Introductions to Re Introduction to the Complex Components

Introductions to Re Introduction to the complex components

General

The study of complex numbers and their characteristics has a long history. It all started with questions about how to understand and interpret the solution of the simple quadratic equation z2 -1.

It was clear that 12 -1 2 1. But it was not clear how to get Ð1 from something squared.

This problem was intensively discussed in the 16th, 17th, and 18th centuries. As a result, mathematicians proposed a special symbol—theH imaginaryL unit ä, which is represented by ä = -1 :

ä2 -1.

L. Euler (1755) introduced the word "complex" (1777) and first used the letter i for denoting -1 . Later, C. F. Gauss (1831) introduced the name "imaginary unit" for i.

Accordingly, ä2 -1 and -ä 2 -1 and the above quadratic equation has two solutions as is expected for a quadratic polynomial:

2 z -1 ; z z1 ä z z2H L-ä.

The imaginary unit ä was interpreted in a geometrical sense as the point with coordinates 0, 1 in the Cartesian (Euclidean) x,y-planeì with the vertical y-axis upward and the origin 0, 0 . This geometric interpretation established the following representations of the complex number z through two real numbers x and y as: 8 < z x + ä y ; x Î R y Î R x, y 8 <

z r cos j + ä r sin j ; r Î R r > 0 j Î R, ì H L where r = x2 + y2 is the distance between points x, y and 0, 0 , and j is the angle between the line connecting H L H L ß ì the points 0, 0 and x, y and the positive x-axis direction (the so-called polar representation).

The last formula lead to the basic relations: 8 < 8 < 8 < 8 < r x2 + y2

x r cos j

y r sin j H L y j tan-1 ; x > 0, H xL

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which describe the main characteristics of the complex number z x + ä y—the so-called modulus (absolute value) r, the real part x, the imaginary part y, and the argument j.

A new era in the theory of complex numbers and functions of complex arguments (analytic functions) arose from the investigations of L. Euler (1727, 1728). In a letter to Goldbach (1731) L. Euler introduced the notation ã for the base of the natural logarithm ã2.71828182…, and he proved that ã is irrational. Later on L. Euler (1740Ð1748) found a series expansion for ãz, which lead to the famous very basic formula, connecting exponential and trigono- metric functions:

ãä j cos j + ä sin j .

This is known as the Euler formula (although it was already derived by R. Cotes in 1714). H L H L The Euler formula allows presentation of the complex number z, using polar coordinates r, j in the more com- pact form:

z r ãä j ; r Î R r ³ 0 j Î 0, 2 Π . H L

It also expressed the logarithm of complex numbers through the formula: ß ì @ L log z log r + ä j ; r Î R r > 0 j Î R.

Taking into account that the cosine and sine have period 2 Π, it follows that ãä j has period 2 Π ä: H L H L ß ì ãä j+2 Π ä ãä j+2 Π cos j + 2 Π + ä sin j + 2 Π cos j + ä sin j ãä j.

Generically,H the Llogarithm function log z is the multivalued function: H L H L H L H L log z log r + ä j + 2 Π k ; r Î R r > 0 j Î R k Î Z. H L For specifying just one value for the logarithm log z and one value of the argument j for a given complex number z, theH L restrictionH L -H Π < j ² L Π for theß argumentì j isì generally used.

During the 18th and 19th centuries many mathematiciansH L worked on building the theory of the functions of complex variables, which was called the theory of analytic functions. Today this is a widely used theory, not only for the above-mentioned four complex components (absolute value, argument, real and imaginary parts), but for complimentary characteristics of a complex number z x + ä y r ãä j such as the conjugate complex number z x - ä y and the signum (sign) z r. J. R. Argand (1806, 1814) introduced the word "module" for the absolute value, and A. L. Cauchy (1821) was the first to use the word "conjugate" for complex numbers in the modern sense. Later K. Weierstrass (1841) introduced the notation z for the absolute value. It was shown that the set of complex numbers and the set of real numbers have basic properties in common—they both are fields because they satisfy so-called field axioms. ÈComplexÈ and real numbers exhibit commutativity under addition and multiplication described by the formulas:

a + b b + a

a b b a.

Complex and real numbers also have associativity under addition and multiplication described by the formulas: http://functions.wolfram.com 3

a + b + c a + b + c

a b c a b c , H L H L and distributivity described by the formulas: H L H L a b + c a b + a c

a + b c a c + b c. H L (The set of rational numbers p q ; p Î Z q Î Z also satisfies all of the previous field axioms and is also a field. ThisH setL is countable, which means that each rational number can be numerated and placed in a definite position with a corresponding integer number n ; n 1, 2, 3, ¼ . But the set of rational numbers does not include so- ì called irrational numbers like 2 or Π. The set of irrational numbers is much larger and cannot be numerated. The sets of all real and complex numbers form uncountable sets.) The great success and achievements of the complex number theory stimulated attempts to introduce not only the imaginary unit ä -1 in the Cartesian (Euclidean) plane x, y , but a similar special third unit ü in Cartesian (Euclidean) three-dimensional space x, y, z , which can be used for building a similar theory of (hyper)complex numbers w x + ä y + ü z: H L w x + ä y + ü z ; x Î R y Î R z Î R8 x,< y, z .

Unfortunately, such an attempt fails to fulfill the field axioms. Further generalizations to build the so-called quaternions and octonionsì are neededì to obtain8 mathematically< interesting and rich objects.

