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## The Closed Range Property for $$ \bar \partial $$ on Domains with Pseudoconcave Boundary

### Complex Analysis (2010-01-01): 307-320 , January 01, 2010

In this paper we study the
$$
\bar \partial
$$
-equation on domains with pseudoconcave boundary. When the domain is the annulus between two pseudoconvex domains in ℂ^{n}, we prove *L*^{2} existence theorems for
$$
\bar \partial
$$
for any
$$
\bar \partial
$$
-closed (*p, q*)- form with 1 ≤ *q* < *n*−1. We also study the critical case when *q* = *n*−1 on the annulus? and show that the space of harmonic forms is infinite dimensional. Some recent results and open problems on pseudoconcave domains in complex projective spaces are also surveyed.

## Zeros of Holomorphic Functions

### Complex Analysis (2007-01-01) 245: 191-211 , January 01, 2007

There are certain (classical families of) functions of a complex variable that mathematicians have studied frequently enough for them to acquire their own names. These functions are, of course, ones that develop naturally and repeatedly in various mathematical settings. Examples of such *named* functions include Euler’s Г-function, the Riemann ζ-function, and the Euler Φ-function. We will study only the first of these functions. There is a long history of synergy between the understanding of such functions and the development of complex analysis.

## Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions

### Complex Analysis (2013-01-01) 245: 139-169 , January 01, 2013

In this chapter we use the Cauchy theory to study functions that are holomorphic on an annulus, and analytic functions with isolated singularities. We describe a classification for isolated singularities. Functions that are holomorphic on an annulus have *Laurent series* expansions, an analogue of power series expansions for holomorphic functions on discs. Holomorphic functions with a finite number of isolated singularities in a domain can be integrated using the *residue theorem*, an analogue of the Cauchy integral formula. We discuss local properties of these functions. The study of zeros and poles of meromorphic functions leads to a theorem of Rouché that connects the number of zeros and poles to an integral. The theorem is not only aesthetically pleasing in its own right but also allows us to give alternate proofs of many important results. In the penultimate section of this chapter we illustrate the use of complex function theory in the evaluation of real definite integrals. The appendix discusses Cauchy principal values, a way to integrate functions with certain singularities.

## Approximation Theorems

### Complex Analysis (2001-01-01): 342-360 , January 01, 2001

In this chapter we prove two fundamental theorems, one “additive” and the other “multiplicative,” on prescribing zeros, and poles of meromorphic functions. The first is the Mittag-Leffler theorem, which asserts that we can prescribe the poles and principal parts of a meromorphic function. The second is the Weierstrass product theorem, which asserts that we can prescribe the zeros and poles, including orders, of a meromorphic function. The theorems are closely related. Both theorems are proved by the same type of approximation procedure, which depends on Runge’s theorem on approximation by rational functions. We prove Runge’s theorem in Section 1, followed by the Mittag-Leffler theorem in Section 2. In Section 3 we introduce infinite products, which can always be converted to infinite series by taking logarithms. We prove the Weierstrass product theorem in Section 4.

## Applications of Contour Integration

### Complex Analysis (2003-01-01): 153-181 , January 01, 2003

One of the very attractive features of complex analysis is that it can provide elegant and easy proofs of results in real analysis. Let us look again at Example 8.16.

## More about C(G) and H(G)

### Complex Analysis (1984-01-01): 33-37 , January 01, 1984

It is commonly prove in courses of advanced calculus that compact sets in $$ \mathbb{R} $$ (or more generally $$ {\mathbb{R}^n} $$ ) are characterized by being closed and bounded. In a general topological vector space only one implication is correct. Here and in the future we will assume that our topological vector spaces are Hausdorff.

## Final Remarks

### Complex Analysis (2003-01-01): 217-224 , January 01, 2003

The purpose of this very brief final chapter is to make the point that complex analysis is a living topic. The first section describes the Riemann Hypothesis, perhaps the most remarkable unsolved problem in mathematics. Because it requires a great deal of mathematical background even to understand the conjecture, it is not as famous as the Goldbach Conjecture (every even number greater than 2 is the sum of two prime numbers) or the Prime Pairs Conjecture (there are infinitely many pairs (*p, q*) of prime numbers with *q* = *p* + 2) but it is hugely more important than either of these, for a successful proof would have many, many consequences in analysis and number theory.

## Properties of Entire Functions

### Complex Analysis (2010-01-01) 0: 59-75 , January 01, 2010

We now show that if f is entire and if
$$ g(z) = \left\{ {\begin{array}{*{20}{c}} {f(z) - f(a)} & {z \ne a} \\{f'(a)} & {z = a} \\ \end{array} } \right. $$
then the Integral Theorem (4.15) and Closed Curve Theorem (4.16) apply to *g* as well as to *f*. (Note that since *f* is entire, *g* is continuous; however, it is not obvious that *g* is entire.)We begin by showing that the Rectangle Theorem applies to *g*.

