# A Guide to Complex Variables

Steven G. Krantz
Volume: 32
Edition: 1
Pages: 202
https://www.jstor.org/stable/10.4169/j.ctt6wpwc3

1. Front Matter
(pp. I-VIII)
2. Preface
(pp. IX-X)
Steven G. Krantz
(pp. XI-XVIII)
4. CHAPTER 1 The Complex Plane
(pp. 1-18)

We assume that the reader is familiar with the real number system R. We let${R^2} = \{ (x,y):x\varepsilon R,y\varepsilon R\}$(Figure 1.1). These are ordered pairs of real numbers.

As we shall see, the complex numbers are nothing other than${R^2}$equipped with a special algebraic structure.

The complex numbers C consist of${R^2}$equippedwith some binary algebraic operations. One defines

$(x,y) + (x',y') = (x + x',y + y')$,

$(x,y)\cdot(x',y') = (xx' - yy',xy' + yx')$.

These operations of + and$\cdot$are commutative and associative.

We denote (1; 0) by 1 and (0,1) byi. If$\alpha \varepsilon R$, then we identify$\alpha$with the complex number$(\alpha ,0)$Using this notation, we see that

$\alpha \cdot (x,y) = (\alpha ,0) \cdot (x,y) = (\alpha x,\alpha y)$. (1.1.2.1)

As if...

5. CHAPTER 2 Complex Line Integrals
(pp. 19-32)

In this section we shall recast the line integral from calculus in complex notation. The result will be the complex line integral. The complex line integral is essential to the Cauchy theory, which we develop below, and that in turn is key to the argument principle and many of the other central ideas of the subject.

It is convenient to think of acurveas a continuous function$\gamma$from a closed interval$[a,b] \subseteq R$into${R^2} \approx C$. We sometimes let${\tilde \gamma }$denote theimageof the mapping. Thus

$\tilde \gamma = \{ \gamma (t):t \in [a,b]\}$.

Often we follow the custom of referring toeitherthe function or...

6. CHAPTER 3 Applications of the Cauchy Theory
(pp. 33-42)

Let$U \subseteq C$be an open set and letfbe holomorphic onU. then$f \in {C^\infty }(U)$. Moreover, if$\bar D(P,r) \subseteq U$and$z \in D(P,r)$, then

${\left( {\frac{\partial }{{\partial z}}} \right)^k}f(z) = {\rm{ }}\frac{k}{{2\pi i}}!\oint_{|\zeta - P| = r} {\frac{{f(\zeta )}}{{{{(\zeta - z)}^{k + 1}}}}} d\zeta$, k = 0,1,2, . . . . (3.1.1.1)

This formula is obtained by differentiating the standard Cauchy formula (2.3.2.1) under the integral sign.

Iffis a holomorphic on a region containing the closed disc$\bar D (P,r)$and if$|f| \le M$on$\bar D (P,r)$, then

$\left| {\frac{{{\partial ^k}}}{{\partial {z^k}}}f(P)} \right| \le \frac{{M \cdot k!}}{{{r^k}}}$. (3.1.2.1)

This is proved by direct estimation of the cauchy formula (3.1.1.1).

A functionfis said to beentireif it is defined and holomorphic on all of C, i.e.,f: C→C is...

7. CHAPTER 4 Isolated Singularities and Laurent Series
(pp. 43-70)

It is often important to consider a function that is holomorphic on a punctured open set$U\backslash \{ P\} \subset C$. Refer to Figure 4.1.

In this chapter we shall obtain a new kind of infinite series expansion that generalizes the idea of the power series expansion of a holomorphic function about a (nonsingular) point—see §§3.1.6. We shall in the process completely classify the behavior of holomorphic functions near an isolated singular point (§§4.1.3).

Let$U \subseteq C$be an open set and$P \in U$. We call the domain$U\backslash \{ P\}$]apunctured domain. Suppose that$f:U\backslash \{ P\} \to C$is holomorphic. Then we say thatfhas anisolated...