Definitions of complex components

The complex components include six basic characteristics describing complex numbers-absolute value (modulus) z , argument (phase) Arg z , real part Re z , imaginary part Im z , complex conjugate z, and sign function (signum) sgn z . It is impossible to define real and imaginary parts of the complex number z through other functions or complex characteristics. They are too basic, so their symbols can be described by simple sentences, for example, ¤ H L H L H L "Re z gives the real part of the number z," and "Im z gives the imaginary part of the number z." H L All other complex components are defined by the following formulas: H L H L x x ; x Î R x ³ 0

x -x ; x Î R x < 0 ¤ ß z Re z 2 + Im z 2 ¤ ß z Arg z -ä log ¤ H L z H L z Re z - ä Im z H L ¤ sgn x 1 ; x Î R x > 0 H L H L sgn x -1 ; x Î R x < 0 H L ß

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sgn 0 0

z sgn z ; z ¹ 0. H L z

Geometrically, the absolute value (or modulus) of a complex number z x + ä y ; x Î R y Î R is the Euclidean H L distance from¤ z to the origin, which can also be described by the formula:

ì x + ä y x2 + y2 .

Geometrically, the argument of a complex number z is the phase angle (in radians) that the line from 0 to z makes with the¤ positive real axis. So, the complex number z x + ä y ; x Î R y Î R can be presented by the formulas:

z z ãä Arg z ì x + ä y HxL2 + y2 ãä Arg x+ä y ¤

-H1 L x + ä y x2 + y2 ãä tan x,y .

Geometrically, the real partH L of a complex number z is the projection of the complex point z on the real axis. So, the real part of the complex number z x + ä y ; x Î R y Î R can be presented by the formulas:

Re z z cos Arg z ì Re x + ä y x2 + y2 cos tan-1 x, y . H L ¤ H H LL Geometrically, the imaginary part of a complex number z is the projection of complex point z on the imaginary axis.H So, Lthe imaginary partI of Hthe complexLM number z x + ä y ; x Î R y Î R can be presented by the formulas:

Im z z sin Arg z ì Im x + ä y x2 + y2 sin tan-1 x, y . H L ¤ H H LL Geometrically, the complex conjugate of a complex number z is the complex point z, which is symmetrical to z withH respectL to the real axis.I So,H theLM conjugate value z of the complex number z x + ä y ; x Î R y Î R can be presented by the formulas:

z z ã-ä Arg z ì

x - ä y x -HäLy. ¤ Geometrically, the sign function (signum) is the complex point sgn z that lays on the intersection of the unit circle z 1 and the line from 0 to z (if z ¹ 0). So, the conjugate value sgn z of the complex number z x + ä y ; x Î R y Î R can be presented by the formulas: H L ¤ z H L sgn z ; z ¹ 0 ì Re z 2 + Im z 2

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x + ä y sgn x + ä y ; x, y ¹ 0, 0 . x2 + y2

H L 8 < 8 < A quick look at the complex components

Here is a quick look at the graphics for the complex components of the complex components over the complex z- plane. The empty graphic indicates that the function value is not real.

Re Re z Im Re z Abs Re z Arg Re z Conjugate Re z Sign Re z

2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 0 0 0 0 0 0 -2 2 -2 2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 0 -2 0 -2 0 -2 0 -2 0 -2 0 -2 2 2 2 2 2 2

Re Im z Im Im z Abs Im z Arg Im z Conjugate Im z Sign Im z

2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 0 0 0 0 0 0 -2 2 -2 2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 0 -2 0 -2 0 -2 0 -2 0 -2 0 -2 2 2 2 2 2 2

Re Abs z Im Abs z Abs Abs z Arg Abs z Conjugate Abs z Sign Abs z

2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 0 0 0 0 0 0 -2 2 -2 2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 0 -2 0 -2 0 -2 0 -2 0 -2 0 -2 2 2 2 2 2 2

Re Arg z Im Arg z Abs Arg z Arg Arg z Conjugate Arg z Sign Arg z

2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 0 0 0 0 0 0 -2 2 -2 2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 0 -2 0 -2 0 -2 0 -2 0 -2 0 -2 2 2 2 2 2 2

Re Conjugate z Im Conjugate z Abs Conjugate z Arg Conjugate z Conjugate Conjugate z Sign Conjugate z

2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 0 0 0 0 0 1 0 1 -2 2 -2 2 -2 2 -2 2 -2 0.75 -2 0.75 0.5 0.5 0 0 0 0 0.250 0 -2 -2 -2 -2 0.5 0.25 0.250.5 0.25 0 -2 0 -2 0 -2 0 -2 0.75 0 0.75 0 2 2 2 2 1 1

Re Sign z Im Sign z Abs Sign z Arg Sign z Conjugate Sign z Sign Sign z

2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 2 @ @ DD 0 0 0 0 0 1 0 1 -2 2 -2 2 -2 2 -2 2 -2 0.75 -2 0.75 0 0 0 0 0 0.5 0 0.5 -2 -2 -2 -2 0.250.5 0.25 0.250.5 0.25 0 -2 0 -2 0 -2 0 -2 0.75 0 0.75 0 2 2 2 2 1 1

Connections within the group of complex components and with other function groups

Representations through more general functions All six complex component functions z ® z , z ® Arg z , z ® Re z , z ® Im z , z ® z, and sz ® gn z cannot be easily represented by more generalized functions because most of them are analytic functions of their arguments. But sometimes such representations can be found through Meijer G functions, for example: ¤ H L H L H L H L a+1 Π a Π a + 1, a 1,1 2 x sec G2,2 1 - x ; x < 1. G -a 2 0, a+1 2

