Complex conjugateFrom Wikipedia, the free encyclopediaContents1 Absolute value 11.1 Terminology and notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Absolute value function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Relationship to the sign function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.3 Antiderivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5.1 Ordered rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5.3 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Analytic function 102.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Alternative characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Properties of analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Analyticity and dierentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Real versus complex analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Analytic functions of several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Argument (complex analysis) 143.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14iii CONTENTS3.2 Principal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Covering space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Complex analysis 214.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4 Major results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Complex conjugate 265.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Use as a variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Complex number 306.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.1.3 Complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.1.4 History in brief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2.1 Equality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2.2 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.3 Elementary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.3.1 Conjugation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.3.2 Addition and subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.3.3 Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37CONTENTS iii6.3.4 Square root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.4 Polar form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.4.1 Absolute value and argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.4.2 Multiplication and division in polar form. . . . . . . . . . . . . . . . . . . . . . . . . . . 406.5 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.5.1 Eulers formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.5.2 Natural logarithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.5.3 Integer and fractional exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.6 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.6.1 Field structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.6.2 Solutions of polynomial equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.6.3 Algebraic characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.6.4 Characterization as a topological eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.7 Formal construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.7.1 Formal development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.7.2 Matrix representation of complex numbers. . . . . . . . . . . . . . . . . . . . . . . . . . 446.8 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.8.1 Complex exponential and related functions. . . . . . . . . . . . . . . . . . . . . . . . . . 456.8.2 Holomorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.9 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.9.1 Control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.9.2 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.9.3 Fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.9.4 Dynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.9.5 Electromagnetism and electrical engineering . . . . . . . . . . . . . . . . . . . . . . . . . 486.9.6 Signal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.9.7 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.9.8 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.9.9 Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.9.10 Algebraic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.9.11 Analytic number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.10History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.11Generalizations and related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.12See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.13Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.14.1 Mathematical references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.14.2 Historical references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.15Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Complex plane 56iv CONTENTS7.1 Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567.2 Stereographic projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.3 Cutting the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.3.1 Multi-valued relationships and branch points . . . . . . . . . . . . . . . . . . . . . . . . . 597.3.2 Restricting the domain of meromorphic functions . . . . . . . . . . . . . . . . . . . . . . 607.3.3 Specifying convergence regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.4 Gluing the cut plane back together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.5 Use of the complex plane in control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.6 Other meanings of complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.7 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 Complex-valued function 658.1 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668.2 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 Holomorphic function 689.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709.5 Several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709.6 Extension to functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7210Imaginary number 7510.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7510.2Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7510.3Multiplication of square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7710.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7810.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7810.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7810.7Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7810.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7811Imaginary unit 79CONTENTS v11.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8011.2i and i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8011.3Proper use. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8111.4Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8211.4.