## The Riemann Mapping Theorem

### Complex Analysis (1985-01-01) 103: 340-358 , January 01, 1985

In this chapter we give the general proof of the Riemann mapping theorem, and also state results concerning the behavior at the boundary.

## Values and growth of functions regular in the unit disk

### Complex Analysis (1977-01-01) 599: 68-75 , January 01, 1977

## On the distribution of values of meromorphic functions of slow growth

### Complex Analysis (1977-01-01) 599: 17-21 , January 01, 1977

## Power Series

### Complex Analysis (2013-01-01) 245: 39-80 , January 01, 2013

This chapter is devoted to an important tool for constructing holomorphic functions: convergent power series. It is the basis for the introduction of new non-algebraic holomorphic functions, the elementary transcendental functions. It turns out that power series play an even more central role in the theory of holomorphic functions, a role beyond enabling the construction of complex transcendental functions that are the extension of the real transcendental functions. A much stronger result holds. All holomorphic functions are (at least locally) convergent power series. This will be proven in the next chapter.

## The Residue Calculus

### Complex Analysis (2001-01-01): 195-223 , January 01, 2001

Section 1 is devoted to the residue theorem and to techniques for evaluating residues. In the remaining sections we apply the residue theorem to evaluate various real integrals. This material provides a good training ground for the techniques of complex integration. The student who is anxious to move on can skip the final several sections of the chapter at first reading.

## Complex Numbers and Functions

### Complex Analysis (1993-01-01) 103: 3-36 , January 01, 1993

One of the advantages of dealing with the real numbers instead of the rational numbers is that certain equations which do not have any solutions in the rational numbers have a solution in real numbers. For instance, *x*^{2} = 2 is such an equation. However, we also know some equations having no solution in real numbers, for instance *x*^{2} = −1, or *x*^{2} = −2. We define a new kind of number where such equations have solutions. The new kind of numbers will be called *complex* numbers.

## Cauchy’s Theorem

### Complex Analysis (2003-01-01): 107-117 , January 01, 2003

Prom Theorem 5.19, which can be seen as a complex version of the Fundamental Theorem of Calculus, we discern a strong tendency, when “reasonable” functions *f* and contours γ are involved, for ∫γ *f*(*z*) *dz* to be zero.

## Preliminaries: Set Theory and Topology

### Complex Analysis (1984-01-01): 1-12 , January 01, 1984

We assume familiarity with the rudiments of informal set theory including such notions as set, subset, superset, the null set Ɉ, the union or intersection of a family of sets, set difference (A\B), complement (compl A), Cartesian product; functions, domain, range, one-to-one, onto, image, inverse, restriction; partial ordering, linear (or total) ordering, and equivalence relation.

## Some Special Functions

### Complex Analysis (2001-01-01): 361-389 , January 01, 2001

Our aim in this chapter is to illustrate the power of complex analysis by proving a deep theorem in number theory, the prime number theorem, which does not appear at first glance to be related to complex analysis. Along the way we introduce various functions that play an important role in complex analysis. In Section 1 we introduce the gamma function Γ(*z*), which provides a meromorphic extension of the factorial function. We derive the asymptotic properties of the gamma function in Section 2 by viewing it as a Laplace transform. This yields Stirling’s asymptotic formula for *n*!. In Section 3 we study the zeta function, which is a meromorphic function whose zeros are related to the asymptotic distribution of prime numbers. In Section 4 we study Dirichlet series associated with various number-theoretic functions, thereby giving a strong hint of the fecund relationship between complex analysis and number theory. The proof of the prime number theorem is given in Section 5.

## Functions of the Complex Variable z

### Complex Analysis (2010-01-01) 0: 21-34 , January 01, 2010

We wish to examine the notion of a “function of z” where *z* is a complex variable. To be sure, a complex variable can be viewed as nothing but a pair of real variables so that in one sense a function of *z* is nothing but a function of two real variables. This was the point of view we took in the last section in discussing continuous functions. But somehow this point of view is too general. There are some functions which are “direct” functions of *z* = *x* + *iy* and not simply functions of the separate pieces *x* and *y*.

## Solutions to Exercises

### Complex Analysis (2003-01-01): 225-253 , January 01, 2003

## The Residue Theorem

### Complex Analysis (2010-01-01) 0: 129-142 , January 01, 2010

We now seek to generalize the Cauchy Closed Curve Theorem (8.6) to functions which have isolated singularities.