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Representations through other functions All six complex components z , Arg z , Re z , Im z , z, and sgn z satisfy numerous internal relations of the type

f z g f1 z , z , where f z and f1 z are different complex components and g is a basic arithmetic operation or (composition of) elementary functions. The most important of these relations are represented in the following table: ¤ H L H L H L H L zH L H z H L L H L Arg z H L Re z Im z z z z z ã-ä Arg z z 2 z Re z - z2 z z2 - 2 ä z Im z z z z

¤ z H L H L -1 H L -1 1 Arg z Arg z -ä log H L Arg z tan Re z , Arg z tan z - Arg z ä log z 2 ¤ ¤ ¤ -ä z - ReHzL ¤ ä Im z , Im z H L ¤ - 2 log z 2 2 ¤ Re zH L Re z z + z Re z z ã-2 ä Arg z 1 + Re z z - ä Im z Re z z+z H L 2 z J N 2 H L H H L H L H H L 2 H H ã2 ä Arg z H H LLL H L H LL H LL ¤ H L ä z 2-z2 Im HzL Im HzL Im HzL z ä ã-2 ä Arg zI 1 - Im z ä Re z - z H L H L Im HzL z-z 2 z 2 H L 2 ä ã2 ä Arg z M I ¤ M H L z 2 z H L z H L z HãL-2 ä Arg z z I z H2LRe zH - zH L L z z - 2 ä Im z H L z H L z ä Arg zM z z z sgn z sgn z¤ ; z ¹ 0 sgn z ã H L sgn z sgn z sgn z ; z z 2 z Re z -z2 z2-2 ä z Im z z z H L H L H L ¤ OtherH LinternalH Lrelations between complexH L components of theH L type f z H L g f1 z , Hf2L z , z , whereH L f z andH L fj z are different complex components that also exist. Some of them are shown here:

z z Arg z Re z H L ImH z H L H L L zH L H L z z cos Arg z - z Re z z Im z cos Arg z sin Arg z ä sin Arg z ¤ H L H L H L H L H L Re z Im z 1 Arg z Arg z ä log z - cos ArgH z H LL sin ArgH z H LL Arg z ä log z z ¤ H H H LL ¤ z ¤ z 2 log z 2 log z H H LLL H L H L ¤ ¤ Re zH L H L H H ¤L Re z z cos Arg z H H LL Re Hz HImLL ä z Re zH L z + ä ImH zH H LL H LL Im z Im z z sin Arg z Im z -Re ä z Im z ä z - Re z H L H L ¤ H H LL H L H L H L H L z 2 z z z z ã-ä Arg z z Re z - ä Im z z H L H L ¤ H H LL H L H L z H L H H LL sgn z ¤ H L sgn z ; Im z 2+Re z 2 ¤ H L H L z ¹ 0 H L H L H L H L Here are some more formulas of the last type:

z Re z 2 + Im z 2

Arg z tan-1 Re z , Im z ¤ H L H L Im z Arg z tan-1 ; Re z > 0 H L H ReH zL H LL z z cos Arg z -H äL z sin Arg z H L H L H L

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Re z + ä Im z sgn z Re z 2 + Im z 2 H L H L sgnHxL Θ x - Θ -x ; x Î R H L H L sgn x 2 Θ x - 1 ; x Î R x ¹ 0, H L H L H L (here Θ x is the Heaviside theta function, also called the unit step function). H L H L ß The first table can be rewritten using the notations z x + ä y ; x Î R y Î R: H L x + ä y ; x + ä y Arg x + ä y Re x + ä y Im x + ä y x Î R y Î R ì

x + ä y ¤ x +Hä y L x +H ä y L x +H ä y L x + ä y ã-ä Arg x+ä y ì 2 x + ä y x - x + ä y 2 x + ä y 2 - 2 ä x

¤ x+ä y ¤ H L ¤ ¤ Arg x + ä y Arg x + ä y -ä log Arg x + ä y tan-1 x, Arg x + ä y tan-1 x2+y2 H L -äH x + ä yL- x H L H ä y, y L H

x+ä y 2+x2+y2 x+ä y Re xH + ä y L Re xH + ä y L Re x + ä y H L H Re xH + ä y L x + ä y 2 x+ä y 2 -2 ä Arg x+ä y H LL L H L ã 1 + H L ã2 ä Arg x+ä y H L H L H H L L H L H ä x2+y2- x+ä y 2 x+ä y Im x + ä y Im x + ä y Im x + ä y H I L ä Im x + ä y ä x - x + ä y 2 x+ä y 2 -2 ä Arg x+ä y M I H L M ã 1 - H L ã2 ä Arg x+ä y H L H L H H L L H L H H LL x2+y2 x + ä y x + ä y x + ä y ã-2 ä Arg x+ä y x + ä y 2 x - x + ä y x + ä y x + ä y - 2 x+ä y HI L x + ä y M x+ä y H L sgn x + ä y sgn x + ä y ; sgn x + ä y ãä Arg x+ä y sgn x + ä y sgn x + ä y 2 2 H L x +y x+ä y x+ä y H L H L x, y ¹ 0, 0 2 x+ä y x- x2+y2 x+ä y 2-2 ä x+ä y y H L H L H L H L H L

H L I M H L H L The best-known properties8 < 8 < and formulas for complex components

Real values for real arguments For real values of argument z, the values of all six complex components z , Arg z , Re z , Im z , z, and sgn z are real.

Simple values at zero ¤ H L H L H L H L The six complex components z , Arg z , Re z , Im z , z, and sgn z have the following values for the argument z 0:

0 0 ¤ H L H L H L H L

Arg 0 -Π, Π ¤ Re 0 0 H L H D

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Im 0 0

0 0 H L sgn 0 0.