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8211.4.2 Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8411.4.3 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8411.4.4 Factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8511.4.5 Other operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8511.5Alternative notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8611.6Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8611.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8611.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8611.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8711.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8711.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8712Meromorphic function 8812.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8912.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8912.2.1 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8912.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8912.4On Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9012.5By extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9012.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9013Real number 9213.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9313.2Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9413.2.1 Axiomatic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9413.2.2 Construction from the rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 9513.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9513.3.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9513.3.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9513.3.3 The complete ordered eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9613.3.4 Advanced properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9613.4Applications and connections to other areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9713.4.1 Real numbers and logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9713.4.2 In physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9813.4.3 In computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9813.4.4 Reals in set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9813.5Vocabulary and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98vi CONTENTS13.6Generalizations and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9913.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9913.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9913.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10013.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10013.11Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 10113.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10113.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10413.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Chapter 1Absolute valueFor other uses, see Absolute value (disambiguation).In mathematics, the absolute value (or modulus) |x| of a real number x is the non-negative value of x without regardThe absolute value of a number may be thought of as its distance from zero.to its sign. Namely, |x| = x for a positive x, |x| = x for a negative x (in which case x is positive), and |0| = 0. Forexample, the absolute value of 3 is 3, and the absolute value of 3 is also 3. The absolute value of a number may bethought of as its distance from zero.Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example,an absolute value is also dened for the complex numbers, the quaternions, ordered rings, elds and vector spaces.The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical andphysical contexts.1.1 Terminology and notationIn 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, specically for thecomplex absolute value,[1][2] and it was borrowed into English in 1866 as the Latin equivalent modulus.[1] The termabsolute value has been used in this sense from at least 1806 in French[3] and 1857 in English.[4] The notation |x|, witha vertical bar on each side, was introduced by Karl Weierstrass in 1841.[5] Other names for absolute value includenumerical value[1] and magnitude.[1]The same notation is used with sets to denote cardinality; the meaning depends on context.1.2 Denition and properties12 CHAPTER 1. ABSOLUTE VALUE1.2.1 Real numbersFor any real number x theabsolutevalue ormodulus of x is denoted by |x| (a vertical bar on each side of thequantity) and is dened as[6]|x| =_x, if x 0x, if x < 0.As can be seen from the above denition, the absolute value of x is always either positive or zero, but never negative.From an analytic geometry point of view, the absolute value of a real number is that numbers distance from zeroalong the real number line, and more generally the absolute value of the dierence of two real numbers is the distancebetween them. Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisationof the absolute value of the dierence (see Distance below).Since the square root notation without sign represents the positive square root, it follows thatwhich is sometimes used as a denition of absolute value of real numbers.[7]The absolute value has the following four fundamental properties:Other important properties of the absolute value include:Two other useful properties concerning inequalities are:|a| bb a b|a| ba bor b aThese relations may be used to solve inequalities involving absolute values. For example:Absolute value is used to dene the absolute dierence, the standard metric on the real numbers.1.2.2 Complex numbersSince the complex numbers are not ordered, the denition given above for the real absolute value cannot be directlygeneralised for a complex number. However the geometric interpretation of the absolute value of a real number as itsdistance from 0 can be generalised. The absolute value of a complex number is dened as its distance in the complexplane from the origin using the Pythagorean theorem. More generally the absolute value of the dierence of twocomplex numbers is equal to the distance between those two complex numbers.For any complex numberz= x + iy,where x and y are real numbers, the absolute value or modulus of z is denoted |z| and is given by[8]|z| =x2+ y2.1.2. DEFINITION AND PROPERTIES 3ImReyy0xrrz=x+iyz=xiyThe absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complexconjugate z have the same absolute value.When the imaginary part y is zero this is the same as the absolute value of the real number x.When a complex number z is expressed in polar form as4 CHAPTER 1. ABSOLUTE VALUEz= reiwith r 0 and real, its absolute value is|z| = rThe absolute value of a complex number can be written in the complex analogue of equation (1) above as:|z| =z zwhere z is the complex conjugate of z. Notice that, contrary to equation (1):|z| =z2The complex absolute value shares all the properties of the real absolute value given in equations (2)(11) above.Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolutevalue as an endomorphism of the multiplicative group of the complex numbers.