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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

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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

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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.

## Properties of Entire Functions

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

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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*.

## Functions of the Complex Variable z

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

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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*.

## The Residue Theorem

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

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

## Differential Calculus in the Complex Plane ℂ

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

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

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

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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.

## Nine Lectures on Comples Analysis

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

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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 (2009-01-01) , January 01, 2009

## Solutions to the Exercises

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

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## Harmonic Functions

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

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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.

## Front Matter - Complex Analysis

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

## Invariance of the Parametric Oka Property

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

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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

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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

## Integral Calculus in the Complex Plane ℂ

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

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## Elliptic Functions

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

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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

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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.

## Subelliptic Estimates

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

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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.

## Introduction to Conformal Mapping

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

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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

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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

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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.

## Further Properties of Analytic Functions

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

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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.

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

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

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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.

## The Riemann Mapping Theorem

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

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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.

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

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

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## Line Integrals and Entire Functions

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

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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.

## Maximum-Modulus Theorems for Unbounded Domains

### Complex Analysis (2010-01-01) 0: 215-223 , January 01, 2010

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The Maximum-Modulus Theorem (6.13) shows that a function which is *C*-analytic in a compact domain *D* assumes its maximum modulus on the boundary. In general, if we consider unbounded domains, the theorem no longer holds. For example,
$$ f(z) = e^z $$
is analytic and unbounded in the right half-plane despite the fact that on the boundary
$$ |e^z | = e^{iy} | = 1 $$
. Nevertheless, given certain restrictions on the growth of the function, we can conclude that it attains its maximum modulus on the boundary. The most natural such condition is that the function remain bounded throughout *D*.

## The Complex Numbers

### Complex Analysis (2010-01-01) 0: 1-20 , January 01, 2010

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Numbers of the form
$$ a + b\sqrt { - 1} $$
, where *a* and *b* are real numbers—what we call complex numbers.appeared as early as the 16th century. Cardan (1501–1576) worked with complex numbers in solving quadratic and cubic equations. In the 18th century, functions involving complex numberswere found by Euler to yield solutions to differential equations. As more manipulations involving complex numbers were tried, it became apparent that many problems in the theory of real-valued functions could be most easily solved using complex numbers and functions. For all their utility, however, complex numbers enjoyed a poor reputation and were not generally considered legitimate numbers until the middle of the 19th century. Descartes, for example, rejected complex roots of equations and coined the term “gimaginary” for such roots. Euler, too, felt that complex numbers “exist only in the imagination” and considered complex roots of an equation useful only in showing that the equation actually has *no* solutions.

## Front Matter - Complex Analysis

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

## Back Matter - Complex Analysis

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

## Remarks on the Homogeneous Complex Monge-Ampère Equation

### Complex Analysis (2010-01-01): 175-185 , January 01, 2010

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We refine the arguments in [12] to show that the extended norm of Bedford-Taylor is in fact exact the same as the original Chern-Levine-Nirenberg intrinsic norm, thus provides a proof of the Chern-Levine-Nirenberg conjecture. The result can be generalized to deal with homogeneous Monge-Ampère equation on any complex manifold.

## The Mixed Case of the Direct Image Theorem and its Applications

### Complex Analysis (2011-01-01) 62: 281-463 , January 01, 2011

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In these lectures we will discuss the so-called mixed case of the direct image theorem and its applications. The starting point of the direct image theorem is the following finiteness theorem.

## Harmonic Functions

### Complex Analysis (2010-01-01) 0: 225-239 , January 01, 2010

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In this chapter, we focus on the real parts of analytic functions and their connection with real harmonic functions.

## Construction of Analytic Functions

### Complex Analysis (2005-01-01): 195-255 , January 01, 2005

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## Front Matter - Complex Analysis

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

## A Radó Theorem for Locally Solvable Structures of Co-rank One

### Complex Analysis (2010-01-01): 187-203 , January 01, 2010

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We extend the classical theorem of Radó to locally solvable structures of co-rank one. One of the main tools in the proof is a refinement of the Baouendi-Treves approximation theorem that may be of independent interest.