## Finite-type conditions for real hypersurfaces in ℂn

### Complex Analysis (1987-01-01) 1268: 83-102 , January 01, 1987

## Differential Calculus in the Complex Plane ℂ

### Complex Analysis (2005-01-01): 9-69 , January 01, 2005

## Complex Integration and Analyticity

### Complex Analysis (2001-01-01): 102-129 , January 01, 2001

In this chapter we take up the complex integral calculus. In Section 1 we introduce complex line integrals, and in Section 2 we develop the complex integral calculus, emphasizing the analogy with the usual one-variable integral calculus. In Section 3 we lay the cornerstone of the complex integral calculus, which is Cauchy’s theorem. The version we prove is an immediate consequence of Green’s theorem. In Section 4 we derive the Cauchy integral formula and use it to show that analytic functions have analytic derivatives. Each of the final four sections features a “named201” theorem. In Section 5 we prove Liouville’s theorem. In Section 6 we give a version of Morera’s theorem that provides a useful criterion for determining whether a continuous function is analytic. Sections 7 and 8, on Goursat’s theorem and the Pompeiu formula, can be omitted at first reading.

## Prelude to Complex Analysis

### Complex Analysis (2003-01-01): 35-49 , January 01, 2003

The development of real analysis (sequences, series, continuity, differentiation, integration) depends on a number of properties of the real number system. First, ℝ is a *field*, a set in which one may add, multiply, subtract and (except by 0) divide. Secondly, there is a notion of *distance*: given two numbers *a* and *b*, the distance between *a* and *b* is |*a* — *b*|. Thirdly, to put it very informally, ℝ has no gaps.

## Harmonic Functions and the Reflection Principle

### Complex Analysis (2001-01-01): 274-288 , January 01, 2001

In Section 1 we introduce the Poisson kernel function and we develop the Poisson integral representation for harmonic functions on the open unit disk. The Poisson kernel is the analogue for harmonic functions of the Cauchy kernel for analytic functions, and the Poisson integral formula solves the Dirichlet problem for the unit disk. In Section 2 we use this solution to characterize harmonic functions by the mean value property. This characterization is the analogue of Morera’s theorem characterizing analytic functions. In Section 3 we apply the characterization of harmonic functions to establish the Schwarz reflection principle for harmonic functions. The reflection principle plays a key role in the study of boundary behavior of conformal maps.

## Runge’s Theorem

### Complex Analysis (1984-01-01): 77-83 , January 01, 1984

If f ∈ *H*(G), G a connected open set, it is a consequence of the power series expansion for holomorphic functions that if f(z_{n}) = 0, z_{n} → z_{0} ∈ G then f = 0 in G. It is also a consequence that if f^{(n)}(z_{0}) = 0 for n = 0,1,2,…, then f = 0 in G. We adopt conventions about “sets with multiplicity” that allow us to treat both cases as one.

## Compact Hausdorff Transversally Holomorphic Foliations

### Complex Analysis (1982-01-01) 950: 360-376 , January 01, 1982

This is a revised version of part of the lectures given by the author at Trieste Seminar on Complex Analysis and its Applications. The last part of this paper gives a report on the results obtained by Girbau-Haefliger-Sundararaman subsequent to the Seminar. The author would like to thank Professor A. Haefliger for his suggestions. The remaining part of the lectures of the author has appeared in [58]. The author thanks Centro de Investigación del I.P.N., México City, for hospitality during the writing of the paper.

## Recent progress and future directions in several complex variables

### Complex Analysis (1987-01-01) 1268: 1-23 , January 01, 1987

## Winding Numbers and Cauchy’s Theorem

### Complex Analysis (1993-01-01) 103: 133-155 , January 01, 1993

We wish to give a general global criterion when the integral of a holomorphic function along a closed path is O. In practice, we meet two types of properties of paths: (1) properties of homotopy, and (2) properties having to do with integration, relating to the number of times a curve “winds” around a point, as we already saw when we evaluated the integral
$$ \int {\frac{1}{{\zeta - z}}} d\zeta $$
along a circle centered at *z*. These properties are of course related, but they also exist independently of each other, so we now consider those conditions on a closed path *γ* when
$$ \int_{\gamma } {f = 0} $$
for all holomorphic functions *f*, and also describe what the value of this integral may be if not 0.