[9]1.3 Absolute value function12343 2 1 1 2 3 0y = |x|The graph of the absolute value function for real numbersThe real absolute value function is continuous everywhere. It is dierentiable everywhere except for x = 0. It ismonotonically decreasing on the interval (,0] and monotonically increasing on the interval [0,+). Since a realnumber and its opposite have the same absolute value, it is an even function, and is hence not invertible.Both the real and complex functions are idempotent.It is a piecewise linear, convex function.1.3. ABSOLUTE VALUE FUNCTION 5xyf (| x |)|f (x) |f (x)Composition of absolute value with a cubic function in dierent orders1.3.1 Relationship to the sign functionThe absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum)function returns a numbers sign irrespective of its value. The following equations showthe relationship between thesetwo functions:|x| = xsgn(x),6 CHAPTER 1. ABSOLUTE VALUEor|x| sgn(x) = x,and for x 0,sgn(x) = |x|x.1.3.2 DerivativeThe real absolute value function has a derivative for every x 0, but is not dierentiable at x = 0. Its derivative for x 0 is given by the step function[10][11]d|x|dx=x|x|=_1 x < 01 x > 0.The subdierential of |x| at x = 0 is the interval [1,1].[12]The complex absolute value function is continuous everywhere but complex dierentiable nowhere because it violatesthe CauchyRiemann equations.[10]The second derivative of |x| with respect to x is zero everywhere except zero, where it does not exist. As a generalisedfunction, the second derivative may be taken as two times the Dirac delta function.1.3.3 AntiderivativeThe antiderivative (indenite integral) of the absolute value function is|x|dx =x|x|2+ C,where C is an arbitrary constant of integration.1.4 DistanceSee also: Metric spaceThe absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complexnumber is the distance from that number to the origin, along the real number line, for real numbers, or in the complexplane, for complex numbers, and more generally, the absolute value of the dierence of two real or complex numbersis the distance between them.The standard Euclidean distance between two pointsa = (a1, a2, . . . , an)andb = (b1, b2, . . . , bn)1.5. GENERALIZATIONS 7in Euclidean n-space is dened as:_ni=1(aibi)2.This can be seen to be a generalisation of |a b|, since if a and b are real, then by equation (1),|a b| =(a b)2.While ifa = a1 + ia2andb = b1 + ib2are complex numbers, thenThe above shows that the absolute value distance for the real numbers or the complex numbers, agrees with thestandard Euclidean distance they inherit as a result of considering them as the one and two-dimensional Euclideanspaces respectively.The properties of the absolute value of the dierence of two real or complex numbers: non-negativity, identity ofindiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion ofa distance function as follows:A real valued function d on a set X X is called a metric (or a distance function) on X, if it satises the followingfour axioms:[13]1.5 Generalizations1.5.1 Ordered ringsThe denition of absolute value given for real numbers above can be extended to any ordered ring. That is, if a is anelement of an ordered ring R, then the absolute value of a, denoted by |a|, is dened to be:[14]|a| =_a, if a 0a, if a 0where a is the additive inverse of a, and 0 is the additive identity element.1.5.2 FieldsMain article: Absolute value (algebra)8 CHAPTER 1. ABSOLUTE VALUEThe fundamental properties of the absolute value for real numbers given in (2)(5) above, can be used to generalisethe notion of absolute value to an arbitrary eld, as follows.A real-valued function v on a eld F is called an absolute value (also a modulus, magnitude, value, or valuation)[15] ifit satises the following four axioms:Where 0 denotes the additive identity element of F. It follows from positive-deniteness and multiplicativeness thatv(1) = 1, where 1 denotes the multiplicative identity element of F. The real and complex absolute values dened aboveare examples of absolute values for an arbitrary eld.If v is an absolute value on F, then the function d on F F, dened by d(a, b) = v(a b), is a metric and the followingare equivalent:d satises the ultrametric inequality d(x, y) max(d(x, z), d(y, z)) for all x, y, z in F._v_nk=11_: n N_is bounded in R. v_nk=11_ 1for every n N. v(a) 1 v(1 + a) 1for all a F. v(a + b) max{v(a), v(b)}for all a, b F.An absolute value which satises any (hence all) of the above conditions is said to be non-Archimedean, otherwiseit is said to be Archimedean.[16]1.5.3 Vector spacesMain article: Norm (mathematics)Again the fundamental properties of the absolute value for real numbers can be used, with a slight modication, togeneralise the notion to an arbitrary vector space.A real-valued function on a vector space V over a eld F, represented as , is called an absolute value, but moreusually a norm, if it satises the following axioms:For all a in F, and v, u in V,The norm of a vector is also called its length or magnitude.In the case of Euclidean space Rn, the function dened by(x1, x2, . . . , xn) =_ni=1x2iis a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector spaceR1, the absolute value is a norm, and is the p-norm (see Lp space) for any p. In fact the absolute value is the onlynorm on R1, in the sense that, for every norm on R1, x = 1 |x|. The complex absolute value is a special caseof the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identied withthe Euclidean plane R2.1.6. NOTES 91.6 Notes[1] Oxford English Dictionary, Draft Revision, June 2008[2] Nahin, O'Connor and Robertson, and functions.Wolfram.com.; for the French sense, see Littr, 1877[3] Lazare Nicolas M. Carnot, Mmoire sur la relation qui existe entre les distances respectives de cinq point quelconques prisdans l'espace, p. 105 at Google Books[4] James Mill Peirce, A Text-book of Analytic Geometry at Google Books. The oldest citation in the 2nd edition of the OxfordEnglish Dictionary is from 1907. The term absolute value is also used in contrast to relative value.[5] Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, p. 25[6] Mendelson, p. 2.[7] Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN 0-534-37718-1., p. A5[8] Gonzlez, Mario O. (1992). Classical Complex Analysis. CRC Press. p. 19. ISBN 9780824784157.[9] Lorenz, Falko (2008), Algebra. Vol. II. Fields with structure, algebras and advanced topics, Universitext, New York:Springer, p. 39, doi:10.1007/978-0-387-72488-1, ISBN 978-0-387-72487-4, MR 2371763.[10] Weisstein, Eric W. Absolute Value. From MathWorld A Wolfram Web Resource.[11] Bartel and Sherbert, p. 163[12] Peter Wriggers, Panagiotis Panatiotopoulos, eds., New Developments in Contact Problems, 1999, ISBN 3-211-83154-1, p.3132[13] These axioms are not minimal; for instance, non-negativity can be derived from the other three: 0 = d(a, a) d(a, b) +d(b, a) = 2d(a, b).[14] Mac Lane, p. 264.