## Compactness Estimates for the $$ \bar \partial $$ -Neumann Problem in Weighted L 2-spaces

### Complex Analysis (2010-01-01): 159-174 , January 01, 2010

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In this paper we discuss compactness estimates for the
$$
\bar \partial
$$
-Neumann problem in the setting of weighted *L*^{2}-spaces on ℂ^{n}. For this purpose we use a version of the Rellich-Lemma for weighted Sobolev spaces.

## Analytic Number Theory

### Complex Analysis (2009-01-01): 1-68 , January 01, 2009

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Analytic number theory contains one of the most beautiful applications of complex analysis. In the following sections we will treat some of the distinguished pearls of this fascinating subject.

We have already seen in Sect. VI.4 that quadratic forms or the corresponding lattices can serve to construct modular forms. The Fourier coefficients of the theta series associated to quadratic forms or to lattices have number theoretical importance. They appear as representation numbers for quadratic forms, and respectively as numbers of lattice points of a given norm. Due to the general structure theorems for modular forms, one can bring together theta series and Eisenstein series. We will compute the Fourier coefficients of the Eisenstein series, obtaining thus number theoretical applications. In particular, in Sect. VII.1 we will find the number of representations of a natural number as a sum of four or eight squares by purely function theoretical means.

Starting with the second section, we will be concerned with Dirichlet series, including the Riemann *ζ*-function. There is a strong connection between modular forms and Dirichlet series (Sect. VII.3). We prove Hecke’s Theorem, claiming a one-to-one correspondence between Dirichlet series with a functional equation of special type and Fourier series with special transformation property under the substitution
$$z \mapsto - 1/z$$
and with certain asymptotic growth conditions. This correspondence will be obtained by means of the Mellin transform of the *G*-function. As an application, we obtain in particular the analytic continuation of the *ζ*-function into the plane, and also its functional equation.

## Conformal Equivalence

### Complex Analysis (2007-01-01) 245: 147-172 , January 01, 2007

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In this chapter we study *conformal maps* between domains in the extended complex plane. These maps are one-to-one meromorphic functions. Our goal is characterize all simply connected domains in the 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

## Different Forms of Analytic Functions

### Complex Analysis (2010-01-01) 0: 241-256 , January 01, 2010

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The analytic functions we have encountered so far have generally been defined either by power series or as a combination of the elementary polynomial, trigonometric and exponential functions, alongwith their inverse functions. In this chapter, we consider three different ways of representing analytic functions. We begin with infinite products and then take a closer look at functions defined by definite integrals, a topic touched upon earlier in Chapter 7 and in Chapter 12.2. Finally, we define Dirichlet series, which provide a link between analytic functions and number theory.

## Applications of a Parametric Oka Principle for Liftings

### Complex Analysis (2010-01-01): 205-211 , January 01, 2010

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A parametric Oka principle for liftings, recently proved by Forstnerič, provides many examples of holomorphic maps that are fibrations in a model structure introduced in previous work of the author. We use this to show that the basic Oka property is equivalent to the parametric Oka property for a large class of manifolds. We introduce new versions of the basic and parametric Oka properties and show, for example, that a complex manifold *X* has the basic Oka property if and only if every holomorphic map to *X* from a contractible submanifold of ℂ^{n} extends holomorphically to ℂ^{n}.

## New Normal Forms for Levi-nondegenerate Hypersurfaces

### Complex Analysis (2010-01-01): 321-340 , January 01, 2010

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In this paper we construct a large class of new normal forms for Levi-nondegenerate real hypersurfaces in complex spaces. We adopt a general approach illustrating why these normal forms are natural and which role is played by the celebrated Chern-Moser normal form. The latter appears in our class as the one with the “maximum normalization” in the lowest degree. However, there are other natural normal forms, even with normalization conditions for the terms of the same degree. Some of these forms do not involve the cube of the trace operator and, in that sense, are simplier than the one by Chern-Moser. We have attempted to give a complete and self-contained exposition (including proofs of well-known results about trace decompositions) that should be accessible to graduate students.