## The Dirichlet Problem

### Complex Analysis (2001-01-01): 390-417 , January 01, 2001

In Chapter X we used the Poisson kernel to solve the Dirichlet problem for the unit disk. In this chapter we study the Dirichlet problem for more general domains in the plane. The basic method, due to O. Perron, is to look for the solution of the Dirichlet problem as the upper envelope of a family of subsolutions. In Section 2 we introduce subharmonic functions, which play the role of the subsolutions. In Section 3 we derive Harnack’s inequality, which provides a compactness criterion for families of harmonic functions. Perron’s procedure for solving the Dirichlet problem is developed in Section 4. We apply the method to give another proof of the Riemann mapping theorem in Section 5. In Sections 6 and 7 we introduce Green’s function.

## Back Matter - Complex Analysis

### Complex Analysis (2003-01-01) , January 01, 2003

## On the zeros of the successive derivatives of integral functions II

### Complex Analysis (1977-01-01) 599: 109-116 , January 01, 1977

In 1936 the author proved [2] the following: *Theorem 1*. *If* f(z) (≢ 0) *is entire of exponential type δ such that each* f^{(v)}(x) (*v*=0, 1, …) *vanishes somewhere in the interval* I_{1}=[0,½] *of the real axis, then*
1
$$\delta \geqslant \pi ,$$
*and the function*
2
$$f(z) = CoS \pi z$$
*shows that π is the best constant, because* cos πz *satisfies all conditions*.

## Front Matter - Complex Analysis

### Complex Analysis (1985-01-01): 103 , January 01, 1985

## Applications of the Residue Theorem to the Evaluation of Integrals and Sums

### Complex Analysis (2010-01-01) 0: 143-160 , January 01, 2010

In the next section, we will see how various types of (real) definite integrals can be associated with integrals around closed curves in the complex plane, so that the Residue Theorem will become a handy tool for definite integration.

## Front Matter - Complex Analysis

### Complex Analysis (1993-01-01): 103 , January 01, 1993

## The Complex Plane and Elementary Functions

### Complex Analysis (2001-01-01): 1-32 , January 01, 2001

In this chapter we set the scene and introduce some of the main characters. We begin with the three representations of complex numbers: the Cartesian representation, the polar representation, and the spherical representation. Then we introduce the basic functions encountered in complex analysis: the exponential function, the logarithm function, power functions, and trigonometric functions. We view several concrete functions *w = f (z)* as mappings from the *z*-plane to the *w*-plane, and we consider the problem of describing the inverse functions.

## Nine Lectures on Comples Analysis

### Complex Analysis (2011-01-01) 62: 1-175 , January 01, 2011

a) Let Ω be an open set in ℂ^{n}, the Cartesian product of n copies of the complex field ℂ, with coordinate functions Z_{1},.…,Z_{n}.

## Front Matter - Complex Analysis

### Complex Analysis (2001-01-01) , January 01, 2001

## Front Matter - Complex Analysis

### Complex Analysis (2009-01-01) , January 01, 2009

## Conformal Mapping

### Complex Analysis (2001-01-01): 289-314 , January 01, 2001

In this chapter we will be concerned with conformal maps from domains onto the open unit disk. One of our goals is the celebrated Riemann mapping theorem: Any simply connected domain in the complex plane, except the entire complex plane itself, can be mapped conformally onto the open unit disk. We begin in Section 1 by reviewing and enlarging our repertoire of conformal maps onto the open unit disk, or equivalently, onto the upper half-plane. In Section 2 we state and discuss the Riemann mapping theorem. Before embarking on the proof, we give some applications to the conformal mapping of polygons in Section 3 and to fluid dynamics in Section 4. In Section 5 we develop some prerequisite material concerning compactness of families of analytic functions, which is at a deeper level than the analysis used up to this point. The proof of the Riemann mapping theorem follows in Section 6.

## Solutions to the Exercises

### Complex Analysis (2009-01-01): 1-59 , January 01, 2009

## Harmonic Functions

### Complex Analysis (2007-01-01) 245: 173-190 , January 01, 2007

This chapter is devoted to the study of harmonic functions. These functions are closely connected to holomorphic maps since the real and imaginary parts of a holomorphic function are harmonic functions. The study of harmonic functions is important in physics and engineering, and there are many results in the theory of harmonic functions that are not connected directly with complex analysis. However, in this chapter we consider that part of the theory of harmonic functions that grows out of the Cauchy Theory. Mathematically this is quite pleasing.

## The Cauchy Theorem

### Complex Analysis (1984-01-01): 84-95 , January 01, 1984

Runge’s Theorem can be used to prove Cauchy’s Theorem. This will require the elements of integration theory described in Chapter 2. Recall that for a rectifiable curve γ: [0,1] → ℂ, we let ‖γ‖ denote its length, i.e. ‖γ‖ = <Inline>1</Inline> |γ′(t)|dt, so that $$ \left| {\int_{\gamma } {f(z)dz} } \right| \leqslant \left\| \gamma \right\| \cdot \left\| f \right\| $$ where ‖f‖ = sup{| f(z) |: z ∈ γ}. The reader is reminded that γ^ denotes the “physical curve”, that is the image of γ.

