Complex-Valued Functions of a Complex Variable

The following sections are included:

• Introduction

• Limits of Functions

• Continuous Functions

• Uniform Convergence of a Sequence or Series of Functions

• Curves

• Elementary Rational Functions

  • Polynomials

  • The power function w = zⁿ (n ≥ 2)

  • Rational functions

  • Linear fractional (or bilinear or Möbius) transformations

  • Joukowski function

• Elementary Transcendental Functions

  • The exponential function e^z

  • Trigonometric functions cos z, sin z, and tan z

• Elementary Multiple-Valued Functions

  • The origin of multiple-valuedness: arg z

  • and its Riemann surface (etc.)

  • w = logz (the natural logarithm function with base e) and its Riemann surface

  • w = cos⁻¹ z and w = tan⁻¹ z and their Riemann surfaces

• Differentiation in Complex Notation

• Integration in complex notation