DOCSLIB.ORG

3 Elementary Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
3 Elementary Functions 3 Elementary Functions We already know a great deal about polynomials and rational functions: these are analytic on their entire domains. We have thought a little about the square-root function and seen some difficulties. The remaining elementary functions are the exponential, logarithmic and trigonometric functions. 3.1 The Exponential and Logarithmic Functions (§30–32, 34) We have already defined the exponential function exp : C ! C : z 7! ez using Euler’s formula ez := ex cos y + iex sin y (∗) and seen that its real and imaginary parts satisfy the Cauchy–Riemann equations on C, whence exp C d z = z is entire (analytic on ). Indeed recall that dz e e . We have also seen several of the basic properties of the exponential function, we state these and several others for reference. Lemma 3.1. Throughout let z, w 2 C. 1. ez 6= 0. ez 2. ez+w = ezew and ez−w = ew 3. For all n 2 Z, (ez)n = enz. 4. ez is periodic with period 2pi. Indeed more is true: ez = ew () z − w = 2pin for some n 2 Z Proof. Part 1 follows trivially from (∗). To prove 2, recall the multiple-angle formulae for cosine and sine. Part 3 requires an induction using part 2 with z = w. Part 4 is more interesting: certainly ew+2pin = ew by the periodicity of sine and cosine. Now suppose ez = ew where z = x + iy and w = u + iv. Then, by considering the modulus and argument, ( ex = eu exeiy = eueiv =) y = v + 2pin for some n 2 Z We conclude that x = u and so z − w = i(y − v) = 2pin. Example 3.2. Find all z 2 C such that ez = 5(−1 + i). Following the Lemma, write z = x + iy and take the polar form of 5(−1 + i) to see that ( p p x = ln(5 2) z = (− + ) () x iy = 3pi/4 () e 5 1 i e e 5 2e 3p y = 4 + 2pn for some n 2 Z p 3p () z = ln(5 2) + + 2pn i for some n 2 Z 4 We see that there are infinitely many suitable z! Duplicate Notation Warning! When n 2 N, the expression e1/n can now mean two things. For instance e1/3 can mean: p p p 1. The set of cube roots of e, namely f 3 e, 3 ee2pi/3, 3 ee−2pi/3g; p 2. The real value 3 e 2 R+. Given that ez is such a common function, in both real and complex analysis, we default to the second meaning: if you mean the set of nth roots, say so! The periodicity of the exponential leads to the far more interesting notion of the complex logarithm. Definition 3.3. Let z = reiq be a non-zero complex number with principal argument q = Arg z. The principal logarithm of z is the value Log z := ln r + iq = ln jzj + i Arg z where ln is the usual natural logarithm. The logarithm of z is any (and all!) of the valuesa log z = ln jzj + i arg z = ln r + i(q + 2pn) : n 2 Z aThis is identical to how we use arg z, which, depending on context, means either the set fArg z + 2pnig or some particular value from this set. We’ll more formally discuss such multi-valued functions in the next section. Examples 3.4. 1. Since −4 = 4epi, we see that Log(−4) = ln 4 + pi and log(−4) = ln 4 + (1 + 2n)pi 2. Again write in polar form to compute: p p p p Log( 3 − i) = Log(2e−pi/6) = ln 2 − i and log( 3 − i) = ln 2 − i + 2pni 6 6 These examples involve solving equations of the form ew = z: writing z = reiq = eln r+iq as above, and appealing to part 4 of Lemma 3.1, we instantly see that ew = z () w = log z Read this carefully, remembering that the logarithm is multi-valued and the exponential periodic: elog z = z and log(ew) = w + 2pni where n 2 Z Before moving on, we clear up some of the basic properties of the principal logarithm function. All parts of this should be clear from Definition 3.3. 