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## Carathéodory Kernels and Farrell’s Theorem

### Complex Analysis (1984-01-01): 124-129 , January 01, 1984

*Given a sequence* {G_{n}} *of regions and a region* G ≠ Ɉ *such that* G ⊆ G_{n+1} ⊆ G_{n}*for* n = 1,2,3,.., *we say that a superset* G’ *of* G *is**suitable**if* G’ *is connected and* G’ ⊆ ∩G_{n}. *Then* ker[G_{n}: G], *the kernel**of* {G_{n}} *with respect to* G, *is defined as the union of all suitable supersets of* G.

## Solutions to the Exercises

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

## 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.

## Elliptic Modular Forms

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

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.

## The Riemann Mapping Theorem

### Complex Analysis (2010-01-01): 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 Fundamental Theorem in Complex Function Theory

### Complex Analysis (2013-01-01) 245: 1-14 , January 01, 2013

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 what we regard as the most important result in the theory of functions of one complex variable, which we label the fundamental theorem of complex function theory, in a form suggested by the teaching and exposition style of Lipman Bers; its proof will occupy most of this volume. The next two sections of this chapter include an outline of our plan for the proof and an outline for the text, respectively; in subsequent chapters we will define all the concepts encountered in the statement of the theorem in this chapter. The reader may not be able at this point to understand all (or any) of the statements in our fundamental theorem or to appreciate its depth and might choose initially to skim this material. All readers should periodically, throughout their journey through this book, return to this chapter, particularly to the last section, that contains an interesting account of part of the history of the subject.

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

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

## Conformal Mappings onto Nonoverlapping Regions

### Complex Analysis (1988-01-01): 27-39 , January 01, 1988

Let *f*(ζ) = *a* + *d*ζ… be analytic and univalent in the unit disk |ζ| < 1, mapping it conformally onto some domain *D*. We shall call *a* = *f*(0) the *center* and |*d*| = |*f′*(0)| the *inner radius* of *D* with respect to *a*. Roughly speaking, our problem is to find *n* functions
(1)
$$f_j(\zeta) = a_j+d_j \zeta + \ldots, \quad j=1,2,\ldots,n,$$
which map the disk conformally onto nonoverlapping regions *D*_{j} whose union has prescribed transfinite diameter *R*, with the centers *a*_{j} as far apart as possible and the inner radii |*d*_{j}| as large as possible. Here only *n* and *R* are specified in advance.

## Analytic Functions

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

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.

## Olomorphic Vectorbundles and Yang Mills Fields

### Complex Analysis (1982-01-01) 950: 377-401 , January 01, 1982

In terms of differential geometry a potential should be interpreted as a connection and its field as the curvature associated to the connection. In gauge theory one is lead to consider connections and curvatures in vectorbundles. The topic of these lectures is to describe the self-dual curvatures of SU(2)-connections of vectorbundles on S^{4}, which are called self-dual euclidean SU(2)-Yang Mills fields. In [1] it was shown that such fields are in a one to one correspondence with certain holomorphic vectorbundles on ℙ_{3}(ℂ), which are now called instantonbundles. By using the theory of moduli for algebraic vectorbundles on complex projective space explicit expressions for the euclidean SU(2)-Yang Mills fields can be derived from this correspondence. This procedure is described here only in the case of the instanton number c^{2}=1.

## Differentiation

### Complex Analysis (2003-01-01): 51-78 , January 01, 2003

The definition of differentiability of a complex function presents no problem, since it is essentially the same as for a real function: a complex function *f* is said to be *differentiable* at a point *c* in ℂ if
$$ \mathop {\lim }\limits_{z \to c} \frac{{f(z) - f(c)}}{{z - c}} $$
exists. The limit is called the *derivative of**f* at *c* and is denoted by *f*′(*c*). Although the definition is formally identical to that used in real analysis, the fact that differentiability requires the rate of change of *f* to be the same in all possible directions means that its consequences are much more far-reaching. Certain things, however, do not change: the standard “calculus” rules for differentiation of sums, products and quotients, and the “chain rule” (*fog*)′(*z*) = *f*′(*g*(*z*))*g*′(*z*) are all valid for complex functions, and the proofs are in essence formally identical to those in real analysis. (See [9, Chapter 4].) Since (trivially) *z* ↦ *z* is differentiable, with derivative 1, it follows that polynomials are differentiable at every point in the plane, and that a rational function *p*(*z*)/*q*(*z*) (where *p* and *q* are polynomials with no common factor) is differentiable except at the zeros of *q*.

## Back Matter - Complex Analysis

### Complex Analysis (1998-01-01) , January 01, 1998

## 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.

## Conformal Equivalence

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

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

## On the Extremality and Unique Extremality of Certain Teichmüller Mappings

### Complex Analysis (1988-01-01): 225-238 , January 01, 1988

*1.* A quasiconformal mapping *f* of a domain *G* onto a domain *G’* with maximal dilatation *K* is called extremal, if every *qc* mapping
$${\tilde f}$$
which agrees with *f* on the boundary of *G* and is homotopic to *f* has a maximal dilatation
$$\tilde K \geqslant K$$
It is called uniquely extremal, if the strict inequality
$$\tilde K > K$$
holds whenever
$$\tilde f \ne f$$
Since the maximal dilatations of a *qc* mapping *f* and of its inverse *f*^{−1} are the same, the mapping *f*^{−1} is extremal (uniquely extremal) if and only if *f* is.