## On the Subellipticity of Some Hypoelliptic Quasihomogeneous Systems of Complex Vector Fields

### Complex Analysis (2010-01-01): 109-123 , January 01, 2010

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For about twenty five years it was a kind of folk theorem that complex vector-fields defined on Ω × ℝ_{t} (with Ω open set in ℝ^{n}) by
$$
L_j = \frac{\partial }
{{\partial t_j }} + i\frac{{\partial \phi }}
{{\partial t_j }}(t)\frac{\partial }
{{\partial x}},j = 1, \ldots ,n, t \in \Omega ,x \in \mathbb{R},
$$
with φ analytic, were subelliptic as soon as they were hypoelliptic. This was indeed the case when *n* = 1 [Tr1] but in the case *n* > 1, an inaccurate reading of the proof (based on a non standard subelliptic estimate) given by Maire [Mai1] (see also Trèves [Tr2]) of the hypoellipticity of such systems, under the condition that φ does not admit any local maximum or minimum, was supporting the belief for this folk theorem. This question reappears in the book of [HeNi] in connection with the semi-classical analysis of Witten Laplacians. Quite recently, J.L. Journé and J.M. Trépreau [JoTre] show by explicit examples that there are very simple systems (with polynomial φ’s) which were hypoelliptic but not subelliptic in the standard *L*^{2}-sense. But these operators are not quasihomogeneous.

In [De] and [DeHe] the homogeneous and the quasihomogeneous cases were analyzed in dimension 2. Large classes of systems for which subellipticiity can be proved were exhibited. We will show in this paper how a new idea for the construction of escaping rays permits to show that in the analytic case *all* the quasihomogeneous hypoelliptic systems in the class above considered by Maire are effectively subelliptic in the 2-dimensional case. The analysis presented here is a continuation of two previous works by the first author for the homogeneous case [De] and the two authors for the quasihomogeneous case [DeHe].

## Front Matter - Complex Analysis

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

## Construction of Analytic Functions

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

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In this (central) chapter, we are concerned with the *construction of analytic functions*. We will meet three different construction principles.
(1)

We first study in detail a classical function using methods of the function theory of one complex variable, namely the *Gamma function*.

We treat the theorems of WEIERSTRASS and MITTAG-LEFFLER for the construction of analytic functions with prescribed zeros and respectively poles with specified principle parts.

(3)We prove the REIMANN Mapping Theorem, which claims that an elementary domain *D* ≠ C can be conformally mapped onto the unit disk E. In this context we will once more review the Cauchy Integral Theorem, prove more general variants of it, and gain different topological characterizations of elementary domains as regions “without holes”.

## The Cauchy Theory–Key Consequences

### Complex Analysis (2007-01-01) 245: 83-99 , January 01, 2007

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This chapter is devoted to some immediate consequences of the Fundamental Theorem for Cauchy Theory, Theorem 4.52, of the last chapter. Although the chapter is very short, it includes proofs of many of the implications of the Fundamental Theorem 1.1. We point out that these relatively compact proofs of a host of major theorems result from the work put in Chapter 4 and earlier chapters.

## Back Matter - Complex Analysis

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

## Analytic Continuation; The Gamma and Zeta Functions

### Complex Analysis (2010-01-01) 0: 257-272 , January 01, 2010

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Suppose we are given a function *f* which is analytic in a region *D*. We will say that *f* can be continued analytically to a region *D*_{1} that intersects *D* if there exists a function *g*, analytic in *D*_{1} and such that *g* = *f* throughout
$$ D_1 \cap D_2 $$
. By the Uniqueness Theorem (6.9) any such continuation of *f* is uniquely determined. (It is possible, however, to have two analytic continuations *g*_{1} and *g*_{2} of a function *f* to regions *D*_{1} and *D*_{2} respectively with
$$ g_1 \ne g_2 $$
throughout
$$ D_1 \cap D_2 $$
. See Exercise 1.)