2 Lemma 3.5. Throughout, z and w are complex numbers with z 6= 0, and n 2 Z. • Log : C n f0g ! fw 2 C : Im w 2 (−p, p]g is a bijection with inverse exp. • Log(ew) = w + 2pni where n 2 Z is chosen such that Im w + 2pn 2 (−p, p]. • If z 2 R+, then Log z = ln z is the usual natural logarithm. The Logarithm Laws Just as the standard rules for exponentiation (Lemma 3.1 parts 2 and 3) apply to the complex expo- nential, something similar works for the log laws. However, the multi-valued nature of the logarithm makes this a little more subtle. Suppose non-zero z, w are given: since jzwj = jzj jwj and arg zw = arg z + arg w, we conclude that log zw = ln jzj jwj + i(arg z + arg w) = ln jzj + i arg z + ln jwj + i arg w = log z + log w Be very careful with this expression, since it is not an identity of functions. Indeed it means two things: • We have set equality: in particular, the following sets are identical: log z + log w = fa + b : a 2 log z, b 2 log wg = fjzj + i Arg z + 2pki + jwj + i Arg w + 2pmi : k, m 2 Zg log zw = fln jzj jwj + i Arg zw + 2pni : n 2 Zg • There exist particular choices of the arguments of z, w and zw so that arg zw = arg z + arg w. Unless you are sure you won’t make a mistake, it is therefore safer to write log zw = log z + log w + 2pni where n 2 Z Given its restricted range, we can be more precise for the principal logarithm: 9n 2 f0, ±1g such that Log zw = Log z + Log w + 2pni p p Example 3.6. Let z = − 3 + i = 2e5p/6 and w = 2(1 + i) = 2epi/4. Then 5p p log z = ln 2 + i + 2pki, log w = ln 2 + i + 2pmi 6 4 5p + p 13p/12 13p log zw = log(4e 6 4 ) = log(4e ) = ln 4 + i + 2pni 12 For the above boxed formula to make sense for particular choices of logarithms, we’d need to select k + m = n + 1. More explicitly, 5p p Log z = ln 2 + i, Log w = ln 2 + i 6 4 11p Log zw = Log(4e−11p/12) = Log(4e−11p/12) = ln 4 − i = Log z + Log w − 2pi 12 3 We can similarly demonstrate the other log law, with the same caveat: z log = log z − log w w You might assume that the final log law (log zn = n log z whenever n 2 N) also makes sense, but you’d be very wrong: an example should explain why you should ignore this law! p Example 3.7. Let z = − 3 + i = 2e5p/6 and compute what is generally meant by the set 2 log z: 5p 5p 2 log z = 2 ln 2 + i + 2pmi = ln 4 + i + 4pmi 6 3 that is, we double the value of everything in the set log z. This is different from the set 5p log z2 = log(4e10p/6) = ln 4 + i + 2pki 3 However, in the language of the previous exercise, there would be no problem if each copy of log z gots its own multiples of 2pi: 5p log z + log z = ln 4 + i + 2p(l + m)i where l, m 2 Z 3 = log z2 Since this is not how 2 log z is generally interpreted, it is safer to state that log z2 6= 2 log z. Since the principal logarithm is a function rather than a set, we can be more precise: for any n 2 N, n Log zn = n Log z + 2pki for some integer k with jkj ≤ 2 Example 3.8. Let z = e−13pi/16 and consider z16. We see that Log z16 = Log e−13pi = Log epi = ip, 16 Log z = −13pi =) Log z16 = 16 Log z + 14pi 4 Exercises. 3.1.1. Compute the following: p 2 (a) exp(3 − 2 i) (b) Log(ie) (c) log(3 − 4i) (d) Log[(−1 + i) ] 3.1.2. (a) If ez is real, show that Im z = np for some integer n. (b) If ez is imaginary, what restriction is placed on z? 3.1.3. Show in two ways that the function f (z) = exp(z2) is entire, and find its derivative. 2 3.1.4. Prove, for any z 2 C, that exp(z2) ≤ exp jzj . What must z satisfy if this is to be equality? 3.1.5. Find Re(e1/z) in terms of x and y. Why is this function harmonic in every domain that does not contain the origin? 3.1.6. Show that Log i3 6= 3 Log i. p p 3.1.7. The square roots of i are i = epi/4 and − i = e−3pi/4. p p p 1 (a) Compute Log i and Log(− i) and check that Log i = 2 Log i. (b) Show that the set of all logarithms of all square roots of i is 1 log i1/2 = n + pi where n 2 Z 4 1/2 1 and therefore deduce that log i = 2 log i as sets. 3.2 Multi-valued functions, Branch Cuts and the Power Function (§33, 35, 36) The complex logarithm is often called a multi-valued function, since each log z represents a set of complex numbers.