[15] Shechter, p. 260. This meaning of valuation is rare. Usually, a valuation is the logarithm of the inverse of an absolutevalue[16] Shechter, pp. 260261.1.7 ReferencesBartle; Sherbert; Introduction to real analysis (4th ed.), John Wiley & Sons, 2011 ISBN 978-0-471-43331-6.Nahin, Paul J.; An Imaginary Tale; Princeton University Press; (hardcover, 1998). ISBN 0-691-02795-1.Mac Lane, Saunders, Garrett Birkho, Algebra, American Mathematical Soc., 1999. ISBN978-0-8218-1646-2.Mendelson, Elliott, Schaums Outline of Beginning Calculus, McGraw-Hill Professional, 2008. ISBN 978-0-07-148754-2.O'Connor, J.J. and Robertson, E.F.; Jean Robert Argand.Schechter, Eric; Handbook of Analysis and Its Foundations, pp. 259263, Absolute Values, Academic Press(1997) ISBN 0-12-622760-8.1.8 External linksHazewinkel, Michiel, ed. (2001), Absolute value,Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4absolute value at PlanetMath.org.Weisstein, Eric W., Absolute Value, MathWorld.Chapter 2Analytic functionNot to be confused with analytic expression or analytic signal.This article is about both real and complex analytic functions. For analytic functions in complex analysis specically,see holomorphic function.In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist bothreal analytic functions and complex analytic functions, categories that are similar in some ways, but dierent inothers. Functions of each type are innitely dierentiable, but complex analytic functions exhibit properties that donot hold generally for real analytic functions. A function is analytic if and only if its Taylor series about x0 convergesto the function in some neighborhood for every x0 in its domain.2.1 DenitionsFormally, a function is real analytic on an open set D in the real line if for any x0 in D one can writef(x) =n=0an (x x0)n= a0 + a1(x x0) + a2(x x0)2+ a3(x x0)3+ in which the coecients a0, a1, ... are real numbers and the series is convergent to (x) for x in a neighborhood of x0.Alternatively, an analytic function is an innitely dierentiable function such that the Taylor series at any point x0 inits domainT(x) =n=0f(n)(x0)n!(x x0)nconverges to f(x) for x in a neighborhood of x0 pointwise (and locally uniformly). The set of all real analytic functionson a given set D is often denoted by C(D).A function dened on some subset of the real line is said to be real analytic at a point x if there is a neighborhoodD of x on which is real analytic.The denition of a complex analytic function is obtained by replacing, in the denitions above, real with complexand real line with complex plane. A function is complex analytic if and only if it is holomorphic i.e. it iscomplex dierentiable. For this reason the terms holomorphic and analytic are often used interchangeably forsuch functions.[1]2.2 ExamplesTypical examples of analytic functions are:102.3. ALTERNATIVE CHARACTERIZATIONS 11All elementary functions:All polynomials: if a polynomial has degree n, any terms of degree larger than n in its Taylor seriesexpansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore,every polynomial is its own Maclaurin series.The exponential function is analytic. Any Taylor series for this function converges not only for x closeenough to x0 (as in the denition) but for all values of x (real or complex).The trigonometric functions, logarithm, and the power functions are analytic on any open set of theirdomain.Most special functions (at least in some range of the complex plane):hypergeometric functionsBessel functionsgamma functionTypical examples of functions that are not analytic are:The absolute value function when dened on the set of real numbers or complex numbers is not everywhereanalytic because it is not dierentiable at 0. Piecewise dened functions (functions given by dierent formulasin dierent regions) are typically not analytic where the pieces meet.The complex conjugate function z z* is not complex analytic, although its restriction to the real line is theidentity function and therefore real analytic, and it is real analytic as a function from R to R.See here for another example of a non-analytic smooth function.2.3 Alternative characterizationsIf is an innitely dierentiable function dened on an open set D R, then the following conditions are equivalent.1) is real analytic.2) There is a complex analytic extension of to an open set G C which contains D.3) For every compact set K D there exists a constant C such that for every x K and every non-negativeinteger k the following bound holds[2]dkfdxk (x) Ck+1k!The real analyticity of a function at a given point x can be characterized using the FBI transform.Complex analytic functions are exactly equivalent to holomorphic functions, and are thus much more easily charac-terized.2.4 Properties of analytic functionsThe sums, products, and compositions of analytic functions are analytic.The reciprocal of an analytic function that is nowhere zero is analytic, as is the inverse of an invertible analyticfunction whose derivative is nowhere zero. (See also the Lagrange inversion theorem.)Any analytic function is smooth, that is, innitely dierentiable. The converse is not true for real functions;in fact, in a certain sense, the real analytic functions are sparse compared to all real innitely dierentiablefunctions. For the complex numbers, the converse does hold, and in fact any function dierentiable once on anopen set is analytic on that set (see analyticity and dierentiability below).12 CHAPTER 2. ANALYTIC FUNCTIONFor any open set C, the set A() of all analytic functions u : C is a Frchet space with respect to theuniform convergence on compact sets. The fact that uniform limits on compact sets of analytic functions areanalytic is an easy consequence of Moreras theorem. The set A() of all bounded analytic functions with thesupremum norm is a Banach space.A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number ofzeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If theset of zeros of an analytic function has an accumulation point inside its domain, then is zero everywhere on theconnected component containing the accumulation point.In other words, if (rn) is a sequence of distinct numberssuch that (rn) = 0 for all n and this sequence converges to a point r in the domain of D, then is identically zero onthe connected component of D containing r. This is known as the Principle of Permanence.Also, if all the derivatives of an analytic function at a point are zero, the function is constant on the correspondingconnected component.These statements imply that while analytic functions do have more degrees of freedom than polynomials, they arestill quite rigid.2.5 Analyticity and dierentiabilityAs noted above, any analytic function (real or complex) is innitely dierentiable (also known as smooth, or C).