## Front Matter - Complex Analysis

### Complex Analysis (2005-01-01) , January 01, 2005

## Riemann Surfaces

### Complex Analysis (2001-01-01): 418-446 , January 01, 2001

Our goal in this chapter is to prove the uniformization theorem for Riemann surfaces and to indicate its usefulness as a tool in complex analysis. We begin in Sections 1 and 2 by defining Riemann surfaces, providing examples, and showing how various local notions as analytic function, meromorphic function, and harmonic function carry over to Riemann surfaces. In Sections 3 and 4 we define Green’s function for Riemann surfaces and show that Green’s function is symmetric. In Section 5 we show that every Riemann surface has bipolar Green’s functions. We prove the uniformization theorem in Section 6. The proof depends on Green’s function when it exists and on bipolar Green’s function otherwise. In Section 7 we define covering spaces and covering maps, and we state several results that indicate the power of the uniformization theorem.

## Applications of the Maximum Modulus Principle and Jensen’s Formula

### Complex Analysis (1993-01-01) 103: 323-355 , January 01, 1993

We return to the maximum principle in a systematic way, and give several ways to apply it, in various contexts.

## Conformal Mappings

### Complex Analysis (1993-01-01) 103: 208-236 , January 01, 1993

In this chapter we consider a more global aspect of analytic functions, describing geometrically what their effect is on various regions. Especially important are the analytic isomorphisms and automorphisms of various regions, of which we consider many examples.

## Calculus of Residues

### Complex Analysis (1985-01-01) 103: 165-195 , January 01, 1985

We have established all the theorems needed to compute integrals of analytic functions in terms of their power series expansions. We first give the general statements covering this situation, and then apply them to examples.

## Front Matter - Complex Analysis

### Complex Analysis (1982-01-01): 950 , January 01, 1982

## Factorization of meromorphic functions and some open problems

### Complex Analysis (1977-01-01) 599: 51-67 , January 01, 1977

## Front Matter - Complex Analysis

### Complex Analysis (1984-01-01) , January 01, 1984

## Invariance of the Parametric Oka Property

### Complex Analysis (2010-01-01): 125-144 , January 01, 2010

Assume that *E* and *B* are complex manifolds and that π : *E* → *B* is a holomorphic Serre fibration such that *E* admits a finite dominating family of holomorphic fiber-sprays over a small neighborhood of any point in *B*. We show that the parametric Oka property (POP) of *B* implies POP of *E*; conversely, POP of *E* implies POP of *B* for contractible parameter spaces. This follows from a parametric Oka principle for holomorphic liftings which we establish in the paper.

## Further Contour Integral Techniques

### Complex Analysis (2010-01-01) 0: 161-168 , January 01, 2010

We have already seen how the Residue Theorem can be used to evaluate real line integrals. The techniques involved, however, are in noway limited to real integrals. To evaluate an integral along any contour, we can always switch to a more “convenient” contour as long as we account for the pertinent residues of the integrand.

## Back Matter - Complex Analysis

### Complex Analysis (2007-01-01): 245 , January 01, 2007

## Applications of the Maximum Modulus Principle

### Complex Analysis (1985-01-01) 103: 255-275 , January 01, 1985

We return to the maximum principle in a systematic way, and give several ways to apply it, in various contexts.

## Integral Calculus in the Complex Plane ℂ

### Complex Analysis (2005-01-01): 71-104 , January 01, 2005

## Elliptic Functions

### Complex Analysis (2009-01-01): 1-65 , January 01, 2009

Historically, the starting point of the theory of elliptic functions were the *elliptic integrals*, named in this way because of their direct connection to computing arc lengths of ellipses. Already in 1718 (G.C. FAGNANO), a very special elliptic integral was extensively investigated,
$$E(x) : = \int^{x}_{0} \frac{dt}{\sqrt{1-t^4}}.$$
It represents in the interval ]0, 1[ a strictly increasing (continuous) function. So we can consider its inverse function *f*. A result of N.H. ABEL (1827) affirms that *f* has a meromorphic continuation into the entire C. In addition to an obvious real period, ABEL discovered a hidden complex period. So the function *f* turned out to be a *doubly periodic function*. Nowadays, a meromorphic function in the plane with two independent periods is also called *elliptic*. Many results that were already know for the elliptic integral, as for instance the famous EULER Addition Theorem for elliptic integrals, appeared to be surprisingly simple corollaries of properties of elliptic functions. This motivated K. WEIERSTRASS to turn the tables. In his lectures in the winter term 1862/1863 he gave a purely function theoretical introduction to the theory of elliptic functions. In the center of this new setup, there is a special function, the ℘-function. It satisfies a differential equation which immediately shows the inverse function of ℘ to be an elliptic integral. The theory of elliptic integrals was thus derived as a byproduct of the theory of elliptic functions.

## Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions

### Complex Analysis (2007-01-01) 245: 101-121 , January 01, 2007

In this chapter we use the Cauchy Theory to study functions that are holomorphic on an annulus and analytic functions with isolated singularities. We describe a classification for isolated singularities. Functions that are holomorphic on an annulus have *Laurent series* expansions, an analog of power series expansions for holomorphic functions on disks. Holomorphic functions with a finite number of isolated singularities in a domain can be integrated using the *Residue Theorem*, an analog of the Cauchy Integral Formula. We discuss the local properties of these functions.

## Constructive Function Theory

### Complex Analysis (1984-01-01): 96-107 , January 01, 1984

The goal of this section is the construction, by means of sums and products of simpler functions, of holomorphic functions with prescribed behavior. In what follows, when we speak of a sequence of complex numbers we ordinarily mean a sequence with multiplicity, so that a function taking some value at a point of the sequence must take that value with the appropriate multiplicity. We also exclude the trivial function f ≡ 0 unless otherwise noted.

## First-Order Conformal Invariants

### Complex Analysis (1984-01-01): 168-174 , January 01, 1984

The Theme of this section is the following: Suppose you find yourself on a plane domain, with only a restricted logic at your disposal; how closely can you determine which domain you are on—up to conformal equivalence? This leads to a study of a system of conformal invariants, the first-order conformal invariants (FOCI), which are obtained from the elementary properties of the algebra (or ring) of analytic functions on plane domains. Although the formal definition of FOCI is given in the terminology of mathematical logic, these invariants are nonetheless all included within the framework of classical function theory. Each of the FOCI corresponds to an elementary assertion about analytic functions that can be understood without any knowledge of mathematical logic.

## Cauchy’s Theorem, First Part

### Complex Analysis (1985-01-01) 103: 87-122 , January 01, 1985

Let [*a, b*] be a closed interval of real numbers. By a curve *γ* (defined on this interval) we mean a function
$$\gamma :[a,b] \to C$$
which we assume to be of class *C*^{1}.

## Plurisubharmonic functions on ring domains

### Complex Analysis (1987-01-01) 1268: 111-120 , January 01, 1987

## The Dual of H(G)

### Complex Analysis (1984-01-01): 67-76 , January 01, 1984

We want to prove, as in the case of the disk, that *H*(G)^{*} = *H*_{0}(ℂ \ G). We first study the dual of *C*(G). We change our notation here and write L(f) = ∫ fdμ when L ∈ *C*(G)^{*}. (For the reader unfamiliar with integration theory this is simply a change in notation: The left-hand side defines the right-hand side. There are two advantages to this notation. First, it is the notation in which research papers are written. Second, the reader can call upon her experience with integration for intuition. For the mathematically advanced reader: we are invoking the Riesz Representation Theorem for *C*(G)^{*}.) We call μ the “measure” associated with L, and we may identify μ and L. The collection of all such μ is denoted M_{0}(G), so that M_{0}(G) = *C*(G)^{*}. We also write L(f) = ∫ f(z)dμ(z) when it is necessary to indicate the independent variable. “Measures” have the same properties as continuous linear functionals (which is what they are); for reinforcement, we list them here. Given μ ∈ M_{0}(G):
i)

∫ (f + g)dμ = ∫ fdμ + ∫ gdμ, f, g ∈ *C*(G).

∫ afdμ = a ∫ fdμ, f ∈ *C*(G), a ∈ ℂ.

If f_{n} → f in *C*(G) then ∫ f_{n}dμ → ∫ fdμ.

There is a compact set K ⊆ G such that | ∫ fdμ | ≤ C‖f‖_{K} for all f ∈ *C*(G).

## Front Matter - Complex Analysis

### Complex Analysis (1985-01-01): 103 , January 01, 1985

## Subelliptic Estimates

### Complex Analysis (2010-01-01): 75-94 , January 01, 2010

This paper gathers old and new information about subelliptic estimates for the $$ \bar \partial $$ -Neumann problem on smoothly bounded pseudoconvex domains. It discusses the failure of effectiveness of Kohn’s algorithm, gives an algorithm for triangular systems, and includes some new information on sharp subelliptic estimates.