## Elliptic Functions

### Complex Analysis (2005-01-01): 257-325 , January 01, 2005

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## Isolated Singularities of an Analytic Function

### Complex Analysis (2010-01-01) 0: 117-128 , January 01, 2010

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*Introduction* While we have concentrated until now on the general properties of analytic functions, we now focus on the special behavior of an analytic function in the neighborhood of an “isolated singularity.”

## Gevrey Hypoellipticity for an Interesting Variant of Kohn’s Operator

### Complex Analysis (2010-01-01): 51-73 , January 01, 2010

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In this paper we consider the analogue of Kohn’s operator but with a point singularity, $$ P = BB^* + B^* (t^{2\ell } + x^{2k} )B, B = D_x + ix^{q - 1} D_t . $$ .

We show that this operator is hypoelliptic and Gevrey hypoelliptic in a certain range, namely *k* < ℓ*q*, with Gevrey index
$$
\tfrac{{\ell _q }}
{{\ell _q - k}} = 1 + \tfrac{k}
{{\ell _q - k}}
$$
. Work in progress by the present authors suggests that, outside the above range of the parameters, i.e., when *k* ≥ ℓ*q*, the operator is not even hypoelliptic.

## Stability of the Vanishing of the $$ \bar \partial _b $$ -cohomoloy Under Small Horizontal Perturbations of the CR Structure in Compact Abstract q-concave CR Manifolds

### Complex Analysis (2010-01-01): 213-225 , January 01, 2010

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We consider perturbations of CR structures which preserve the complex tangent bundle. For a compact generic CR manifold its concavity properties and hence the finiteness of some $$ \bar \partial _b $$ -cohomology groups are also preserved by such perturbations of the CR structure. Here we study the stability of the vanishing of these groups.

## Differential Calculus in the Complex Plane C

### Complex Analysis (2009-01-01): 1-60 , January 01, 2009

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In this chapter we shall first give an introduction to *complex numbers* and their *topology*. In doing so we shall assume that this is not the first time the reader has encountered the system C of complex numbers. The same assumption is made for topological notions in C (*convergence, continuity* etc.). For this reason we shall not dwell on these matters. In Sect. I.4 we introduce the notion of *complex derivative*. One can begin reading directly with this section if one is already sufficiently familiar with the complex numbers and their topology. In Sect. I.5 the *relationship* between real *differentiability* and complex *differentiability* will be treated (the *Cauchy-Riemann differential equations*).

The story of the complex numbers from their early beginnings in the 16^{th} century until their eventual full acceptance in the course of the 19^{th} century—probably by the scientific authority of C.F. GAUSS— as well as the rather lengthy period of uncertainty and unclarity about them, is an impressive example in the history of mathematics. The historically interested reader should read [Re2]. For more historical remarks about complex numbers see also [CE].

## Foundations

### Complex Analysis (2007-01-01) 245: 7-21 , January 01, 2007

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The first section of this chapter introduces the complex plane, fixes notation, and discusses some useful concepts from real analysis. Some readers may initially choose to skim this section. The second section contains the definition and elementary properties of the class of holomorphic functions - the basic object of our study.

## Applications to Other Areas of Mathematics

### Complex Analysis (2010-01-01) 0: 273-290 , January 01, 2010

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We have already seen, especially in Chapter 11, howthe methods of complex analysis can be applied to the solution of problems from other area of mathematics. In this chapter we will get some insight into the fantastic breadth of such applications. For that reason, the topics chosen are rather disparate. Section 19.1 involves calculating the total variation of a real function, and illustraties how the methods of Chapter 11 can be applied to yet another nontypical problem. Section 19.2 offers a proof of the classic Fourier Uniqueness Theorem using two preliminary results from real analysis and a surprising application of Liouville's theorem. In Section 19.3 we see how the use of a generating function allows complex analytic results to be applied to an infinite system of (real) equations. Generating functions are also the key to the four different problems in number theory that comprise section 19.4. Finally, in section 19.5, we offer a well-trimmed analytic proof of the prime number theorem based on properties of the Zeta function and another Dirichlet series.

