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Algebraic Extension of the Field RThus, following Euler, we introduce a number i, the imaginary unit, such thati 2 = −1. The interaction between i and the real numbers is to consist of the following. One may multiply i by numbers y ∈ R, that is, numbers of the form iynecessarily arise, and one may add such numbers to real numbers, that is, numbersof the form x + iy occur, where x, y ∈ R.2645Differential CalculusIf we wish to have the usual operations of a commutative addition and a commutative multiplication that is distributive with respect to addition defined on the setof objects of the form x + iy (which, following Gauss, we shall call the complexnumbers), then we must make the following definitions:(x1 + iy1 ) + (x2 + iy2 ) := (x1 + x2 ) + i(y1 + y2 )(5.99)(x1 + iy1 ) · (x2 + iy2 ) := (x1 x2 − y1 y2 ) + i(x1 y2 + x2 y1 ).(5.100)andTwo complex numbers x1 + iy1 and x2 + iy2 are considered equal if and only ifx1 = x2 and y1 = y2 .We identify the real numbers x ∈ R with the numbers of the form x + i · 0, andi with the number 0 + i · 1.
The role of 0 in the complex numbers, as can be seenfrom Eq. (5.99), is played by the number 0 + i · 0 = 0 ∈ R; the role of 1, as can beseen from Eq. (5.100), is played by 1 + i · 0 = 1 ∈ R.It follows from properties of the real numbers and definitions (5.99) and (5.100)that the set of complex numbers is a field containing R as a subfield.We shall denote the field of complex numbers by C and typical elements of itusually by z and w.The only nonobvious point in the verification that C is a field is the assertionthat every non-zero complex number z = x + iy has an inverse z−1 with respect tomultiplication (a reciprocal), that is z · z−1 = 1.
Let us verify this.We call the number x − iy the conjugate of z = x + iy, and we denote it z̃.We observe that z · z = (x 2 + y 2 ) + i · 0 = x 2 + y 2 = 0 if z = 0. Thus z−1 shouldy1xbe taken as x 2 +y2 · z̄ = x 2 +y 2 − i x 2 +y 2 .b. Geometric Interpretation of the Field CWe remark that once the algebraic operations (5.99) and (5.100) on complex numbers have been introduced, the symbol i, which led us to these definitions, is nolonger needed.
We can identify the complex number z = x + iy with the orderedpair (x, y) of real numbers, called respectively the real part and the imaginary partof the complex number z. (The notation for this is x = Re z, y = Im z.)But then, regarding the pair (x, y) as the Cartesian coordinates of a point of theplane R2 = R × R, one can identify complex numbers with the points of this planeor with two-dimensional vectors having coordinates (x, y).In such a vector interpretation the coordinatewise addition (5.99) of complexnumbers corresponds to vector addition. Moreover such an interpretation naturallyleads to the idea of the absolute value or modulus |z| of a complex number as theabsolute value or length of the vector (x, y) corresponding to it, that is|z| = x 2 + y 2 , if z = x + iy,(5.101)5.5 Complex Numbers and Elementary Functions265and also to a way of measuring the distance between complex numbers z1 and z2 asthe distance between the points of the plane corresponding to them, that is, as|z1 − z2 | = (x1 − x2 )2 + (y1 − y2 )2 .(5.102)The set of complex numbers, interpreted as the set of points of the plane, is calledthe complex plane and also denoted by C, just as the set of real numbers and the realline are both denoted by R.Since a point of the plane can also be defined in polar coordinates (r, ϕ) connected with Cartesian coordinates by the relationsx = r cos ϕ,y = r sin ϕ,(5.103)the complex numberz = x + iy(5.104)z = r(cos ϕ + i sin ϕ).(5.105)can be represented in the formThe expressions (5.104) and (5.105) are called respectively the algebraic andtrigonometric (polar) forms of the complex number.In the expression (5.105) the number r ≥ 0 is called the modulus or absolutevalue of the complex number z (since, as one can see from (5.103), r = |z|), and ϕthe argument of z.
The argument has meaning only for z = 0. Since the functionscos ϕ and sin ϕ are periodic, the argument of a complex number is determined onlyup to a multiple of 2π , and the symbol Arg z denotes the set of angles of the formϕ + 2πk, k ∈ Z, where ϕ is any angle satisfying (5.105). When it is desirable forevery complex number to determine uniquely some angle ϕ ∈ Arg z, one must agreein advance on the range from which the argument is to be chosen.
This range isusually either 0 ≤ ϕ < 2π or −π < ϕ ≤ π . If such a choice has been made, we saythat a branch (or the principal branch) of the argument has been chosen. The valuesof the argument within the chosen range are usually denoted arg z.The trigonometric form (5.105) for writing complex numbers is convenient incarrying out the operation of multiplication of complex numbers.
In fact, ifz1 = r1 (cos ϕ1 + i sin ϕ1 ),z2 = r2 (cos ϕ2 + i sin ϕ2 ),thenz1 · z2 = (r1 cos ϕ1 + ir1 sin ϕ1 )(r2 cos ϕ2 + ir2 sin ϕ2 ) == (r1 r2 cos ϕ1 cos ϕ2 − r1 r2 sin ϕ1 sin ϕ2 )++ i(r1 r2 sin ϕ1 cos ϕ2 + r1 r2 cos ϕ2 sin ϕ2 ),266or5Differential Calculusz1 · z2 = r1 r2 cos(ϕ2 + ϕ2 ) + i sin(ϕ1 + ϕ2 ) .(5.106)Thus, when two complex numbers are multiplied, their moduli are multiplied andtheir arguments are added.We remark that what we have actually shown is that if ϕ1 ∈ Arg z1 and ϕ2 ∈Arg z2 , then ϕ1 + ϕ2 ∈ Arg(z1 · z2 ). But since the argument is defined only up to amultiple of 2π , we can write thatArg(z1 · z2 ) = Arg z1 + Arg z2 ,(5.107)interpreting this equality as set equality, the set on the right-hand side being the setof all numbers of the form ϕ1 + ϕ2 , where ϕ1 ∈ Arg z1 and ϕ2 ∈ Arg z2 .
