3][note 2]Imaginary numbers can therefore be thought of as complex numbers whose real part is zero. The name imaginarynumber was coined in the 17th century as a derogatory term, as such numbers were regarded by some as ctitiousor useless. The term imaginary number now means simply a complex number with a real part equal to 0, that is, anumber of the form bi.4.1 HistoryMain article: History of complex numbersAlthough Greek mathematician and engineer Heron of Alexandria is noted as the rst to have conceived thesenumbers,[4][5] Rafael Bombelli rst set down the rules for multiplication of complex numbers in 1572. The concepthad appeared in print earlier, for instance in work by Gerolamo Cardano. At the time, such numbers were poorlyunderstood and regarded by some as ctitious or useless, much as zero and the negative numbers once were. Manyother mathematicians were slow to adopt the use of imaginary numbers, including Ren Descartes, who wrote aboutthem in his La Gomtrie, where the term imaginary was used and meant to be derogatory.[6] The use of imaginarynumbers was not widely accepted until the work of Leonhard Euler (17071783) and Carl Friedrich Gauss (17771855). The geometric signicance of complex numbers as points in a plane was rst described by Caspar Wessel(17451818).[7]In 1843 a mathematical physicist, William Rowan Hamilton, extended the idea of an axis of imaginary numbers inthe plane to a three-dimensional space of quaternion imaginaries.With the development of quotient rings of polynomial rings, the concept behind an imaginary number became moresubstantial, but then one also nds other imaginary numbers such as the j of tessarines which has a square of +1. Thisidea rst surfaced with the articles by James Cockle beginning in 1848.[8]4.2 Geometric interpretationGeometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to bepresented perpendicular to the real axis. One way of viewing imaginary numbers is to consider a standard number line,positively increasing in magnitude to the right, and negatively increasing in magnitude to the left. At 0 on this x-axis,a y-axis can be drawn with positive direction going up; positive imaginary numbers then increase in magnitude3940 CHAPTER 4. IMAGINARY NUMBERImReyy0xrrz=x+iyz=xiyAn illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.upwards, and negative imaginary numbers increase in magnitude downwards. This vertical axis is often called theimaginary axis and is denoted i, I , or .In this representation, multiplication by 1 corresponds to a rotation of 180 degrees about the origin. Multiplicationby i corresponds t