(Note that this dierentiability is in the sense of real variables; compare complex derivatives below.) There existsmooth real functions that are not analytic: see non-analytic smooth function. In fact there are many such functions.The situation is quite dierent when one considers complex analytic functions and complex derivatives. It can beproved that any complex function dierentiable (in the complex sense) in an open set is analytic.Consequently, incomplex analysis, the term analytic function is synonymous with holomorphic function.2.6 Real versus complex analytic functionsReal and complex analytic functions have important dierences (one could notice that even from their dierentrelationship with dierentiability). Analyticity of complex functions is a more restrictive property, as it has more re-strictive necessary conditions and complex analytic functions have more structure than their real-line counterparts.[3]According to Liouvilles theorem, any bounded complex analytic function dened on the whole complex plane isconstant. The corresponding statement for real analytic functions, with the complex plane replaced by the real line,is clearly false; this is illustrated byf(x) =1x2+ 1.Also, if a complex analytic function is dened in an open ball around a point x0, its power series expansion at x0 isconvergent in the whole ball (holomorphic functions are analytic). This statement for real analytic functions (withopen ball meaning an open interval of the real line rather than an open disk of the complex plane) is not true ingeneral; the function of the example above gives an example for x0 = 0 and a ball of radius exceeding 1, since thepower series 1 x2+ x4 x6... diverges for |x| > 1.Any real analytic function on some open set on the real line can be extended to a complex analytic function onsome open set of the complex plane. However, not every real analytic function dened on the whole real line canbe extended to a complex function dened on the whole complex plane. The function (x) dened in the paragraphabove is a counterexample, as it is not dened for x = i. This explains why the Taylor series of (x) diverges for |x| >1, i.e., the radius of convergence is 1 because the complexied function has a pole at distance 1 from the evaluationpoint 0 and no further poles within the open disc of radius 1 around the evaluation point.2.7. ANALYTIC FUNCTIONS OF SEVERAL VARIABLES 132.7 Analytic functions of several variablesOne can dene analytic functions in several variables by means of power series in those variables (see power series).Analytic functions of several variables have some of the same properties as analytic functions of one variable. How-ever, especially for complex analytic functions, new and interesting phenomena show up when working in 2 or moredimensions. For instance, zero sets of complex analytic functions in more than one variable are never discrete.2.8 See alsoCauchyRiemann equationsHolomorphic functionPaleyWiener theoremQuasi-analytic functionInnite compositions of analytic functions2.9 Notes[1] A function f of the complex variable z is analytic at point z0 if its derivative exists not only at z but at each point z insome neighborhood of z0. It is analytic in a region R if it is analytic as every point in R. The term holomorphic is also usedin the literature do denote analyticity. Churchill, Brown, and Verhey Complex Variables and Applications McGraw-Hill1948 ISBN 0-07-010855-2 pg 46[2] Komatsu 1960.[3] Krantz & Parks 2002.2.10 ReferencesConway, John B. (1978). Functions of One Complex Variable I. Graduate Texts in Mathematics 11. Springer-Verlag. ISBN 0-387-90328-3.Krantz, Steven; Parks, Harold R. (2002). A Primer of Real Analytic Functions (2nd ed.).Birkhuser.ISBN0-8176-4264-1.Komatsu, Hikosaburo (1960). A characterization of real analytic functions. Proc. Japan Acad. 36 (3):9093. doi:10.3792/pja/1195524081.2.11 External linksHazewinkel, Michiel, ed.(2001), Analytic function, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Weisstein, Eric W., Analytic Function, MathWorld.Analytic Functions Module by John H. MathewsSolver for all zeros of a complex analytic function that lie within a rectangular region by Ivan B. IvanovChapter 3Argument (complex analysis)Arg (mathematics)" redirects here. For argument of a function, see Argument of a function.In mathematics, arg is a function operating on complex numbers (visualized in a complex plane). It gives the anglebetween the positive real axis to the line joining the point to the origin, shown as in gure 1, known as an argumentof the point.3.1 DenitionAn argument of the complex number z = x + iy, denoted arg z, is dened in two equivalent ways:1. Geometrically, in the complex plane, as the angle from the positive real axis to the vector representing z.The numeric value is given by the angle in radians and is positive if measured counterclockwise.2. Algebraically, as any real quantity such thatz= r(cos + i sin )for some positive real r. The quantity r is the modulus of z, denoted |z|:r =x2+ y2.The names amplitude[1] for the modulus and phase[2] for the argument are sometimes used equivalently.Under both denitions, it can be seen that the argument of any (non-zero) complex number has many possible values:rstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles diering by aninteger multiple of 2 radians (a complete circle) are the same.Similarly, from the periodicity of sin and cos, thesecond denition also has this property.3.2 Principal valueBecause a complete rotation around 0 leaves a complex number unchanged, there are many choices which could bemade for by circling the origin any number of times. This is shown in gure 3, a representation of the multi-valued(set-valued) function, where a vertical line cuts the surface at heights representing all the possible choices of anglefor that point.When a well-dened function is required then the usual choice, known as the principal value, is the value in theopen-closed interval ( rad, rad], that is from to radians, excluding rad itself (equivalently from 180 to+180 degrees, excluding 180 itself). This represents an angle of up to half a complete circle from the positive realaxis in either direction.Some authors dene the range of the principal value as being in the closed-open interval [0, 2).143.3. COVERING SPACE 15ImRe0rxyFigure 1. This Argand diagram represents the complex numbers lying on a plane. For each point on the plane, arg is the functionwhich returns the angle .3.2.1 NotationThe principal value sometimes has the initial letter capitalized as in Arg z, especially when a general version of theargument is also being considered. Note that notation varies, so arg and Arg may be interchanged in dierent texts.The set of all possible values of the argument can be written in terms of Arg as:arg z= {Arg z + 2n | n Z}.3.3 Covering spaceIn informal situations, arg may be left not well-dened, for instance arg z(t) where z depends on a parameter t maychange by 2 every time z goes around the origin. This idea can be made more precise by considering z(t) as16 CHAPTER 3. ARGUMENT (COMPLEX ANALYSIS)ImRe0=+2Figure 3. Two choices for the argument being dened not on the complex plane but on a covering space. Polar coordinates excluding the