Appendix B. Elementary Complex Analysis

Abstract

This appendix treats some aspects of elementary complex analysis that are useful as tools in real analysis. It assumes knowledge of Appendix A and much of Chapters I to III.

Section B1 Introduces the complex derivative of a complex-valued function defined on an open subset of $\mathbb{C}$, and it relates the notion to differentiability in the sense of Chapter III. The Cauchy–Riemann equations are part of this relationship. An analytic function on a region in $\mathbb{C}$ is a function with a complex derivative at each point.

Section B2 introduces complex line integrals and relates them to the traditional line integrals in the last three sections of Chapter III. An important result is that a continuous complex-valued function on a region in $\mathbb{C}$ is the complex derivative of an analytic function if and only if its complex line integral over every piecewise $C^1$ closed curve in the region is zero.

Section B3 proves Goursat's Lemma and a local form of the Cauchy Integral Theorem. Goursat's Lemma says that the complex line integral of a function over a rectangle is 0 if the function is analytic on a region containing the rectangle and its inside. The local form of the Cauchy Integral Theorem that follows says that for an analytic function in an open disk, the complex line integral is zero over every piecewise $C^1$ closed curve.

Section B4 obtains a simple form of the Cauchy Integral Formula for a disk and derives from it the corresponding formula for complex derivatives, Morera's Theorem, Cauchy's estimate, Liouville's Theorem, and the Fundamental Theorem of Algebra.

Section B5 establishes two versions of the complex-variable form of Taylor's Theorem. The first form includes a remainder term, and the second form asserts a convergent power series expansion.

Section B6 treats various local properties of analytic functions in regions. If the complex derivatives of all orders of such a function are zero, then the function is 0. Consequently if the function is not identically 0, then any zero has a nonnegative integer order, and the zeros of the function are isolated. Other consequences are the Maximum Modulus Theorem, a description of the behavior at poles, Weierstrass's result on essential singularities, and the Inverse Function Theorem.

Section B7 examines the exponential function and its local invertibility. This examination leads to the definition of winding number for a closed curve about a point, and the general form of the Cauchy Integral Formula for a disk follows.

Section B8 discusses operations on Taylor series and methods for computing such series.

Section B9 gives a first form of the Argument Principle relating the integral of $f'(z)/f(z)$ to the zeros and poles of $f(z)$.

Section B10 states and proves a first form of the Residue Theorem for evaluating the complex line integral of a function analytic except for poles.

Section B11 uses the the first form of the Residue Theorem to evaluate a number of examples of real definite integrals.

Section B12 extends the Cauchy Integral Theorem from closed curves in disks to cycles in simply connected regions, and it derives a corresponding version of the Residue Theorem.

Section B13 examines the extent to which the results of Section B12 extend to general regions when the cycle is assumed to be a boundary cycle.

Section B14 develops the Laurent series expansion of a function analytic in an annulus (washer). As a consequence the nature of essential singularities becomes a little clearer.

Section B15 introduces holomorphic functions of several variables, showing the equivalence of various definitions of such functions. This material is not used until Advanced Real Analysis.