Thus it isuseful to interpret the sum of the arguments in the sense of the set equality (5.107).With this understanding of equality of arguments, one can assert, for example,that two complex numbers are equal if and only if their moduli and arguments areequal.The following formula of de Moivre22 follows by induction from formula(5.106):if z = r(cos ϕ + i sin ϕ),then zn = r n (cos nϕ + i sin nϕ).(5.108)Taking account of the explanations given in connection with the argument of acomplex number, one can use de Moivre’s formula to write out explicitly all thecomplex solutions of the equation zn = a.Indeed, ifa = ρ(cos ψ + i sin ψ)and, by formula (5.108)zn = r n (cos nϕ + i sin nϕ),√we have r = n ρ and nϕ = ψ + 2πk, k ∈ Z, from which we have ϕk = ψn + 2πn k.Different complex numbers are obviously obtained only for k = 0, 1, .
. . , n − 1.Thus we find n distinct roots of a: ψ 2πψ 2π√n+k + i sin+k(k = 0, 1, . . . , n − 1).zk = ρ cosnnnnIn particular, if a = 1, that is, ρ = 1 and ψ = 0, we have#2π2πzk = n k 1 = cosk + i sink(k = 0, 1, . . . , n − 1).nnThese points are located on the unit circle at the vertices of a regular n-gon.22 A.de Moivre (1667–1754) – British mathematician.5.5 Complex Numbers and Elementary Functions267In connection with the geometric interpretation of the complex numbers themselves, it is useful to recall the geometric interpretation of the arithmetic operationson them.For a fixed b ∈ C, the sum z + b can be interpreted as the mapping of C into itselfgiven by the formula z → z + b. This mapping is a translation of the plane by thevector b.For a fixed a = |a|(cos ϕ + i sin ϕ) = 0, the product az can be interpreted as themapping z → az of C into itself, which is the composition of a dilation by a factorof |a| and a rotation through the angle ϕ ∈ Arg a.
This is clear from formula (5.106).5.5.2 Convergence in C and Series with Complex TermsThe distance (5.102) between complex numbers enables us to define the εneighborhood of a number z0 ∈ C as the set {z ∈ C | |z − z0 | < ε}. This set isa disk (without the boundary circle) of radius ε centered at the point (x0 , y0 ) ifz0 = xi + iy0 .We shall say that a sequence {zn } of complex numbers converges to z0 ∈ C iflimn→∞ |zn − z0 | = 0.It is clear from the inequalitiesmax |xn − x0 |, |yn − y0 | ≤ |zn − z0 | ≤ |xn − x0 | + |yn − y0 |(5.109)that a sequence of complex numbers converges if and only if the sequences of realand imaginary parts of the terms of the sequence both converge.By analogy with sequences of real numbers, a sequence of complex numbers {zn }is called a fundamental or Cauchy sequence if for every ε > 0 there exists an indexN ∈ N such that |zn − zm | < ε for all n, m > N .It is clear from inequalities (5.109) that a sequence of complex numbers is aCauchy sequence if and only if the sequences of real and imaginary parts of itsterms are both Cauchy sequences.Taking the Cauchy convergence criterion for sequences of real numbers into account, we conclude on the basis of (5.109) that the following proposition holds.Proposition 1 (The Cauchy criterion) A sequence of complex numbers converges ifand only if it is a Cauchy sequence.If we interpret the sum of a series of complex numbersz1 + z2 + · · · + zn + · · ·(5.110)as the limit of its partial sums sn = z1 + · · · + zn as n → ∞, we also obtain theCauchy criterion for convergence of the series (5.110).2685Differential CalculusProposition 2 The series (5.110) converges if and only if for every ε > 0 there existsN ∈ N such that|zm + · · · + zn | < ε(5.111)for any natural numbers n ≥ m > N .From this one can see that a necessary condition for convergence of the series(5.110) is that zn → 0 as n → ∞.
(This, however, is also clear from the very definition of convergence.)As in the real case, the series (5.110) is absolutely convergent if the series|z1 | + |z2 | + · · · + |zn | + · · ·(5.112)converges.It follows from the Cauchy criterion and the inequality|zm + · · · + zn | ≤ |zm | + · · · + |zn |that if the series (5.110) converges absolutely, then it converges.Examples The series1 nz + ···,1) 1 + 1!1 z + 2!1 z2 + · · · + n!1 31 52) z − 3! z + 5! z − · · · , and3) 1 − 2!1 z2 + 4!1 z4 − · · ·converge absolutely for all z ∈ C, since the series1 ) 1 + 1!1 |z| + 2!1 |z|2 + · · · ,2 ) |z| + 3!1 |z|3 + 5!1 |z|5 + · · · ,3 ) 1 + 2!1 |z|2 + 4!1 |z|4 + · · · ,all converge for any value of |z| ∈ R.
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