origin and with anunconstrained angle provide such a space, in this case arg is dened byarg : R+{0} R R(r,) .The covering space is equivalent to the punctured complex plane:C {0}and has as base space the product of a positive non-zero radius and an angle on the unit circle:R+{0} S1.The principal value Arg then maps the covering space of this representation to the interval (, ]:3.4. COMPUTATION 170/2/21i1iThe limit of Arg 1from this side is /4 1 + i/4Figure 4. The principal value Arg of the blue point at 1 + i is /4. The red line here is the branch cut and corresponds to the twored lines in gure 2 seen vertically above each other).Arg : R+{0} R (,](r,) .3.4 ComputationThe principal value Arg of a complex number given as x + iy is normally available in math libraries of many program-ming languages using the function atan2 or some language specic variant. The value of atan2(y, x) is the principalvalue in the range (, ].Many texts say the value is given by arctan(y/x), as y/x is slope, and arctan converts slope to angle.This is correctonly when x > 0, so the quotient is dened and the angle lies between /2 and /2, but extending this denition tocases where x is not positive is relatively involved. Specically, one may dene the principal value of the argumentseparately on the two half-planes x > 0 and x < 0 (separated into two quadrants if one wishes a branch cut on thenegative x-axis), y > 0, y < 0, and then patch together.18 CHAPTER 3. ARGUMENT (COMPLEX ANALYSIS)Figure 2. The arguments of the complex plane are plotted vertically. arg measures the angle of points, shown by the fact that theoutward radial lines, which have constant angle to the real axis, lie on the surface. The layered structure shows that each point hasinnitely many arguments, each one corresponding an intersection between a vertical line through the point and the sheet. The redhatching indicates the surface corresponding to the principal value. See larger version.Arg(x + iy) = atan2(y,x) =___arctan(yx) ifx > 0,arctan(yx) + ifx < 0 and y 0,arctan(yx) ifx < 0 and y< 0,+2ifx = 0 and y> 0,2ifx = 0 and y< 0,undened ifx = 0 and y= 0.For the variant where Arg is dened to lie in the interval [0, 2), the value can be found by adding 2 to the valueabove when it is negative.Alternatively, the principal value can be calculated in a uniformway using the tangent half-angle formula, the functionbeing dened over the complex plane but excluding the origin:3.5. IDENTITIES 19Arg(x + iy) =___2 arctan_yx2+y2+x_ifx > 0 or y = 0, ifx < 0 and y= 0,undened ifx = 0 and y= 0.This is based on a parametrization of the circle (except for the negative x-axis) by rational functions. This version ofArg is not stable enough for oating point computational use (it may overow near the region x < 0, y = 0) but canbe used in symbolic calculation.A variant of the last formula which avoids overow is sometimes used in high precision computation:Arg(x + iy) =___2 arctan_x2+y2xy_ify = 0,0 ifx > 0 and y= 0, ifx < 0 and y= 0,undened ifx = 0 and y= 0.3.5 IdentitiesOne of the main motivations for dening the principal value Arg is to be able to write complex numbers in modulus-argument form. Hence for any complex number z,z= |z| ei Arg z.This is only really valid if z is non-zero but can be considered as valid also for z = 0 if Arg(0) is considered as beingan indeterminate form rather than as being undened.Some further identities follow. If z1 and z2 are two non-zero complex numbers, thenArg(z1z2) Arg(z1) + Arg(z2) (mod(, ]),Arg_z1z2_ Arg(z1) Arg(z2) (mod(, ]).If z 0 and n is any integer, thenArg (zn) nArg(z) (mod(, ]).3.5.1 ExampleArg_1 ii_= Arg(1 i) Arg(i) = 342= 54=34(mod(, ]).3.6 References3.6.1 Notes[1] Knopp, Konrad; Bagemihl, Frederick (1996). Theory of Functions Parts I and II. Dover Publications. p. 3. ISBN 0-486-69219-1.[2] Dictionary of Mathematics (2002). phase.20 CHAPTER 3. ARGUMENT (COMPLEX ANALYSIS)3.6.2 BibliographyAhlfors, Lars (1979). Complex Analysis: An Introduction to the Theory of Analytic Functions of One ComplexVariable (3rd ed.). New York;London: McGraw-Hill. ISBN 0-07-000657-1.Beardon, Alan (1979). Complex Analysis: The Argument Principle in Analysis and Topology. Chichester:Wiley. ISBN 0-471-99671-8.Borowski, Ephraim; Borwein, Jonathan (2002) [1st ed. 1989 as Dictionary of Mathematics]. Mathematics.Collins Dictionary (2nd ed.). Glasgow: HarperCollins. ISBN 0-00-710295-X.3.7 External linksWeisstein, Eric W., Complex Argument, MathWorld.Chapter 4Complex analysisComplex analytic redirects here. For the class of functions often called complex analytic, see Holomorphic func-tion.Complexanalysis, traditionally known as thetheoryoffunctionsofacomplexvariable, is the branch ofmathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics,including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamicsand thermodynamics and also in engineering elds such as nuclear, aerospace, mechanical and electrical engineering.Murray R. Spiegel described complex analysis as one of the most beautiful as well as useful branches of Mathemat-ics.Complex analysis is particularly concerned with analytic functions of complex variables (or, more generally, meromorphicfunctions).Because the separate real and imaginary parts of any analytic function must satisfy Laplaces equation,complex analysis is widely applicable to two-dimensional problems in physics.4.1 HistoryComplex analysis is one of the classical branches in mathematics with roots in the 19th century and just prior. Im-portant mathematicians associated with complex analysis include Euler, Gauss, Riemann, Cauchy, Weierstrass, andmany more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many phys-ical applications and is also used throughout analytic number theory. In modern times, it has become very popularthrough a newboost fromcomplex dynamics and the pictures of fractals produced by iterating holomorphic functions.Another important application of complex analysis is in string theory which studies conformal invariants in quantumeld theory.4.2 Complex functionsA complex function is one in which the independent variable and the dependent variable are both complex numbers.More precisely, a complex function is a function whose domain and range are subsets of the complex plane.For any complex function, both the independent variable and the dependent variable may be separated into real andimaginary parts:z= x + iy andw = f(z) = u(x, y) + iv(x, y)where x, y R and u(x, y), v(x, y) are real-valued functions.In other words, the components of the function f(z),u = u(x, y)2122 CHAPTER 4. COMPLEX ANALYSISPlot of the function f(x) = (x2 1)(x 2 i)2/ (x2+ 2 + 2i). The hue represents the function argument, while the brightnessrepresents the magnitude.v= v(x, y),can be interpreted as real-valued functions of the two real variables, x and y.The basic concepts of complex analysis are often introduced by extending the elementary real functions (e.g., exponentialfunctions, logarithmic