Load more

Recommended publications
  • Lecture 5: Complex Logarithm and Trigonometric Functions
    LECTURE 5: COMPLEX LOGARITHM AND TRIGONOMETRIC FUNCTIONS Let C∗ = C \{0}. Recall that exp : C → C∗ is surjective (onto), that is, given w ∈ C∗ with w = ρ(cos φ + i sin φ), ρ = |w|, φ = Arg w we have ez = w where z = ln ρ + iφ (ln stands for the real log) Since exponential is not injective (one one) it does not make sense to talk about the inverse of this function. However, we also know that exp : H → C∗ is bijective. So, what is the inverse of this function? Well, that is the logarithm. We start with a general definition Definition 1. For z ∈ C∗ we define log z = ln |z| + i argz. Here ln |z| stands for the real logarithm of |z|. Since argz = Argz + 2kπ, k ∈ Z it follows that log z is not well defined as a function (it is multivalued), which is something we find difficult to handle. It is time for another definition. Definition 2. For z ∈ C∗ the principal value of the logarithm is defined as Log z = ln |z| + i Argz. Thus the connection between the two definitions is Log z + 2kπ = log z for some k ∈ Z. Also note that Log : C∗ → H is well defined (now it is single valued). Remark: We have the following observations to make, (1) If z 6= 0 then eLog z = eln |z|+i Argz = z (What about Log (ez)?). (2) Suppose x is a positive real number then Log x = ln x + i Argx = ln x (for positive real numbers we do not get anything new).
    [Show full text]
  • Complex Analysis
    Complex Analysis Andrew Kobin Fall 2010 Contents Contents Contents 0 Introduction 1 1 The Complex Plane 2 1.1 A Formal View of Complex Numbers . .2 1.2 Properties of Complex Numbers . .4 1.3 Subsets of the Complex Plane . .5 2 Complex-Valued Functions 7 2.1 Functions and Limits . .7 2.2 Infinite Series . 10 2.3 Exponential and Logarithmic Functions . 11 2.4 Trigonometric Functions . 14 3 Calculus in the Complex Plane 16 3.1 Line Integrals . 16 3.2 Differentiability . 19 3.3 Power Series . 23 3.4 Cauchy's Theorem . 25 3.5 Cauchy's Integral Formula . 27 3.6 Analytic Functions . 30 3.7 Harmonic Functions . 33 3.8 The Maximum Principle . 36 4 Meromorphic Functions and Singularities 37 4.1 Laurent Series . 37 4.2 Isolated Singularities . 40 4.3 The Residue Theorem . 42 4.4 Some Fourier Analysis . 45 4.5 The Argument Principle . 46 5 Complex Mappings 47 5.1 M¨obiusTransformations . 47 5.2 Conformal Mappings . 47 5.3 The Riemann Mapping Theorem . 47 6 Riemann Surfaces 48 6.1 Holomorphic and Meromorphic Maps . 48 6.2 Covering Spaces . 52 7 Elliptic Functions 55 7.1 Elliptic Functions . 55 7.2 Elliptic Curves . 61 7.3 The Classical Jacobian . 67 7.4 Jacobians of Higher Genus Curves . 72 i 0 Introduction 0 Introduction These notes come from a semester course on complex analysis taught by Dr. Richard Carmichael at Wake Forest University during the fall of 2010. The main topics covered include Complex numbers and their properties Complex-valued functions Line integrals Derivatives and power series Cauchy's Integral Formula Singularities and the Residue Theorem The primary reference for the course and throughout these notes is Fisher's Complex Vari- ables, 2nd edition.