Arg 0 is not a uniquely defined number. Depending on the argument of z, the limit lim z ®0 Arg z can take any valueH L in the interval -Π, Π . È È SpecificH L values for specialized variable H L H D The six complex components z , Arg z , Re z , Im z , z, and sgn z have the following values for some concrete numeric arguments:

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z z Arg z Re z Im z z sgn z 0 0 Î -Π, Π 0 0 0 0 1 1 0 1 0 1 1 ¤ H L H L H L H L -1 1 Π -1 0 -1 -1 Π H D ä 1 0 1 -ä ä 2 -ä 1 - Π 0 -1 ä -ä 2 1 + ä 2 Π 1 1 1 - ä 1+ä 4 2 -1 + ä 2 3 Π -1 1 -1 - ä -1+ä 4 2 -1 - ä 2 - 3 Π -1 -1 -1 + ä - 1+ä 4 2 1 - ä 2 - Π 1 -1 1 + ä 1-ä 4 2

ä Π Π 3 + ä 2 3 1 3 - ä ã 6 6 ä Π Π 1 + ä 3 2 1 3 1 - ä 3 ã 3 3 2 ä Π 2 Π -1 + ä 3 2 -1 3 -1 - ä 3 ã 3 3 5 ä Π 5 Π - 3 + ä 2 - 3 1 - 3 - ä ã 6 6 5 ä Π 5 Π - - 3 - ä 2 - - 3 -1 - 3 + ä ã 6 6 2 ä Π 2 Π - -1 - ä 3 2 - -1 - 3 -1 + ä 3 ã 3 3 ä Π Π - 1 - ä 3 2 - 1 - 3 1 + ä 3 ã 3 3 ä Π Π - 3 - ä 2 - 3 -1 3 + ä ã 6 6 2 2 0 2 0 2 1 -2 2 Π -2 0 -2 -1 Π Π 0 Π 0 Π 1 3 ä 3 Π 0 3 -3 ä ä 2 -2 ä 2 - Π 0 -2 2 ä -ä 2 2 + ä 5 ArcTan 1 2 1 2 - ä 2+ä 2 5

Restricted arguments haveB theF following formulas for the six complex components z , Arg z , Re z , Im z , z, and sgn z :

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z z Arg z Re z Im z z sgn z x ; x Î R x > 0 x 0 x 0 x 1 x ; x Î R x < 0 -x Π x 0 x -1 ¤ H L H L H L H L - x x ; x Î R ì x2 tan 1 x, 0 x 0 x ; x2 ì x ¹ 0 H L ä y ; y Î R y2 tan-1 0, y 0 y -ä y ä ; y > 0 0 ; y 0 -ä ; y < 0 H L 2 2 -1 x+ä y x + ä y ; x + y tan x, y x y x - ä y ; x2+y2 x Î R y Î R x, y ¹ 0, 0 Π-j r ãä j ; r Î R r j + 2 HΠ L r cos j r sin j r ã-ä j ãä j ì 2 Π r > 0 j Î R 8 < 8 < The values of ìcomplex components z , Argf zv, Re z H, ImL z , Hz, Land sgn z at any infinity can be described through ì the following:

z z Arg z Re z Im z z ¤ sgnH L z H L H L H L ¥ ¥ 0 ¥ 0 ¥ 1 -¥ ¥ Π -¥ 0 -¥ -1 ¤ H L H L H L H L ä ¥ ¥ Π 0 ¥ -ä ¥ ä 2 -ä ¥ ¥ - Π 0 -¥ ä ¥ -ä 2 ¥ ¥ Î -Π, Π È È ¥ È È È È È È È È

Analyticity H D All six complex components z , Arg z , Re z , Im z , z, and sgn z are not analytical functions. None of them fulfills the CauchyÐRiemann conditions and as such the value of the derivative depends on the direction. The functions z , Arg z , Re z , and Im z are real-analytic functions of the variable z (except, maybe, z ¹ 0). The real ¤ H L H L H L H L and the imaginary parts of z and sgn z are real-analytic functions of the variable z.

Sets of discontinuity ¤ H L H L H L H L The four complex components z , Re z , Im z , and z are continuous functions in C. The function sgn z has discontinuity at point z 0.

The function Arg z is a single- valued,¤ H LcontinuousH L function on the z-plane cut along the interval -¥, 0 , where it is continuous fromH L above. Its behavior can be described by the following formulas:

lim Arg x + ä Ε HArgL x Π ; x < 0 H L Ε®+0

lim Arg x - ä Ε -Π ; x < 0. Ε®+0 H L H L

Periodicity H L http://functions.wolfram.com 11

All six complex components z , Arg z , Re z , Im z , z, and sgn z do not have any periodicity.

Parity and symmetry ¤ H L H L H L H L All six complex components z , Arg z , Re z , Im z , z, and sgn z have mirror symmetry:

z z Arg z -Arg z ; z Ï -¥, 0 ¤ H L H L H L H L Re z Re z Im z -Im z ¤ ¤ H L H L H L z z sgn z sgn z . H L H L H L H L The absolute value z is an even function. The four complex components Re z , Im z , z, and sgn z are odd functions. The HargumentL H LArg z is an odd function for almost all z:

-z z ¤ H L H L H L H L Arg -z -Arg z ; z Ï -¥, 0 ¤ ¤ -z ArgH-zL Arg zH -L H ä Π L z

Re -z -Re z Im -z -Im z H L H L -z -z sgn -z -sgn z . H L H L H L H L Homogeneity H L H L The six complex components z , Arg z , Re z , Im z , z, and sgn z have the following homogeneity properties:

Λ z Λ z ¤ H L H L H L H L Arg a z Arg z ; a Î R a > 0 ¤ ¤ ¤ Re a z a Re z ; a Î R H L H L ß Im a z a Im z ; a Î R H L H L a z a z H L H L sgn a z sgn a sgn z .