functions, and trigonometric functions) into the complex domain.4.3 Holomorphic functionsMain article: Holomorphic functionHolomorphic functions are complex functions dened on an open subset of the complex plane that are dierentiable.Complex dierentiability has much stronger consequences than usual (real) dierentiability. For instance, holo-morphic functions are innitely dierentiable, whereas some real dierentiable functions are not. Most elementaryfunctions, including the exponential function, the trigonometric functions, and all polynomial functions, are holomor-phic.See also: analytic function, holomorphic sheaf and vector bundles.4.4. MAJOR RESULTS 23The Mandelbrot set, a fractal.4.4 Major resultsOne central tool in complex analysis is the line integral. The integral around a closed path of a function that is holo-morphic everywhere inside the area bounded by the closed path is always zero; this is the Cauchy integral theorem.The values of a holomorphic function inside a disk can be computed by a certain path integral on the disks bound-ary (Cauchys integral formula). Path integrals in the complex plane are often used to determine complicated realintegrals, and here the theory of residues among others is useful (see methods of contour integration). If a functionhas a pole or isolated singularity at some point, that is, at that point where its values blow up and have no nitebound, then one can compute the functions residue at that pole. These residues can be used to compute path integralsinvolving the function; this is the content of the powerful residue theorem. The remarkable behavior of holomorphicfunctions near essential singularities is described by Picards Theorem. Functions that have only poles but no essentialsingularities are called meromorphic. Laurent series are similar to Taylor series but can be used to study the behaviorof functions near singularities.A bounded function that is holomorphic in the entire complex plane must be constant; this is Liouvilles theorem. Itcan be used to provide a natural and short proof for the fundamental theorem of algebra which states that the eld ofcomplex numbers is algebraically closed.If a function is holomorphic throughout a connected domain then its values are fully determined by its values onany smaller subdomain. The function on the larger domain is said to be analytically continued from its values on thesmaller domain. This allows the extension of the denition of functions, such as the Riemann zeta function, which areinitially dened in terms of innite sums that converge only on limited domains to almost the entire complex plane.Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic functionto a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on aclosely related surface known as a Riemann surface.All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more thanone complex dimension in which the analytic properties such as power series expansion carry over whereas most ofthe geometric properties of holomorphic functions in one complex dimension (such as conformality) do not carry24 CHAPTER 4. COMPLEX ANALYSISover. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane,which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions.4.5 See alsoComplex dynamicsList of complex analysis topicsReal analysisRunges theoremSeveral complex variablesReal-valued functionFunction of a real variableReal multivariable function4.6 ReferencesAhlfors, L., Complex Analysis, 3 ed. (McGraw-Hill, 1979).Stephen D. Fisher, Complex Variables, 2 ed. (Dover, 1999).Carathodory, C., Theory of Functions of a Complex Variable (Chelsea, New York). [2 volumes.]Henrici, P., Applied and Computational Complex Analysis (Wiley). [Three volumes: 1974, 1977, 1986.]Kreyszig, E., Advanced Engineering Mathematics, 10 ed., Ch.13-18 (Wiley, 2011).Markushevich, A.I.,Theory of Functions of a Complex Variable (Prentice-Hall, 1965). [Three volumes.]Marsden & Homan, Basic Complex Analysis. 3 ed. (Freeman, 1999).Needham, T., Visual Complex Analysis (Oxford, 1997).Rudin, W., Real and Complex Analysis, 3 ed. (McGraw-Hill, 1986).Scheidemann, V., Introduction to complex analysis in several variables (Birkhauser, 2005)Shaw, W.T., Complex Analysis with Mathematica (Cambridge, 2006).Spiegel, Murray R. Theory and Problems of Complex Variables - with an introduction to Conformal Mappingand its applications (McGraw-Hill, 1964).Stein & Shakarchi, Complex Analysis (Princeton, 2003).4.7 External linksComplex Analysis -- textbook by George CainComplex analysis course web site by Douglas N. ArnoldExample problems in complex analysisA collection of links to programs for visualizing complex functions (and related)Complex Analysis Project by John H. MathewsHans Lundmarks complex analysis page (many links)4.7. EXTERNAL LINKS 25Wolfram Researchs MathWorld Complex Analysis PageComplex function demosApplication of Complex Functions in 2D Digital Image TransformationComplex Visualizer - Java applet for visualizing arbitrary complex functionsComplex Map - iOS app for visualizing complex functions and iterationsJavaScript complex function graphing toolEarliest Known Uses of Some of the Words of Mathematics: Calculus & AnalysisChapter 5Complex conjugateIn mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary partequal in magnitude but opposite in sign.[1][2] For example, the complex conjugate of 3 + 4i is 3 4i.In polar form, the conjugate of eiis ei. This can be shown using Eulers formula.Complex conjugates are important for nding roots of polynomials. According to the complex conjugate root theorem,if a complex number is a root to a polynomial in one variable with real coecients (such as the quadratic equation orthe cubic equation), so is its conjugate.5.1 NotationThe complex conjugate of a complex number z is written as z or z. The rst notation avoids confusion with thenotation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate.The second is preferred in physics, where dagger is used for the conjugate transpose, while the bar-notation is morecommon in pure mathematics. If a complex number is represented as a 22 matrix, the notations are identical.In some texts, the complex conjugate of a previous known number is abbreviated as c.c.. For example, writingei+ c.c. means ei+ ei5.2 PropertiesThe following properties apply for all complex numbers z and w, unless stated otherwise, and can be proven by writingz and w in the form a + ib.A signicant property of the complex conjugate is that a complex number is equal to its complex conjugate if itsimaginary part is zero, that is, if the complex number is real. For any two complex numbers w,z:z + w = z + wz w = z wzw = zwz/w = z/w, ifz/wdened isz= zz Rzn= (z)n, n Z|z| = |z||z|2= zz= zzz= zz1=z|z|2, z = 0265.2. PROPERTIES 27ImReyy0xrrz=x+iyz=xiyGeometric representation of z and its conjugate z in the complex plane. The complex conjugate is found by reecting z across thereal axis.The