Analytic Functions of a Complex Variable

Analytic Functions of a Complex Variable

David Raymond Curtiss
Volume: 2
Copyright Date: 1926
Edition: 1
Pages: 184
https://www.jstor.org/stable/10.4169/j.ctt5hh97z
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    Analytic Functions of a Complex Variable
    Book Description:

    The MAA is pleased to re-issue the early Carus Mathematical Monographs in ebook format. Readers with an interest in the history of the undergraduate curriculum or the history of a particular field will be rewarded by study of these very clear and approachable little volumes. The theory of functions of a complex variable has been developed by the efforts of thousands of workers through the last hundred years. To give even the briefest account of the present state of that theory in all its branches would be impossible within the limits of this book. What is attempted here is a presentation of fundamental principles with sufficient details of proof and discussion to avoid the style of a mere summary or synopsis. In various places there are indications of directions in which special portions of the subject branch off from the main stem. The reader is assumed to have an acquaintance with elementary differential and integral calculus. Without such knowledge one may, however, obtain some idea of the scope and purposes of the theory of functions from this monograph. Those who are familiar with more than the elements of the calculus should profit most.

    eISBN: 978-1-61444-002-4
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. PREFACE
    (pp. v-vi)
    D. R. Curtiss
  3. Table of Contents
    (pp. vii-x)
  4. CHAPTER I ORIGIN AND APPLICATIONS OF THE THEORY
    (pp. 1-5)

    The first step toward a theory of functions of a complex variable was the introduction of the complex number in the solution of algebraic equations with real coefficients. The equation${x^2} - 2x + 5 = 0$, for example, has no real roots, but is formally satisfied by the expressions$1 \pm \sqrt { - 4} $, or$1 \pm 2i$, whereistands for$\sqrt { - 1} $, if the fundamental operations are suitably defined for these complex numbers. Soon it was perceived, and finally proved in a satisfactory way by Gauss shortly before the year 1800, that every algebraic equation with real coefficients has complex roots, real numbers being a special kind of complex numbers...

  5. CHAPTER II COMPLEX NUMBERS
    (pp. 6-17)

    When real numbers are combined by addition, subtraction, multiplication, or division with non-vanishing divisor, the results are real numbers; such numbers therefore form aclosed systemfor these operations. But this is not always true when we pass to root extraction. No real number can be the square root of a negative real number.

    The situation is analogous to one which exists for the number system composed of the positive integers. Here we have a system closed for addition, but in which subtraction of a number from one not greater than itself is impossible. When it seems desirable to allow...

  6. CHAPTER III REAL FUNCTIONS OF REAL VARIABLES
    (pp. 18-42)

    Before we take up functions of a complex variable we must consider some of the fundamental properties of functions of real variables. It is assumed that the reader has a knowledge of the calculus which will enable us to abbreviate the discussion in places.

    According to the definition of Dirichlet (1805–59), adependent variable yis a function of anindependent variable xif to each value ofxthere corresponds one or more values fory.When this relation is expressed in the form of an equation in mathematical notation, the function isexplicitif the equation is...

  7. CHAPTER IV COMPLEX FUNCTIONS THAT HAVE DERIVATIVES
    (pp. 43-59)

    With the complex variable we repeat almost word for word the definition of a function which was used for real variables:The dependent variable w is a function of the independent variable z if to each value of the latter corresponds one or more values of the former.It is implied here thatwis a complex variable, as well asz,and we should recall that a real variable is a special case of a complex variable. A function may be defined for all values ofz, but more frequently the region of definition will be taken as a...

  8. CHAPTER V APPLICATIONS IN GEOMETRY AND PHYSICS
    (pp. 60-82)

    It is not our object in this chapter to give a general survey of the geometrical and physical applications of the theory of functions of a complex variable, but only to consider certain especially interesting cases. On the geometrical side we shall confine ourselves to the study of equations$\omega = f(z)$regarded as transformations which bring about a correspondence between points of the ω- andz-planes.

    In a map of a piece of the earth’s surface small enough to be thought of as plane, a straight line on the earth will in general be represented by a curved line on the...

  9. CHAPTER VI INTEGRALS OF ANALYTIC FUNCTIONS
    (pp. 83-105)

    Following the familiar terminology of elementary calculus we define an indefinite integral of a single-valued continuous function of a complex variable$f(z)$as a function$f(z)$whose derivative is$f(z)$. Thus the two following identities are equivalent:

    $F(z) = \int {f(z)dz} $,$\frac{{dF(z)}}{{dz}} = f(z)$.

    With real functions of a real variable,everysingle-valued continuous function has an indefinite integral, but the corresponding statement is not true for functions of a complex variable. For example,$x - iy$is a continuous function ofzwhich has no indefinite integral. For if there were a function$\omega = u + iv$whose derivative is$x - iy$we would have

    $\frac{{\partial w}}{{\partial z}} = \frac{{\partial w}}{{\partial x}} = \frac{{\partial u}}{{\partial x}} + i\frac{{\partial v}}{{\partial x}} = x - iy.$,

    or

    $\frac{{\partial u}}{{\partial x}} = x$,$\frac{{\partial v}}{{\partial x}} = - y$.

    If...

  10. CHAPTER VII INFINITE SERIES
    (pp. 106-130)

    In every book on the calculus there will be found at least a brief treatment of infinite series of real numbers or functions. Such a series being defined as an array of the form

    ${u_0} + {u_1} + {u_2} + .... + {u_n} + ....$,

    where thenth term is given by some formula, we designate by${s_n}$the sum of the firstnterms for each positive integern, and consider the sequence

    ${s_1},{s_2},....,{s_n},.....$

    If this sequence has a limitUasnbecomes infinite, i.e., if

    $\mathop {\lim }\limits_{n = \infty } {s_n} = U$,

    the series is said to beconvergent, otherwise it isdivergent.In the former case it is sometimes said to have...

  11. CHAPTER VIII SINGULARITIES OF SINGLE-VALUED ANALYTIC FUNCTIONS
    (pp. 131-146)

    A point at which a single-valued function has not a derivative, or in every neighborhood of which there are points at which the function has not a derivative, is called asingular pointof the function. An especially interesting class of such points is composed of those possessing a neighborhood throughout which the function is analytic but which, of course, does not include the point itself. A point answering to this description is anisolated singular point.

    It is useful here, for purposes of comparison, to consider the kinds ofisolated discontinuitieswhich singlevalued real functions of a real variable...

  12. CHAPTER IX ANALYTIC CONTINUATION. MANY-VALUED ANALYTIC FUNCTIONS
    (pp. 147-170)

    A function$f(z)$is defined so as to be single valued and analytic throughout a regionT, and a function$\phi (z)$is single-valued and analytic throughout a region${T'}$that does not coincide completely withT. When shall we say that they represent thesame function$f(z)$over the two regions? As an example let$f(z)$and$\phi (z)$be denned by the following infinite series,

    $f(z) = 1 + z + {z^2} + .... + {x^n} + ....$,

    $\phi (z) = - 1 + (z - 2) - {(z - 2)^2} + .... + {( - 1)^{n + 1}}{(z - 2)^n} + ....$.

    The first of these series converges throughout the interior of the unit circle about the origin, the second throughout the interior of a circle of unit radius whose center is at$z = 2$. These...

  13. INDEX
    (pp. 171-173)