    [Show full text]
  • Inverse Trigonometric Functions
    Chapter 2 INVERSE TRIGONOMETRIC FUNCTIONS vMathematics, in general, is fundamentally the science of self-evident things. — FELIX KLEIN v 2.1 Introduction In Chapter 1, we have studied that the inverse of a function f, denoted by f–1, exists if f is one-one and onto. There are many functions which are not one-one, onto or both and hence we can not talk of their inverses. In Class XI, we studied that trigonometric functions are not one-one and onto over their natural domains and ranges and hence their inverses do not exist. In this chapter, we shall study about the restrictions on domains and ranges of trigonometric functions which ensure the existence of their inverses and observe their behaviour through graphical representations. Besides, some elementary properties will also be discussed. The inverse trigonometric functions play an important Aryabhata role in calculus for they serve to define many integrals. (476-550 A. D.) The concepts of inverse trigonometric functions is also used in science and engineering. 2.2 Basic Concepts In Class XI, we have studied trigonometric functions, which are defined as follows: sine function, i.e., sine : R → [– 1, 1] cosine function, i.e., cos : R → [– 1, 1] π tangent function, i.e., tan : R – { x : x = (2n + 1) , n ∈ Z} → R 2 cotangent function, i.e., cot : R – { x : x = nπ, n ∈ Z} → R π secant function, i.e., sec : R – { x : x = (2n + 1) , n ∈ Z} → R – (– 1, 1) 2 cosecant function, i.e., cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1) 2021-22 34 MATHEMATICS We have also learnt in Chapter 1 that if f : X→Y such that f(x) = y is one-one and onto, then we can define a unique function g : Y→X such that g(y) = x, where x ∈ X and y = f(x), y ∈ Y.
    [Show full text]
  • Complex Analysis
    COMPLEX ANALYSIS MARCO M. PELOSO Contents 1. Holomorphic functions 1 1.1. The complex numbers and power series 1 1.2. Holomorphic functions 3 1.3. Exercises 7 2. Complex integration and Cauchy's theorem 9 2.1. Cauchy's theorem for a rectangle 12 2.2. Cauchy's theorem in a disk 13 2.3. Cauchy's formula 15 2.4. Exercises 17 3. Examples of holomorphic functions 18 3.1. Power series 18 3.2. The complex logarithm 20 3.3. The binomial series 23 3.4. Exercises 23 4. Consequences of Cauchy's integral formula 25 4.1. Expansion of a holomorphic function in Taylor series 25 4.2. Further consequences of Cauchy's integral formula 26 4.3. The identity principle 29 4.4. The open mapping theorem and the principle of maximum modulus 30 4.5. The general form of Cauchy's theorem 31 4.6. Exercises 36 5. Isolated singularities of holomorphic functions 37 5.1. The residue theorem 40 5.2. The Riemann sphere 42 5.3. Evalutation of definite integrals 43 5.4. The argument principle and Rouch´e's theorem 48 5.5. Consequences of Rouch´e'stheorem 49 5.6. Exercises 51 6. Conformal mappings 53 6.1. Fractional linear transformations 56 6.2. The Riemann mapping theorem 58 6.3. Exercises 63 7. Harmonic functions 65 7.1. Maximum principle 65 7.2. The Dirichlet problem 68 7.3. Exercises 71 8. Entire functions 72 8.1. Infinite products 72 Appunti per il corso Analisi Complessa per i Corsi di Laurea in Matematica dell'Universit`adi Milano.
