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    Complex Variables II Essentials
    Complex Variables II Essentials
    Complex Variables II Essentials
    Ebook200 pages2 hours

    Complex Variables II Essentials

    By Alan D. Solomon

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    About this ebook

    REA’s Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Complex Variables II includes elementary mappings and Mobius transformation, mappings by general functions, conformal mappings and harmonic functions, applying complex functions to applied mathematics, analytic continuation, and analytic function properties.
    LanguageEnglish
    PublisherResearch & Education Association
    Release dateJan 1, 2013
    ISBN9780738672175
    Complex Variables II Essentials

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      Book preview

      Complex Variables II Essentials - Alan D. Solomon

      MAPPINGS

      CHAPTER 7

      ELEMENTARY MAPPINGS AND THE MOBIUS TRANSFORMATION

      7.1 REVIEW OF BASIC CONCEPTS

      1. A function w = f(z) defines a mapping of its domain in the z-plane, onto its range R in the w-plane.

      2. The image of a set S of the z-plane under a mapping w =f(z) is the set T of all points of the form w =f(z) for every point z of S. (Figure 7.1)

      Fig. 7.1f(z) maps S onto T

      3. The level lines of a function u = u(x, y) are the curves in the z-plane along which u is constant.

      Fig. 7.2Level lines of a functionu(x,y)

      In describing the effect of a mapping we will represent the images of representative points A, B, ... by A’, B’, ... as in Figure 7.1.

      7.2 EXPANSIONS AND CONTRACTIONS

      4. A function of the form

      (7.1)

      where c , a positive real number, is an expansion if c > 1 and a contraction if c < 1. The number c is referred to as the contraction or expansion factor for the mapping ƒ.

      EXAMPLE 7.1

      Under the expansion ƒ(z) = 2z, the rectangle

      R: – 1

      of the z-plane is mapped onto the rectangle

      S: – 2

      of the w = u + iv plane (Figure 7.3).

      Fig. 7.3 The Image of a Rectangle Under w = 2z

      EXAMPLE 7.2

      Under the contraction w = ƒ(z) = z/2, the circle lzl = 3 of the z-plane is mapped onto the circle lwl = 3/2 of the w-plane (Figure 7.4).

      Contractions and expansions map circles onto circles. Any circle of the z-plane has a polar representation

      z = z0 + R0eiθ,     0≤θ ≤ 2π

      with center z0 and radius R0. Under the contraction or expansion w =ƒ(z) = cz, real c > 0, the image of this circle is the set of points

      w = cz = cz0 + cR0eiθ, 0≤θ ≤ 2π.

      This is a circle in the w-plane with center cZ0 and radius cR0.

      7.3 ROTATIONS

      5. The mapping

      (7.2)

      for a given angle ø, is a rotation. This maps a point z onto a point w obtained by rotating the ray from the origin to z through the angle ø but conserving the distance to the origin (the modulus) (Figure 7.5). The ø is positive (negative) if the rotation is counterclockwise (clockwise).

      EXAMPLE 7.3

      In its polar representation,

      Thus the mapping

      is a rotation through π / 4= 45°. Therefore the image of the real axis z = x is the line u = ν of the w = u + iv plane, all bodies are rotated by 45° and the imaginary axis z = iy is mapped into the line u = −v (Figure 7.6).

      A rotation through an angle ø has the same effect as that of rotating the entire z- plane through ø and placing it over the w-plane to obtain its u, v coordinates.

      EXAMPLE 7.4

      is a rotation through the angle 30°, or π / 6, since

      7.4 TRANSLATIONS

      6. For a given complex number zo the mapping

      (7.3)

      is a translation through the value zo

      A translation moves the entire z-plane such that its axes maintain their original directions but the origin is now placed at the point zo of the w-plane (Figure 7.7); z = 0 is mapped into the point w = zo, circles are mapped into circles, lines into lines, and all orientations are maintained.

      Fig. 7.7 The mapping w = z − (2 + 3i)

      EXAMPLE 7.5

      Under the translation w = ƒ(z) = z − (2 + 3i) the origin z = 0 is mapped onto the point w = −(2 + 3i), the circle

      is mapped onto the circle

      C ′: w = 2i +3eiθ – (2+3i)

      = – (2+i) + 3eiθ ,

      and lines are mapped onto lines preserving their original directions (Figure 7.7).

      7.5 COMPOUND MAPPINGS

      7. The compound function

      (7.4)

      with w = F(Z), and Z= G(z), defines a compound mapping from the z-plane to the w-plane. The mapping may be considered as one of two stages:

      followed by

      (Figure 7.8).

      Fig. 7.8The compound functionw = F(G(z))

      THEOREM

      Let C be any complex number. Then the mapping

      (7.5)

      may be considered as the compound mapping of a rotation and a contraction or expansion.

      PROOF

      Suppose that C has the polar representation

      C = coeiθ , co ≥ 0, 0 ≤

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