## Proper mappings between balls in Cn

### Complex Analysis (1987-01-01) 1268: 66-82 , January 01, 1987

## Front Matter - Complex Analysis

### Complex Analysis (1977-01-01): 599 , January 01, 1977

## Front Matter - Complex Analysis

### Complex Analysis (2003-01-01) , January 01, 2003

## Back Matter - Complex Analysis

### Complex Analysis (1987-01-01): 1268 , January 01, 1987

## Introduction to Conformal Mapping

### Complex Analysis (2010-01-01) 0: 169-194 , January 01, 2010

In this chapter, we take a closer look at themapping properties of an analytic function. Throughout the chapter, all curves *z*(*t*) are assumed to be such that
$$ z(t) \ne 0 $$
for all *t*.

## Integral Calculus in the Complex Plane C

### Complex Analysis (2009-01-01): 1-33 , January 01, 2009

In Section I.5 we already encountered the problem of finding a primitive function for a given analytic function $$f:D \to \mathcal{C}, D\subset \mathcal{C}$$ open, i. e., an analytic function $$F:D\to \mathcal{C}$$ such that $$F^\prime = f.$$

In general, one may ask: Which functions
$$f : D \to \mathcal{C}, D \subset \mathcal{C}$$
open, have a primitive? Recall that in the real case any *continuous* function
$$f : [a, b] \to \mathcal{R}, a < b$$
, has a primitive, namely, for example the integral
$$F(x): = \int^{x}_{a} f(t) dt.$$
Whether one uses the notion of a RIEMANN integral or the integral for regulated functions is irrelevant in this connection.

## Power Series

### Complex Analysis (2007-01-01) 245: 23-57 , January 01, 2007

This chapter is devoted to an important method for constructing holomorphic functions. The tool is convergent power series. It is the basis for the introduction of new non-algebraic holomorphic functions, called elementary transcendental functions. It will turn out that all holomorphic functions are described (at least locally) by this tool. This will be proven in the next chapter.

## The points of maximum modulus of a univalent function

### Complex Analysis (1977-01-01) 599: 96-100 , January 01, 1977

## Conformal Equivalence and Hyperbolic Geometry

### Complex Analysis (2013-01-01) 245: 199-228 , January 01, 2013

In this chapter, we study *conformal maps* between domains in the extended complex plane
$$\widehat{\mathbb{C}}$$
; these are one-to-one meromorphic functions. Our goal here is to characterize all simply connected domains in the extended complex plane. The first two sections of this chapter study the action of a quotient of the group of two-by-two nonsingular complex matrices on the extended complex plane, namely, the group PSL(2, *ℂ*) and the projective special linear group. This group is also known as the Möbius group. In the third section we characterize simply connected proper domains in the complex plane by establishing the Riemann mapping theorem (RMT). This extraordi- nary theorem tells us that there are conformal maps between any two such domains.

## Values shared by an entire function and its derivative

### Complex Analysis (1977-01-01) 599: 101-103 , January 01, 1977

## Deformations of Complex Structures and Holomorphic Vector Bundles

### Complex Analysis (1982-01-01) 950: 196-209 , January 01, 1982

The aim of these lectures is to give an introduction to the theory of deformations of complex structures as developed by Kodaira and Spencer. The theory studies complex structures which are near a given complex structure on a compact differentiable manifold. One also has an analogous theory of deformations of holomorphic vector bundles on a compact complex manifold.

## Ideals in H(G)

### Complex Analysis (1984-01-01): 108-116 , January 01, 1984

We use the results of the previous section to derive some descriptive results on ideals of holomorphic functions.

## Duality of H(G)—The Case of the Unit Disc

### Complex Analysis (1984-01-01): 44-50 , January 01, 1984

We begin with a general result about linear functionals on a locally convex topological vector space. Let E have the topology generated by a family *P* of seminorms. For each non-empty finite set A = {‖•‖_{1}, ‖•‖_{2},…, ‖•‖_{n}} ⊂ *P*, define
$$ {\left\| x \right\|_A} = \mathop{{\max }}\limits_{{l \leqslant j \leqslant n}} \,{\left\| x \right\|_j} $$
, x ∈ E. Then ‖•‖_{A} is a seminorm. Let *P̃* = *P* ∪ {‖•‖_{A}: A is a non empty finite subset of *P*}; then *P* and *P̃* generate the same topology on E (Exercise 2). Consequently, we may assume *P* = *P̃* in the following proposition.

## Further Properties of Analytic Functions

### Complex Analysis (2010-01-01) 0: 93-105 , January 01, 2010

The Uniqueness Theorem (6.9) states that a non-constant analytic function in a region cannot be constant on any open set. Similarly, according to Proposition 3.7, |*f*| cannot be constant. Thus a non-constant analytic function cannot map an open set into a point or a circular arc. By applying the Maximum-Modulus Theorem, we can derive the following sharper result on the mapping properties of an analytic function.