    [Show full text]
  • Complex Numbers and Functions
    Complex Numbers and Functions Richard Crew January 20, 2018 This is a brief review of the basic facts of complex numbers, intended for students in my section of MAP 4305/5304. I will discuss basic facts of com- plex arithmetic, limits and derivatives of complex functions, power series and functions like the complex exponential, sine and cosine which can be defined by convergent power series. This is a preliminary version and will be added to later. 1 Complex Numbers 1.1 Arithmetic. A complex number is an expression a + bi where i2 = −1. Here the real number a is the real part of the complex number and bi is the imaginary part. If z is a complex number we write <(z) and =(z) for the real and imaginary parts respectively. Two complex numbers are equal if and only if their real and imaginary parts are equal. In particular a + bi = 0 only when a = b = 0. The set of complex numbers is denoted by C. Complex numbers are added, subtracted and multiplied according to the usual rules of algebra: (a + bi) + (c + di) = (a + c) + (b + di) (1.1) (a + bi) − (c + di) = (a − c) + (b − di) (1.2) (a + bi)(c + di) = (ac − bd) + (ad + bc)i (1.3) (note how i2 = −1 has been used in the last equation). Division performed by rationalizing the denominator: a + bi (a + bi)(c − di) (ac − bd) + (bc − ad)i = = (1.4) c + di (c + di)(c − di) c2 + d2 Note that denominator only vanishes if c + di = 0, so that a complex number can be divided by any nonzero complex number.
    [Show full text]
  • Universit`A Degli Studi Di Perugia the Lambert W Function on Matrices
    Universita` degli Studi di Perugia Facolta` di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Triennale in Informatica The Lambert W function on matrices Candidato Relatore MassimilianoFasi BrunoIannazzo Contents Preface iii 1 The Lambert W function 1 1.1 Definitions............................. 1 1.2 Branches.............................. 2 1.3 Seriesexpansions ......................... 10 1.3.1 Taylor series and the Lagrange Inversion Theorem. 10 1.3.2 Asymptoticexpansions. 13 2 Lambert W function for scalar values 15 2.1 Iterativeroot-findingmethods. 16 2.1.1 Newton’smethod. 17 2.1.2 Halley’smethod . 18 2.1.3 K¨onig’s family of iterative methods . 20 2.2 Computing W ........................... 22 2.2.1 Choiceoftheinitialvalue . 23 2.2.2 Iteration.......................... 26 3 Lambert W function for matrices 29 3.1 Iterativeroot-findingmethods. 29 3.1.1 Newton’smethod. 31 3.2 Computing W ........................... 34 3.2.1 Computing W (A)trougheigenvectors . 34 3.2.2 Computing W (A) trough an iterative method . 36 A Complex numbers 45 A.1 Definitionandrepresentations. 45 B Functions of matrices 47 B.1 Definitions............................. 47 i ii CONTENTS C Source code 51 C.1 mixW(<branch>, <argument>) ................. 51 C.2 blockW(<branch>, <argument>, <guess>) .......... 52 C.3 matW(<branch>, <argument>) ................. 53 Preface Main aim of the present work was learning something about a not- so-widely known special function, that we will formally call Lambert W function. This function has many useful applications, although its presence goes sometimes unrecognised, in mathematics and in physics as well, and we found some of them very curious and amusing. One of the strangest situation in which it comes out is in writing in a simpler form the function .
    [Show full text]
  • Chapter 2 Complex Analysis
    Chapter 2 Complex Analysis In this part of the course we will study some basic complex analysis. This is an extremely useful and beautiful part of mathematics and forms the basis of many techniques employed in many branches of mathematics and physics. We will extend the notions of derivatives and integrals, familiar from calculus, to the case of complex functions of a complex variable. In so doing we will come across analytic functions, which form the centerpiece of this part of the course. In fact, to a large extent complex analysis is the study of analytic functions. After a brief review of complex numbers as points in the complex plane, we will ¯rst discuss analyticity and give plenty of examples of analytic functions. We will then discuss complex integration, culminating with the generalised Cauchy Integral Formula, and some of its applications. We then go on to discuss the power series representations of analytic functions and the residue calculus, which will allow us to compute many real integrals and in¯nite sums very easily via complex integration. 2.1 Analytic functions In this section we will study complex functions of a complex variable. We will see that di®erentiability of such a function is a non-trivial property, giving rise to the concept of an analytic function. We will then study many examples of analytic functions. In fact, the construction of analytic functions will form a basic leitmotif for this part of the course. 2.1.1 The complex plane We already discussed complex numbers briefly in Section 1.3.5.