## Properties of Analytic Functions

### Complex Analysis (2010-01-01) 0: 77-91 , January 01, 2010

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In the last two chapters, we studied the connection between everywhere convergent power series and entire functions. We now turn our attention to the more general relationship between power series and analytic functions.According to Theorem 2.9 every power series represents an analytic function inside its circle of convergence. Our first goal is the converse of this theorem: we will show that a function analytic in a disc can be represented there by a power series. We then turn to the question of analytic functions in arbitrary open sets and the local behavior of such functions.

## Front Matter - Complex Analysis

### Complex Analysis (2010-01-01) , January 01, 2010

## Coherent Sheaves and Cohesive Sheaves

### Complex Analysis (2010-01-01): 227-244 , January 01, 2010

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We consider coherent and cohesive sheaves of *O* over open sets ω ⊂ ℂ^{n}. We prove that coherent sheaves, and certain other sheaves derived from them, are cohesive; and conversely, certain sheaves derived from cohesive sheaves are coherent. An important tool in all this, also proved here, is that the sheaf of Banach space valued holomorphic germs is flat.

## Analytic Functions

### Complex Analysis (2010-01-01) 0: 35-43 , January 01, 2010

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The direct functions of *z* which we have studied so far—polynomials and convergent power series—were shown to be differentiable functions of *z*. We now take a closer look at the property of differentiability and its relation to the Cauchy-Riemann equations.

## Elliptic Modular Forms

### Complex Analysis (2005-01-01): 327-390 , January 01, 2005

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## Characteristic Classes of the Boundary of a Complex b-manifold

### Complex Analysis (2010-01-01): 245-262 , January 01, 2010

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We prove a classification theorem by cohomology classes for compact Riemannian manifolds with a one-parameter group of isometries without fixed points generalizing the classification of line bundles (more precisely, their circle bundles) over compact manifolds by their first Chern class. We also prove a classification theorem generalizing that of holomorphic line bundles over compact complex manifold by the Picard group of the base for a subfamily of manifolds with additional structure resembling that of circle bundles of such holomorphic line bundles.

## Solvability of Planar Complex Vector Fields with Applications to Deformation of Surfaces

### Complex Analysis (2010-01-01): 263-278 , January 01, 2010

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Properties of solutions of a class of semilinear equations of the form *Lu* = *f(x, y, u)*, where *L* is a ℂ-valued planar vector field is studied. Series and integral representations are obtained in a tubular neighborhood of the characteristic curve of *L.* An application to infinitesimal bendings of surfaces with nonnegative curvature in ℝ^{3} is given.

## Propagation of Singularities for the Cauchy-Riemann Equations

### Complex Analysis (2011-01-01) 62: 177-280 , January 01, 2011

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These lectures are intended as an introduction to the study of several complex variables from the point of view of partial differential equations. More specifically here we take the approach of the calculus of variations known as the ā-Neumann problem. Most of the material covered here is contained in Polland and Kohn, [4], Hörmander [11] and in the more recent work of the author (see [16], [17] and [20]). We consistently use the Laplace operator as in Kohn [l4], since we believe that this method is particularly suitable for the study of regularity and for the study of the induced Cauchy-Riemann equations. Our main emphasis is in finding regular solutions of the inhomogenious Cauchy-Riemann equations. We wish to call attention to the extensive research on this problem by different methods from the ones mentioned above (see Ramirez [29], Grauert and Lieb[8], Kerzman [13], øvrelid [28], Henkin [9], Folland and Stein [5]). It would take us too far afield to present these matters here. Another closely related subject which we cannot take up here is the theory of approximations by holomorphic functions (see R. Nlrenberg and o. Wells [27], R. Nlrenberg [26], Hørmander and Wermer [12], etc.).