penultimate relation is involution; i.e., the conjugate of the conjugate of a complex number z is z. The ultimaterelation is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.28 CHAPTER 5. COMPLEX CONJUGATEexp(z) = exp(z)log(z) = log(z) if z is non-zeroIfp is a polynomial with real coecients, andp(z) =0 , thenp(z) =0 as well. Thus, non-real roots of realpolynomials occur in complex conjugate pairs (see Complex conjugate root theorem).In general, if is a holomorphic function whose restriction to the real numbers is real-valued, and (z) is dened,then(z) = (z).The map (z) = z fromC to Cis a homeomorphism (where the topology on Cis taken to be the standard topology)and antilinear, if one considers C as a complex vector space over itself. Even though it appears to be a well-behavedfunction, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation.It is bijective and compatible with the arithmetical operations, and hence is a eld automorphism. As it keeps thereal numbers xed, it is an element of the Galois group of the eld extension C/R . This Galois group has only twoelements: and the identity on C . Thus the only two eld automorphisms of C that leave the real numbers xed arethe identity map and complex conjugation.5.3 Use as a variableOnce a complex number z=x + iy or z=eiis given, its conjugate is sucient to reproduce the parts of thez-variable:Real part:x = Re (z) =z + z2Imaginary part:y= Im(z) =z z2iModulus (or absolute value): = |z| =zzArgument:ei= ei arg z=zz , so = arg z=1i lnzz= ln z ln z2iThus the pair of variables z and z also serve up the plane as do x,y and and . Furthermore, the z variable is usefulin specifying lines in the plane:{z | zr + zr = 0}is a line through the origin and perpendicular to r since the real part of z r is zero only when the cosine of the anglebetween z and r is zero. Similarly, for a xed complex unit u = exp(b i), the equation:z z0z z0= udetermines the line through z0in the direction of u.These uses of the conjugate of z as a variable are illustrated in Frank Morley's book Inversive Geometry (1933),written with his son Frank Vigor Morley.5.4. GENERALIZATIONS 295.4 GeneralizationsThe other planar real algebras, dual numbers, and split-complex numbers are also explicated by use of complexconjugation.For matrices of complex numbers AB = (A)(B) , where A represents the element-by-element conjugation of A .[3]Contrast this to the property (AB)= BA , where A represents the conjugate transpose of A .Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more generalis the concept of adjoint operator for operators on (possibly innite-dimensional) complex Hilbert spaces. All this issubsumed by the *-operations of C*-algebras.One may also dene a conjugation for quaternions and coquaternions: the conjugate of a+bi+cj+dk is abicjdk.Note that all these generalizations are multiplicative only if the factors are reversed:(zw)= wz.Since the multiplication of planar real algebras is commutative, this reversal is not needed there.There is also an abstract notion of conjugation for vector spaces Vover the complex numbers. In this context, anyantilinear map : V Vthat satises1. 2= idV, where 2= and idVis the identity map on V,2. (zv) = z(v) for all v V, z C , and3. (v1 + v2) = (v1) + (v2) for all v1 V, v2 V,is called a complex conjugation, or a real structure. As the involution is antilinear, it cannot be the identity map onV . Of course, is a R-linear transformation of V , if one notes that every complex space V has a real form obtainedby taking the same vectors as in the original space and restricting the scalars to be real. The above properties actuallydene a real structure on the complex vector spaceV.[4] One example of this notion is the conjugate transposeoperation of complex matrices dened above. It should be remarked that on generic complex vector spaces there isno canonical notion of complex conjugation.5.5 See alsoComplex conjugate vector spaceReal structure5.6 Notes[1] Weisstein, Eric W., Complex Conjugates, MathWorld.[2] Weisstein, Eric W., Imaginary Numbers, MathWorld.[3] Arfken, Mathematical Methods for Physicists, 1985, pg. 201[4] Budinich, P. and Trautman, A. The Spinorial Chessboard. Spinger-Verlag, 1988, p. 295.7 ReferencesBudinich, P. and Trautman, A. The Spinorial Chessboard. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (an-tilinear maps are discussed in section 3.3).Chapter 6Complex 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= 1.306.1. OVERVIEW 31A 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]6.1 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.6.1.1 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 .32 CHAPTER 6. COMPLEX NUMBERImReyy0xrrz=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.6.1.2 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 numbers6.1. OVERVIEW 33are written as a + bj or a + jb.6.1.3 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.6.1.4 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.34 CHAPTER 6. COMPLEX NUMBERWork 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.6.2 Relations6.2.1 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)).6.2.2 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= 1.6.3 Elementary operations6.3.1 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,6.3. ELEMENTARY OPERATIONS 35ImReyy0xrrz=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 by36 CHAPTER 6. COMPLEX NUMBER1z= 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.6.3.2 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 by6.3. ELEMENTARY OPERATIONS 37(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.6.3.3 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).6.3.4 Square rootSee also: Square roots of negative and complex numbersThe square roots of a + bi (with b 0) are ( + i) , where=a +a2+ b2238 CHAPTER 6. COMPLEX NUMBERand= 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]6.4 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.6.4. POLAR FORM 396.4.1 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.40 CHAPTER 6. COMPLEX NUMBER0123451 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.6.4.2 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 formula6.5. EXPONENTIATION 414= 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)) .6.5 Exponentiation6.5.1 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.6.5.2 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}42 CHAPTER 6. COMPLEX NUMBER6.5.3 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.6.6 Properties6.6.1 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.6.6. PROPERTIES 436.6.2 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.6.6.3 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.6.6.4 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.44 CHAPTER 6. COMPLEX NUMBER6.7 Formal construction6.7.1 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.6.7.2 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 valu