    [Show full text]
  • An Example of a Covering Surface with Movable Natural Boundaries
    An example of a covering surface with movable natural boundaries Claudi Meneghin November 13, 2018 Abstract We exhibit a covering surface of the punctured complex plane (with no points over 0) whose natural projection mapping fails to be a topo- logical covering, due to the existence of branches with natural bound- ∗ aries, projecting over different slits in C . In [1], A.F.Beardon shows an example of a ’covering surface’ over a region in the unit disc which is not a topological covering (We recall that a ’covering surface’ of a region D ⊂ C is a Riemann surface S admitting a surjective conformal mapping onto D, among which there are the analytical continua- tions of holomorphic germs; a continuous mapping p of a topological space Y onto another one X is a ’topological covering’ provided that each point x ∈ X admits an open neighbourhood U such that the restriction of p to −1 each connected component Vi of p (U) is a homeomorphism of Vi onto U, see [2, 3]). Beardon’s example takes origin by the analytical continuation of a holo- morphic germ, namely a branch of the inverse of the Blaschke product arXiv:math/0408086v3 [math.CV] 11 Jul 2008 ∞ z − a B(z)= n , n 1 − anz Y=1 with suitable hypotheses about the an’s. It is shown that there is a sequence of inverse branches fk of B at 0 such that the radii of convergence of their Taylor developments tend to 0 as k →∞. This of course prevents the natural projection (taking a germ at z0 to the point z0) of the analytical continuation of any branch of B−1 to be a topological covering.
    [Show full text]
  • On the Lambert W Function 329
    On the Lambert W function 329 The text below is the same as that published in Advances in Computational Mathematics, Vol 5 (1996) 329 – 359, except for a minor correction after equation (1.1). The pagination matches the published version until the bibliography. On the Lambert W Function R. M. Corless1,G.H.Gonnet2,D.E.G.Hare3,D.J.Jeffrey1 andD.E.Knuth4 1Department of Applied Mathematics, University of Western Ontario, London, Canada, N6A 5B7 2Institut f¨ur Wissenschaftliches Rechnen, ETH Z¨urich, Switzerland 3Symbolic Computation Group, University of Waterloo, Waterloo, Canada, N2L 3G1 4Department of Computer Science, Stanford University, Stanford, California, USA 94305-2140 Abstract The Lambert W function is defined to be the multivalued inverse of the function w → wew. It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches of W ,an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing W . 1. Introduction In 1758, Lambert solved the trinomial equation x = q +xm by giving a series develop- ment for x in powers of q. Later, he extended the series to give powers of x as well [48,49]. In [28], Euler transformed Lambert’s equation into the more symmetrical form xα − xβ =(α − β)vxα+β (1.1) by substituting x−β for x and setting m = α/β and q =(α − β)v. Euler’s version of Lambert’s series solution was thus n 1 2 x =1+nv + 2 n(n + α + β)v + 1 n(n + α +2β)(n +2α + β)v3 6 (1.2) 1 4 + 24 n(n + α +3β)(n +2α +2β)(n +3α + β)v +etc.