## Analytic Number Theory

### Complex Analysis (2005-01-01): 391-458 , January 01, 2005

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## Positivity of the $$ \bar \partial $$ -Neumann Laplacian

### Complex Analysis (2010-01-01): 145-158 , January 01, 2010

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We study the $$ \bar \partial $$ -Neumann Laplacian from spectral theoretic perspectives. In particular, we show how pseudoconvexity of a bounded domain is characterized by positivity of the $$ \bar \partial $$ -Neumann Laplacian.

## On the Zariski Closure of a Germ of Totally Geodesic Complex Submanifold on a Subvariety of a Complex Hyperbolic Space Form of Finite Volume

### Complex Analysis (2010-01-01): 279-300 , January 01, 2010

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Let *X* be a complex hyperbolic space form of finite volume, and *W* ⊂ *X* be a complex-analytic subvariety. Let *S* ⊂ *X* be a locally closed complex submanifold lying on *W* which is totally geodesic with respect to the canonical Kähler-Einstein metric on *X*. We prove that the Zariski closure of *S* in *W* is a totally geodesic subset. The latter implies that the Gauss map on any complex-analytic subvariety *W* ⊂ *X* is generically finite unless *W* is totally geodesic.

## Elliptic Modular Forms

### Complex Analysis (2009-01-01): 1-63 , January 01, 2009

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In connection with the question which complex numbers can be written as the absolute invariant of a lattice, we were led to analytic functions with a new type of symmetries. These functions are analytic functions on the upper half-plane with a specific transformation law with respect to the action of the full elliptic modular group (or of certain subgroups) on H, namely
$$f\left(\frac{az+b}{cz+d}\right)=(cz + d)^k f(z).$$
Functions with such a transformation behavior are called *modular forms*.

We will see that the elliptic modular group is generated by the substitutions
$$z \mapsto z + 1\ {\rm{and}}\ z \mapsto -\frac{1}{z}.$$
It is thus enough to check the transformation behavior only for these substitutions. There is an analogy to the transformation behavior of elliptic functions under translations in a lattice *L*, where it was also sufficient to check the invariance under the two generating *translations* ω_{1}, ω_{2}. But in contrast to the translation lattice *L*, the elliptic modular group is *not* commutative. Hence the theory of modular forms is more complicated than the theory of elliptic functions. This could be already observed in the construction of a fundamental domain for the action of the modular group Γ on the upper half-plane H, V.8.7.

## Sequences and Series of Holomorphic Functions

### Complex Analysis (2007-01-01) 245: 123-146 , January 01, 2007

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We now turn from the study of a single holomorphic function to the investigation of collections of holomorphic functions. In the first section we will see that under the appropriate notion of convergence of a sequence of holomorphic functions, the limit function inherits several properties that the approximating functions have, such as being holomorphic, and in the second section, we show that the space of holomorphic functions on a domain can be given the structure of a complete metric space. We then apply these ideas to obtain a series expansion for the cotangent function.

## Invariant CR Mappings

### Complex Analysis (2010-01-01): 95-107 , January 01, 2010

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This paper concerns CR mappings between hyperquadrics. We emphasize the case when the domain is a sphere and the maps are invariant under a finite subgroup of the unitary group. We survey known results, provide some new examples, and use them to prove the following result. For each *d* there is a CR polynomial mapping of degree 2*d* + 1 between hyperquadrics that preserves the number of negative eigenvalues. Thus rigidity fails for CR mappings between hyperquadrics.

## Back Matter - Complex Analysis

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

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

### Complex Analysis (2009-01-01): 1-88 , January 01, 2009

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It is known from real analysis that *pointwise convergence* of a sequence of functions shows certain pathologies. For instance, the pointwise limit of a sequence of continuous functions is not necessarily continuous, and in general we cannot exchange limit processes, et cetera. Therefore we are led to introduce the notion of *uniform convergence*, which has better stability properties. For example, the limit of a uniformly convergent sequence of continuous functions is continuous. Another basic stability theorem holds for the (proper) integral:

*A uniformly convergent sequence of integrable functions converges to an integrable function. The limit and integration can be exchanged*.