Scale symmetry H L H L H L Some complex components have scale symmetry:

za z a ; a Î R

sgn za sgn z a ; a Î R. ¤ ¤ Series representations H L H L The functions x and sgn x with real x have the following series expansions near point x 0:

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4 ¥ -1 k-1 2 x T2 k x + ; x Î R -1 < x < 1 2 Π k=1 4 k - 1 Π

H k L ¤ ¥ â-1 H1 L 1 ß x 2 k + - P2 k x ; x Î R -1 < x < 1 k=0 k + 1 ! 2 2 k

H L k ¤ â1 ¥ -1 1 H L ß x H L - H2 k x ; x Î R -1 < x < 1 Π k=0 2 k ! 2 k

H L k ¤ â1 ¥ -1 H L ß sgn x H L H2 k+1 x ; x Î R -1 < x < 1 2 k Π k=0 2 2 k + 1 k!

k-1H L H L 4 ¥ â-1 H L ß sgn x H T2 k-1 Lx ; x Î R -1 < x < 1 Π k=1 2 k - 1

H k L H L ¥ â-1 4 k + 3 2Hk L! ß sgn x P2 k+1 x ; x Î R -1 < x < 1. 2 k+1 k=0 2 k + 1 ! k! H L H L H L IntegralH L â representations H L ß H L The function sgn x with real x has the following contour integral representation:

1 Γ+ä ¥ G -s x + 1 -s sgn x âs ; 0 < Γ x > -2. Π ä Γ-ä ¥H L G 1 - s

Limit representationsH L H L H L à ì H L The functions x and sgn x with real x have the following limit representations:

n-1 k pn x - pn -x - x lim x ; n Î N -1 < x < 1 pn x x + ã n . n®¥ ¤ H L pn x + pn -x k=0 H L H L ¤ 4 n! G m + 3 G mí+ n + 2 íF -mH ,L 1 - n, m + n + 2; 3 , 3 ; x2 2 3 2 2 ä 2 2 sgn x lim H L H L x ; -1 < x < 1 n Î N m Î N + ®¥ m n Π m! G n + 1 G m + n + 3 F -n, -m - 1 , m + n + 3 ; 1 , 1; x2 2 2 3 2 2 2 2 J N H L J N H L F -m, m + 2, 1 - n, n + 3 ; 3 , 3 , 3 ; x2 ß ß 4 m + 1 2 n + 1 4 3 2 2 2 2 2 sgn x lim J N Jx N J ; -1 N< x < 1 n Î N m Î N. + ®¥ m n Π F -n, n + 1, -m - 1 , m + 3 ; 1 , 1, 1; x2 4 3 2 2 2 H L H L J N TheH lastL two representations are sometimes called generalized Padé approximations. ß ß J N Transformations The values of all complex components z , Arg z , Re z , Im z , z, and sgn z at the points -z, ä z, Ðä z, a z ; a Î R,

1 , za, ãz, and ãä z are given by the following identities: z ¤ H L H L H L H L http://functions.wolfram.com 13

z -z ä z -ä z a z ; a Î z -z z ä z z -ä z z a z a z a z -a 0 3 4 4 -z Π -1 Π ä z Π -1 Π -ä z Arg ¤ z Arg ¤ -z ¤ Arg z - ä Π Arg¤ ä z ¤ Arg z - - Arg ¤-ä z ¤ Arg z + - Arg¤ a z z 2 z 2 z ¤ Re z Re -z -Re z Re ä z -Im z H L Re -ä z Im z Re a z Im zH L Im -Hz L -Im zH L Im äHz L Re zH L Im -Hä z L -Re zH L Im aH z L z -z -z ä z -ä z -ä z ä z a z a z H L H L H L H L H L H L H L H L sgn z sgn -z -sgn z sgn ä z ä sgn z sgn -ä z -ä sgn z sgn a z H L H L H L H L H L H L H L H L sgn a z 0

H L H L H L H L H L H L H L H L z 1 za ãz ãä z z H L z 1 1 za z Re a exp -Im a Arg z ãz ãRe z ãä z ã- z z

Π-Im a log z Arg z Arg 1 - z z-1 Arg z Arg za 2H LΠ + Im a log z Arg ãz H ImL z + 2 Π Π-Im z Arg ãä z z 2 Π 2 Π ¤ ¤ ¢ ¦ 1 ¤ ¤ H H L H LL ¤ ¡ ¥ Arg -Arg z ; Arg z ¹ Π H H LL H L z H L 1 Re z Re z Re J N H L Re Hza L z Refa ã-Im a Argvz H H LL Re ãHz L ãRe zH cosL Im fz v Re ãIä z M z z 2 J N H L H L cos Im a log z + Arg z Re a H L H L H L H L H L 1 Im z a Re a -Im a Arg z z Re z ä z Im z Im - ¤ Im z z ã Im ã ã sin Im z Im ã H L J z N z 2 H L ¤ H L H H LL I M sin Im a log z + Arg z Re a H L H H L H H L¤L H L H LH L H LL H L -a 1 1 a 1 z z ä z -ä z ¤ z ã ã ã ã H L z J zN H Lz ¤ H L H H LL I M 1 z H H L H ¤L H L H LL sgn z sgn sgn za z ä Im a exp ä Re a Arg z sgn ãz ãä Im z sgn ãä z z z J N ¤ H L H L z1 The values of all complex components z , Arg z , Re z , Im z , z, and sgn z at the points z1 + z2, z1 z2, and are H L J N H L ¤ H H L H LL H L z2 I M described by the following table:

¤ H L H L H L H L z1 z z1 + z2 z1 z2 z2

z1 z1 z z1 + z2 z1 - z2 + z1 + z2 - z1 - z2 z1 z2 z1 z2 z2 z2

-1 z1 ¤ Arg z Arg z1 + z2 tan Re z1 + Re z2 , Im z1 + Im z2 Arg z1 z2 Arg z1 + Arg z2 + Arg Arg z1 - z2 ¤ Π-Arg z -Arg z ¤ ¤ ¤ ¤¤ ¤ ¤ ¤ 2¤Π ¤ 1¤ 2 ¢ ¦ 2 Π 2 Π Π - Arg z1 H L H L H H L H L H L H LL H L H L H HL L H L J N H L 1 Re z Re z Re z1 + z2 Re z1 + Re z2 Re z1 z2 Re z1 Re z2 - Im z2 Im z1 Re f v z z 2 1 gK Im z H L w Im z Im z1 + z2 Im z1 + Im z2 Im z1 z2 Im z2 Re z1 + Im z1 Re z2 Im - H L z z 2 H L H L H L H L H L H L H L H L H L J N ¤ 1 1 H L z z + z z + z z z z z 1 2 1 2 1 2 1 2 z z H L H L H L H L H L H L H L H L H L J N ¤ z1+z2 1 z sgn z sgn z1 + z2 sgn z1 z2 sgn z1 sgn z2 sgn z1+z2 z z

¤ Some complex components ¤ can be easily evaluated in more general cases of the points including symbolic sums H L H L H L H L H L J N and products of zk, k = 1, ¼, n, for example: http://functions.wolfram.com 14

n n

zk zk k=1 k=1

n n n Π - Arg zk ä ä ¤ k=1 + Arg zk Arg zk + 2 Π ; n Î N k=1 k=1 2 Π Ú H L ä â H L Π - n Arg z Arg zn n Arg z + 2 Π ; n Î N+ 2 Π

n n H L H L H L Re zk Re zk k=1 k=1

n â â H L 2 n Re zn -1 j Im z 2 j Re z n-2 j ; n Î N+ 2 j fj=0v

H nL âH n L H L H L Im zk Im zk k=1 k=1

â n-1 â H L 2 n Im zn -1 j Im z 2 j+1 Re z n-2 j-1 ; n Î N+ 2 j + 1 fj=0v

n H L nâ H L H L H L zk zk k=1 k=1

ân ân zk zk k=1 k=1

ä n ä n sgn zk sgn zk . k=1 k=1

Theä previousä tablesH andL formulas can be modified or simplified for particular cases when some variables become real or satisfy special restrictions, for example:

z xa ; x Î R x > 0 za ; a Î R ãx+ä y ; x Î R y Î R z xa xRe a za z a ãx+ä y ãx Π-a Arg z Arg z Arg xa tanì -1 cos Im a log x , Arg za a Arg z + 2 Π Arg ãx+ä y tanì-1 cos y , sin y H L 2 Π sin Im a log x ¤ ¤ ¤ ¤ H L ¡ ¥ Re z Re xa xRe a cos Im a log x Re za z a cos a tan-1 Re z , Im z Re ãx+ä y ãx cos y H L H L H H H L H LL H L H L f v I M H H L H LL Im z Im xa xRe a sin Im a log x Im za z a sin a tan-1 Re z , Im z Im ãx+ä y ãx sin y H H LH L H LLL a Re a a a -1 x+ä y x-ä y z H L x H xL cosH L Im Ha logH L x H- LL z H Lz cos¤ a tanI ReHz , HImL z H-LLM ã I ãM H L -1 H L H L ä sin Im a logH Hx L H LL H L ä sin ¤ a tanI Re zH , ImH Lz H LLM I M H L H L a ä Im a a a x+ä y ä y sgn z sgn x xH H H L H LL sgn z ¤IsgnI z H H L H LLM sgn ã ã H H L H LLL H L I H H L H LLMM Arg z1 z2 Arg z1 + Arg z2 ; -Π < Arg z1 + Arg z2 £ Π H L H L H L H L I M

H L H L H L H L H L http://functions.wolfram.com 15

z1 Arg Arg z1 - Arg z2 ; -Π < Arg z1 - Arg z2 £ Π. z2

Taking into account that complex components have numerous representations through other complex components H L H L H L H L and elementary functions such as the logarithm, exponential function, or the inverse tangent function, all of the previous formulas can be transformed into different equivalent forms. Here are some of the resulting formulas for the power function z ® zΑ:

za exp ä a Im log z ; ä a Î R

za exp ä a Arg z ; ä a Î R ¤ H H H LLL za exp Re a log z ¤ H H LL za exp Re a log z - Im a Arg z ¤ H H H LLL za z Re a exp -Im a tan-1 Re z , Im z ¤ H H L H ¤L H L H LL Arg za HaLArg z ; a Î R -Π < a Arg z £ Π ¤ ¤ I H L H H L H LLM Arg za Arg ãä a Arg z ; a Î R H L H L ì H L H L Π Π Arg za Im a log z ; - < Im a £ z Î R z > 0 H L I M log z log z