    [Show full text]
  • Inverse Trigonemetric Function
    1 INVERSE TRIGONEMETRIC FUNCTION 09-04-2020 We know that all trigonometric functions are periodic in nature and hence are not one- one and onto i.e they are not bijections and therefore they are not invertible. But if we restrict their domain and co-domain in such a way that they become one –one and onto, then they become invertible and they have an inverse. Consider the function f:R → R given by f(x) = sinx. The domain of sinx is R and the codomain is π 1 5π 1 R. It is many one(because many values of x give the same image i.e f = and f = 6 2 6 2 and into function because range [-1,1] is the subset of codomain R. If we restrict the domain π π to − , and make the co-domain equal to range[-1,1], sine function becomes one -one and 2 2 onto i.e bijective and hence it becomes invertible. Actually sine function restricted to any of 3π π π π π3 π the intervals − , , − , , , is one –one and its range is [-1,1].We can define the 2 2 2 2 2 2 inverse of sine function in each of these intervals. Corresponding to each such intervals, we get π π a branch of sine inverse function. The branch with range − , is called principal value 2 2 π π branch .Thus f: − , → [-1,1] given by f(x) =sinx is invertible. Similarly all trigonometric 2 2 functions are invertible when their domain and co domain are restricted.
    [Show full text]
  • The Complex Logarithm, Exponential and Power Functions
    Physics 116A Winter 2011 The complex logarithm, exponential and power functions In these notes, we examine the logarithm, exponential and power functions, where the arguments∗ of these functions can be complex numbers. In particular, we are interested in how their properties differ from the properties of the corresponding real-valued functions.† 1. Review of the properties of the argument of a complex number Before we begin, I shall review the properties of the argument of a non-zero complex number z, denoted by arg z (which is a multi-valued function), and the principal value of the argument, Arg z, which is single-valued and conventionally defined such that: π < Arg z π . (1) − ≤ Details can be found in the class handout entitled, The argument of a complex number. Here, we recall a number of results from that handout. One can regard arg z as a set consisting of the following elements, arg z = Arg z +2πn, n =0 , 1 , 2 , 3 ,..., π < Arg z π . (2) ± ± ± − ≤ One can also express Arg z in terms of arg z as follows: 1 arg z Arg z = arg z +2π , (3) 2 − 2π where [ ] denotes the greatest integer function. That is, [x] is defined to be the largest integer less than or equal to the real number x. Consequently, [x] is the unique integer that satisfies the inequality x 1 < [x] x , for real x and integer [x] . (4) − ≤ ∗Note that the word argument has two distinct meanings. In this context, given a function w = f(z), we say that z is the argument of the function f.
    [Show full text]
  • Complex Numbers Sections 13.5, 13.6 & 13.7
    Complex exponential Trigonometric and hyperbolic functions Complex logarithm Complex power function Chapter 13: Complex Numbers Sections 13.5, 13.6 & 13.7 Chapter 13: Complex Numbers Complex exponential Trigonometric and hyperbolic functions Definition Complex logarithm Properties Complex power function 1. Complex exponential The exponential of a complex number z = x + iy is defined as exp(z)=exp(x + iy)=exp(x) exp(iy) =exp(x)(cos(y)+i sin(y)) . As for real numbers, the exponential function is equal to its derivative, i.e. d exp(z)=exp(z). (1) dz The exponential is therefore entire. You may also use the notation exp(z)=ez . Chapter 13: Complex Numbers Complex exponential Trigonometric and hyperbolic functions Definition Complex logarithm Properties Complex power function Properties of the exponential function The exponential function is periodic with period 2πi: indeed, for any integer k ∈ Z, exp(z +2kπi)=exp(x)(cos(y +2kπ)+i sin(y +2kπ)) =exp(x)(cos(y)+i sin(y)) = exp(z). Moreover, | exp(z)| = | exp(x)||exp(iy)| = exp(x) cos2(y)+sin2(y) = exp(x)=exp(e(z)) . As with real numbers, exp(z1 + z2)=exp(z1) exp(z2); exp(z) =0. Chapter 13: Complex Numbers Complex exponential Trigonometric and hyperbolic functions Trigonometric functions Complex logarithm Hyperbolic functions Complex power function 2. Trigonometric functions The complex sine and cosine functions are defined in a way similar to their real counterparts, eiz + e−iz eiz − e−iz cos(z)= , sin(z)= . (2) 2 2i The tangent, cotangent, secant and cosecant are defined as usual. For instance, sin(z) 1 tan(z)= , sec(z)= , etc.
    [Show full text]