8. CHAPTER 5 The Argument Principle
(pp. 71-82)

In this chapter, we shall be concerned with questions that have a geometric, qualitative nature rather than an analytical, quantitative one. These questions center around the issue of the local geometric behavior of a holomorphic function.

Suppose that$f:U \to C$is a holomorphic function on a connected, open set$U \subseteq C$and that$\bar D (P,r) \subseteq U$. We know from the Cauchy integral formula that the values offon$\partial D(P,r)$. In particular, the number and even the location of the zeros offin$D(P,r)$are determined in principal byfon$\partial D(P,r)$. But it is nonetheless a pleasant surprise that there is a...

9. CHAPTER 6 The Geometric Theory of Holomorphic Functions
(pp. 83-94)

The main objects of study in this chapter are holomorphic functions$h:U \to V$, withUandVopen in C, that are one-to-one and onto. Such a holomorphic function is called aconformal(orbiholomorphic) mapping. The fact thathis supposed to be one-to-one implies that${h'}$is nowhere zero on U (remember that if${h'}$vanishes to order$k \geqslant 0$at a point$P\in U$, thenhis$(k + 0)$-to-1 in a small neighborhood ofP—see §§5.2.1). As a result,${h^{ - 1}}:V \to U$is also holomorphic, as we discussed in §§5.2.1. A conformal map$h:U \to V$from one open set to another can be...

10. CHAPTER 7 Harmonic Functions
(pp. 95-110)

We reiterate the definition of “harmonic”. LetFbe a holomorphic function on an open set$U \subseteq C$. Write$F = u + iv$, whereuandvare realvalued. The real partusatisfies a certain partial differential equation known asLaplace’s equation:

$\left( {\frac{{{\partial ^2}}} {{\partial {x^2}}} + \frac{{{\partial ^2}}} {{\partial {y^2}}}} \right)u = 0$. (7.1.1.1)

(The imaginary partvsatisfies the same equation.) In this chapter we shall study systematically those${C^2}$functions that satisfy this equation. They are calledharmonicfunctions. (We encountered some of these ideas already in §1.4.)

Recall the precise definition of harmonic function:

A real-valued function$u:U \to R$on an open set$U \subseteq C$isharmonicif it is${C^2}$on...

11. CHAPTER 8 Infinite Series and Products
(pp. 111-124)

Let$U \subseteq C$be an open set and${g_j}:U \to C$functions. Recall (§§3.1.5) that the sequence$\{ {g_j}\}$is said toconverge uniformly on compact subsets of Uto a functiongif the following condition holds: For each compact$K \subseteq U$(see §§3.1.5) and each ε > 0 there is anN> 0 such that ifj> N, then$|g(z) - {g_j}(z)| < \varepsilon$for all$z\in K$. In general, the choice ofNdepends on εandonK, but not on the particular point$z\in K$.

Because of the comleteness (see [RUD1], [KRA2]) of the complex numbers, a sequence of functions is uniformly convergent on...

12. CHAPTER 9 Analytic Continuation
(pp. 125-142)

Suppose thatVis a connected, open subset of C and that${f_1}:V \to C$and${f_2}:V \to C$are holomorphic functions. If there is an open, non-empty subsetUofVsuch that${f_1} \equiv {f_2}$onU, then${f_1} \equiv {f_2}$on all ofV(see §§3.2.3). Put another way, if we givenfholomorphic onU, then there is at most one way to extend it toVso that the extended function is holomorphic. (There might not even be one such extension: ifVis the unit disc andUthe discD(3/4,1/4), then the function$f\left( z \right) = 1/z$does not extend. Or ifUis...

13. Glossary of Terms from Complex Variable Theory and Analysis
(pp. 143-174)
14. Bibliography
(pp. 175-176)
15. Index
(pp. 177-182)