*However, differentiability in real analysis is not stable with respect to uniform convergence*.

The corresponding stability theorems are more complicated and require additional conditions on the sequence of derivatives. In function theory one introduces the concept of uniform convergence by analogy with real analysis. The stability of continuity and of the integral along curves can be obtained completely analogously to the real case, and in fact can be reduced to that case.

## The Cauchy Theory–A Fundamental Theorem

### Complex Analysis (2007-01-01) 245: 59-82 , January 01, 2007

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As with the theory of differentiation for complex-valued functions of a complex variable, the integration theory of such functions begins by mimicking and extending results from the theory for real-valued functions of a real variable, but again the resulting theory is substantially different, more robust and elegant.

## The Large Time Asymptotics of the Entropy

### Complex Analysis (2010-01-01): 301-306 , January 01, 2010

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In this note we supply the detailed proof of the entropy asymptotics on manifolds with nonnegative Ricci curvature. We also discuss the possible connections between the large time behavior of the entropy and the existence of harmonic functions.

## Simply Connected Domains

### Complex Analysis (2010-01-01) 0: 107-116 , January 01, 2010

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As we have seen, it can happen that a function *f* is analytic on a closed curve *C* and yet
$$ \int c f \ne 0 $$
.

## On Involutive Systems of First-order Nonlinear Partial Differential Equations

### Complex Analysis (2010-01-01): 25-50 , January 01, 2010

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We study the local and microlocal analyticity of solutions *u* of a system of nonlinear pdes of the form
$$
F_j (x,u,u_x ) = 0, 1 \leqslant j \leqslant n
$$
where the *F*_{j} (*x*, ζ_{0}, ζ) are complex-valued, real analytic in an open subset of (*x*, ζ_{0}, ζ) ∈ ℝ^{N}×ℂ×ℂ^{N} and holomorphic in (ζ_{0}, ζ). The function *u* is assumed to be a *C*^{2} solution. The *F*_{j} satisfy an involution condition and *d*_{ζ}*F*_{1} ∧ ... ∧ *d*_{ζ}*F*_{n} ≠ 0 where *d*_{ζ} denotes the exterior derivative in ζ = (ζ_{1}, ..., ζ_{N}).

## Solutions to the Exercises

### Complex Analysis (2005-01-01): 459-521 , January 01, 2005

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## Introduction

### Complex Analysis (2005-01-01): 1-7 , January 01, 2005

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## Introduction

### Complex Analysis (2009-01-01): 1-7 , January 01, 2009

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The complex numbers have their historical origin in the 16^{th} century when they were created during attempts to solve *algebraic equations*. G. CARDANO (1545) has already introduced formal expressions as for instance
$$5 \pm \sqrt{-15}$$
, in order to express solutions of quadratic and cubic equations. Around 1560 R. BOMBELLI computed systematically using such expressions and found 4 as a solution of the equation
$$x^3 = 15x + 4$$
in the disguised form
$$4 = \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}}$$
Also in the work of G.W. LEIBNIZ (1675) one can find equations of this kind, e.g.
$$\sqrt{1 + \sqrt{-3}} + \sqrt{1 - \sqrt{-3}} = \sqrt{6}.$$
In the year 1777 L. EULER introduced the notation
$${\rm{i}} = \sqrt{-1}$$
for the *imaginary* unit.

## The Fundamental Theorem in Complex Function Theory

### Complex Analysis (2007-01-01) 245: 1-5 , January 01, 2007

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This introductory chapter is meant to convey the need for and the intrinsic beauty found in passing from a real variable x to a complex variable z. In the first section we “solve” two natural problems using complex analysis. In the second, we state the most important result in the theory of functions of one complex variable that we call the Fundamental Theorem of complex variables; its proof will occupy most of this volume.