## A uniqueness theorem with application to the Abel series

### Complex Analysis (1977-01-01) 599: 11-16 , January 01, 1977

## Oblique Polar Lines of ∫ X |f|2λ|g|2μ□

### Complex Analysis (2010-01-01): 1-23 , January 01, 2010

Existence of oblique polar lines for the meromorphic extension of the current valued function ∫|*f*|^{2λ}|*g*|^{2μ}□ is given under the following hypotheses: *f* and *g* are holomorphic function germs in ℂ^{n+1} such that *g* is non-singular, the germ Σ := {d*f* ∧ d*g* = 0} is one dimensional, and *g* is proper and finite on S := {d*f* = 0}. The main tools we use are interaction of strata for *f* (see [4]), monodromy of the local system *H*^{n-1} (*u*) on *S* for a given eigenvalue exp(−2*iπu*) of the monodromy of *f*, and the monodromy of the cover *g*S. Two non-trivial examples are completely worked out.

## Back Matter - Complex Analysis

### Complex Analysis (2001-01-01) , January 01, 2001

## Entire and Meromorphic Functions

### Complex Analysis (1993-01-01) 103: 356-373 , January 01, 1993

A function is said to be *entire* if it is analytic on all of *C*. It is said to be *meromorphic* if it is analytic except for isolated singularities which are poles. In this chapter we describe such functions more closely. We develop a multiplicative theory for entire functions, giving factorizations for them in terms of their zeros, just as a polynomial factors into linear factors determined by its zeros. We develop an additive theory for meromorphic functions, in terms of their principal part (polar part) at the poles.

## The Riemann Mapping Theorem

### Complex Analysis (2010-01-01) 0: 195-214 , January 01, 2010

Before proving the Riemann Mapping Theorem, we examine the relation between conformal mapping and the theory of fluid flow. Our main goal is to motivate some of the results of the next section and the treatment here will be less formal than that of the remainder of the book.

## The Cauchy Theory: Key Consequences

### Complex Analysis (2013-01-01) 245: 119-137 , January 01, 2013

This chapter is devoted to some immediate consequences of the fundamental result for the Cauchy theory, Theorem 4.61, of the last chapter. Although the chapter is very short, it includes proofs of many of the implications of the fundamental theorem in complex function theory (Theorem 1.1). We point out that these relatively compact proofs of a host of major theorems result from the work put into Chap. 4 and earlier chapters.

## Sequences and Series of Analytic Functions, the Residue Theorem

### Complex Analysis (2005-01-01): 105-194 , January 01, 2005

## Back Matter - Complex Analysis

### Complex Analysis (1993-01-01): 103 , January 01, 1993

## Laurent Series and the Residue Theorem

### Complex Analysis (2003-01-01): 137-152 , January 01, 2003

In Section 3.5 we looked briefly at functions with isolated singularities. It is clear that a function *f* with an isolated singularity at a point *c* cannot have a Taylor series centred on *c*. What it does have is a *Laurent*^{1} series, a generalized version of a Taylor series in which there are negative as well as positive powers of *z* — *c*.

## Properties of C(G) and H(G)

### Complex Analysis (1984-01-01): 27-32 , January 01, 1984

With little change we can study functions of several variables. To keep the notation simple, we will restrict ourselves to two variables. In that case, G is an open set in ℂ × ℂ = ℂ^{2}. Then *C*(G) is defined just as in the one-variable case. (The distance in ℂ^{2} between two points (z,w) and (z’,w’) will be denoted d((z,w), (z’,w’)) = (|z - z’|^{2} + |w - w’|^{2})^{½}. We also define *H*(G) as before, but must first define *holomorphic*.

## Line Integrals and Entire Functions

### Complex Analysis (2010-01-01) 0: 45-57 , January 01, 2010

Recall that, according to Theorem 2.9, an everywhere convergent power series represents an entire function.Ourmain goal in the next two chapters is the somewhat surprising converse of that result: namely, that *every* entire function can be expanded as an everywhere convergent power series. As an immediate corollary, we will be able to prove that every entire function is infinitely differentiable. To arrive at these results, however, we must begin by discussing integrals rather than derivatives.

## The Riemann Mapping Theorem

### Complex Analysis (1984-01-01): 117-123 , January 01, 1984

The Riemann Mapping Theorem implies that, as far as *H(G)* can tell, all simply connected regions are the “same”. To clarify what this means we need the following notion of equivalence.