Π Π ArgHzaL ImHa logH LzL + Arg z Re a H;L- <íIm a £ í z Î R z > 0 H L log zH L log z

a -1 -1 -1 ArgHz L tanH L cosH a¤tanL ReH Lz , ImH Lz , sin a tan ReH Lz , Im z í í H L H L Π - Im a log z Arg za 2 Π + Im a log z H L I I 2 Π H H L H LLM I H H L H LLMM

Π - ImH a logH LLz - Arg z Re a Arg H zaL 2 Π H H LL + Im a log z + Arg z Re a 2 Π

Arg za tan-1 cos ImH aL logH z¤L + tanH-1L ReHzL, Im z Re a , sin Im a log z + tan-1 Re z , Im z Re a H L H L H ¤L H L H L Re za z a cos a Arg z ; a Î R H L I I H L H ¤L H H L H LL H LM I H L H ¤L H H L H LL H LMM ¥ ¥ 2 j -a j+l Im z ReHzaL ¤ H H LL 1 - Re z l ; a Î R 1 j=0 l=0 l - j ! j! 4 Re z - 1 2 j H L H L H L ââ -1 H H LL Re za z Re a ã-Im a tan Re z ,Im z cos Im a log z + Re a tan-1 Re z , Im z H L J N H H L L

Im za z a sinH L a ArgH L z H; aH ÎL RH LL H L ¤ I H L H ¤L H L H H L H LLM -1 Im za z Re a ã-Im a tan Re z ,Im z sin Im a log z + Re a tan-1 Re z , Im z H L ¤ H H LL za za ; a ÎHRL H L H H L H LL H L ¤ I H L H ¤L H L H H L H LLM xa xa ; x Î R x > 0

ß http://functions.wolfram.com 16

za za ; Arg z ¹ Π

za z Re a exp -Arg z Im a - ä Im a log z + Arg z Re a H L -1 za z ReHaL ã-Im a tan Re z ,Im z cos Im a log z + Re a tan-1 Re z , Im z - ä sin Im a log z + Re a tan-1 Re z , Im z ¤ H H L H L H H L H ¤L H L H LLL sgn za H expL a HReL logH zH L ;H äLLa Î R ¤ I I H L H ¤L H L H H L H LLM I H L H ¤L H L H H L H LLMM sgn za z a ; ä a Î R H L H H H LLL sgn za za exp -Re a log z H L ¤ sgn za z ä Im a exp ä Re a tan-1 Re z , Im z H L H H H LLL sgn za exp äH LIm a log z + Arg z Re a . H L ¤ I H L H H L H LLM Similar identities can be derived for the exponent functions, such as: H L H H H L H ¤L H L H LLL Arg ãz Im z ; -Π < Im z £ Π

Arg ãz Π - Π - Im z mod 2 Π H L H L H L Arg ãä z Π - Π - Re z mod 2 Π . H L H H LL H L Some arithmetical operations involving complex components or elementary functions of complex components are: I M H H LL H L z1 z2 z1 z2

z a za ; a Î R ¤ ¤ ¤ 1 2 2 2 2 z1 + z2 z1 - z2 + z1 + z2 ¤ ¤ 2

Π - Arg z1 - Arg z2 Arg ¤ z1 +¤Arg z2I Arg¤z1 z 2 - 2 Π¤ M 2 Π

Re z + Re z Re z + z H L H L H1 L 2H L 1 H 2 L

Im z1 + Im z2 Im z1 + z2 H L H L H L z1 + z2 z1 + z2 H L H L H L z1 z2 z1 z2

za za ; a Î R z ãä Arg z z H L ãä Arg z cos Arg z + ä sin Arg z ¤ H L Im z Im z Π Π ãä Arg z cos tan-1 + ä sin tan-1 ; - < Arg Re z £ . H H LLRe z H H LL Re z 2 2 H L Complex characteristicsH L H L H H LL H L H L http://functions.wolfram.com 17

The next two tables describe all the complex components applied to all complex components z , Arg z , Re z , Im z , z, and sgn z at the points z and z x + ä y ; x Î R y Î R:

z Abs Arg Re Im Conjugate ¤ H L H L H L H L ì Abs z z Arg z Re z Im z Im z 2 z z z cos Arg z Im z 2 tan-1 Re z , Im z z sin Arg z ¤¤ ¤ H L¤ H L¤ H L¤ H L ¤ ¤ Arg Arg z 0 Arg Arg z Arg ¤ Re zH H LL¤ ArgH ImL¤ z Arg z -1 -1 -1 -1 tan ArgH zH, L0 H LL tan Re z , 0 tan¤ 0H , ImH zLL¤ tan Re x , -Im z Arg Im z H ¤L H H LL H H LL 1H- ΘHImLL z Π H L H H L L H H L L H H LL H H L H LL Re Re z z Re Arg z Arg z Re Re z Re z Re Im z Im z Re z Re z H H LL Im Im z 0 Im Arg z 0 Im Re z 0 ImH Im zH H 0LLL Im z -Im z H ¤L ¤ H H LL H L H H LL H L H H LL H L H L H L Conjugate z z Arg z Arg z Re z Re z Im z Im z z z H ¤L H H LL Arg z H H LL Re z H H LL Im z H L z H L Sign sgn z 1 ; sgn Arg z sgn Re z sgn Im z sgn z ; z ¹ 0 z Arg z 2 2 2 ¤z ¹ 0¤ H L H L H L H L Re z H L H L Im z H L H L H L ¤ H ¤L H H LL H L H H LL H L H H LL H L H L x + ä y ; Abs Arg Re Im Conjugate x Î R y Î R

Abs x + ä y x2 + y2 Arg x + ä y Re x + ä y x2 Im x + ä y y2 x + ä y x2 + ì 2 tan-1 x, y ¤¤ H L¤ H L¤ H L¤ ¤ Arg Arg x + ä y 0 Arg Arg x + ä y Arg Re x + ä y Arg Im x + ä y Arg z -Arg -1 -1 -1 -1 tan tanH xL, y , 0 tan x, 0 tan y, 0 Arg z ¹ Π Arg Re x + ä y Arg Im x + ä y H ¤L H H LL 1 H- UnitStepH LLx Π 1 H- ΘHy Π LL H L H I H L M H L H L H L Re Re x + ä y Re Arg x + ä y Re Re x + ä y x Re Im x + ä y y Re x + ä y x @ @ DD H H LL tan-1 x, y x2 + y2 H @ DL H H LL Im Im H x + ä y¤L 0 Im HArgHx + ä yLL 0 Im HReHx + ä yLL 0 Im HImHx + ä yLL 0 Im Hx + ä yL - H L

H ¤L H H LL H H LL H H LL H L Conjugate x + ä y x2 + y2 Arg x + ä y Re x + ä y x Im x + ä y y x + ä y x + ä y tan-1 x, y Sign sgn x + ä y 1 ; sgn Arg x + ä y sgn Re x + ä y sgn Im x + ä y sgn x + ä y ¤ H L H L x H L y x + ä y ¹ 0 tan-1 x,y sgn x sgn y x-ä y H L 2 2 x y x2+y2 2 tan-1 x,y H ¤L H H L LL H H LL H H LL H L H L H L H L

Differentiation

The derivatives of five complex components x , Re x , Im x , x, and sgn x at the real point x Î R can be interpreted in a real-analytic or distributional sense and are given by the following formulas:

¤ H L H L H L http://functions.wolfram.com 18

¶ x sgn x ¶x

¶Re ¤ x 1H L ¶x

¶ImHxL 0 ¶x

¶ x H L 1 ¶x

¶sgn x 2 ∆ x , ¶x

whereH L∆ x is the Dirac delta function. H L It is impossible to make a classical, direction-independent interpretation of these derivatives for complex values of variableH xL because the complex components do not fulfill the Cauchy-Riemann conditions.

Indefinite integration

The indefinite integrals of some complex components at the real point x Î R can be represented by the following formulas:

x x x â x 2

¤ sgn x â x x . à ¤

Definite integration à H L ¤ The definite integrals of some complex components in the complex plane can also be represented through complex components, for example:

1 t ât 1 -1

z 2 2 à t¤ ât z Im z + Re z -z

z z z à t¤ ât H L H L 0 2

z ¤ à sgn¤ t ât 0 -z

z 2 à sgnHtL ât z . 0

Some definite integrals including absolute values can be easily evaluated, for example (in the Hadamard sense of à H L integration, the next identity is correct for all complex values of n): http://functions.wolfram.com 19

1 1 2 x - y n â x â y ; Re n > -2. 0 0 n + 1 n + 2

Integral transforms à à ¤ H L H L H L Fourier integral transforms of the absolute value and signum functions t and sgn t can be evaluated through generalized functions:

2 1 ¤ H L Ft t z - Π z2

2 1 -@ 1 ¤D H L Ft t z - Π z2

@ ¤D H L 2 1 Fct t z - Π z2

Π @ ¤D H L ¢ Fst t z - ∆ z 2

@ ¤D H L Π H L Fct sgn t z ∆ z 2

@ H LD H L 2 1H L Fst sgn t z . Π z

Laplace integral transforms of these functions can be evaluated in a classical sense and have the following values: @ H LD H L 1 Lt t z z2

1 Lt@sgn ¤D HtL z . z

Differential equations @ H LD H L The absolute value function x for real x Î R satisfies the following simple first-order differential equation under- standable in a distributional sense:

w¢ x sign w x ; w x x . ¤

In a similar manner: H L H H LL H L ¤ w¢ x 1 ; w x Re x

w¢ x 0 ; w x Im x H L H L H L w¢ x 2 Π ∆ x ; w x Arg x . H L H L H L Inequalities H L H L H L H L http://functions.wolfram.com 20

All six complex components z , Arg z , Re z , Im z , z and sgn z satisfy numerous inequalities. The best known are so-called triangle inequalities for absolute values:

z1 + z2 £ z1 + z2 ¤ H L H L H L H L

n n zk £¤ ¤zk . ¤ k=1 k=1

Someâ otherâ inequalities¤ can be described by the following formulas:

z1 - z2 ³ z1 - z2

z1 - z2 £ z1 + z2 £ z1 + z2 ¤ ¤ ¤¤ Arg z £ Π ¤ ¤ ¤ ¤ ¤ Re z £ z H L¤ Im z £ z H L¤ ¤ Re sgn z £ 1 H L¤ ¤ Im sgn z £ 1 H H LL sgn z £ 1. H H LL Zeros H L¤ The six complex components z , Arg z , Re z , Im z , z, and sgn z have the set of zeros described by the following formulas:

z 0 ; z 0 ¤ H L H L H L H L

Arg z 0 ; z Î R z > 0 ¤ Re z 0 ; ä z Î R H L ß Im z 0 ; z Î R H L z 0 ; z 0 H L sgn z 0 ; z 0. Applications of complex components H L All six complex components are used throughout mathematics, the exact sciences, and engineering. http://functions.wolfram.com 21

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