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

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

## On Approximation by Rational Functions of Class L1

### Complex Analysis (1988-01-01): 201-206 , January 01, 1988

Let *S* = {*z*_{k}}, 0 ≤ |*z*_{1}|≤ |*z*_{2}|≤…, be a countably infinite set in the complex plane ℂ with no limit points in ℂ. We denote by *B*_{S} the collection of functions *f*(*z*), analytic in ℂ\*S*, possessing finite *L*^{1} norm,
$$
\parallel f\parallel = \iint\limits_C {|f(z)|}dxdy < \infty \qquad (z = x + iy).
$$

## On Boundary Correspondence for Domains on the Sphere

### Complex Analysis (1988-01-01): 115-119 , January 01, 1988

It is well known that a conformal mapping between domains bounded by Jordan curves can be extended to a homeomorphism of the closures. For many applications what is needed however is a local version of this result. In the present paper such a result is provided in a generalized context.

## 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.

## The Cauchy Theory: A Fundamental Theorem

### Complex Analysis (2013-01-01) 245: 81-117 , January 01, 2013

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 more elegant. Specifically, a curve or path γ in *ℂ* is a continuous *function* from a closed interval in *ℝ* to *ℂ*. Thus the restriction of a complex-valued function *f* on *ℂ* to the range of a curve has real and imaginary parts which can be viewed as real-valued functions of a real variable and thus integrated on the interval. Adding the integral of the real part to *ı* times the integral of the imaginary part defines a complex-valued integral of a complex-valued function (i.e.,
$$\int \nolimits \nolimits f = \int \nolimits \nolimits \mathfrak{R}f + \imath \int \nolimits \nolimits \mathfrak{I}f$$
). In fact, there are several useful ways to employ the ability to integrate a function of a real variable to define complex-valued integrals of a complex variable over certain paths. Among these integrals are those known as line integrals, complex line integrals, and integrals with respect to arc length. One can then use the integration theory of real variables to obtain an integration theory for complex-valued functions along curves in *ℂ*. This extends to a more general theory, the Cauchy theory, which constitutes a main portion of the fundamental theorem (Theorem 1.1). The integration theory depends not just on the integrated function being holomorphic but also upon the topology of the curve over which the integration is being carried out and the topology of the domain in which the curve lies. In the simplest situation Cauchy’s theorem says that the integral of a holomorphic function over a simple closed curve lying in a convex domain is equal to zero.

## Front Matter - Complex Analysis

### Complex Analysis (1999-01-01): 103 , January 01, 1999

## Line Integrals and Harmonic Functions

### Complex Analysis (2001-01-01): 70-101 , January 01, 2001

In Sections 1 and 2 we review multivariable integral calculus in order to prepare for complex integration in the next chapter. The salient features are Green’s theorem and independence of path for line integrals. In Section 3 we introduce harmonic functions, and in Sections 4 and 5 we discuss the mean value property and the maximum principle for harmonic functions. Sections 6 and 7 include various applications to physics. The student may proceed directly to complex integration in the next chapter after paging through the review of multivariable calculus in Sections 1 and 2 and reading about harmonic conjugates in Section 3.

## On Meromorphic Functions with Growth Conditions

### Complex Analysis (1988-01-01): 61-80 , January 01, 1988

We assume that the function *f* is meromorphic in 픻 = {*z* ∈ ℂ : |*z*| < 1}. Let *C*_{1}, *C*_{2},… denote positive constants. We say that *f* has locally bounded characteristic (l.b.c) in 픻 if there exists a function φ analytic and univalent in 픻 with φ(픻) ⊂ 픻, φ(픻 ⋂ ℝ) ⊂ ℝ and
1.1
$$
\varphi (1)=1, \quad \varphi \prime(1) < \infty,
$$
such that
1.2
$$
T_o(r, f(e^{i \Theta} \varphi (w))) \leq C_{1} \quad (0 \leq \Theta \leq 2\pi ).
$$

## Properties of Analytic Functions

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

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.

## Some Consequences of Cauchy’s Theorem

### Complex Analysis (2003-01-01): 119-136 , January 01, 2003

We have already observed in Theorem 5.13 that if σ is a circle with centre 0 then $$ \int_\sigma {\frac{1}{z}} dz = 2\pi i $$ .

## Finite-type conditions for real hypersurfaces in ℂn

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

## 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.

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

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

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.

## Coherent Sheaves and Cohesive Sheaves

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

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.

## 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.

## Propagation of Singularities for the Cauchy-Riemann Equations

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

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.).

## Ring (not Algebra) Isomorphisms of H(G)

### Complex Analysis (1984-01-01): 130-135 , January 01, 1984

We return here to the ring structure of *H*(G). A *ring homomorphism* of R_{1} to R_{2} is a function *φ*: R_{1} → R_{2} which preserves multiplication i.e. *φ*(rs) = *φ*(r)*φ*(s) and *φ*(r + s) = *φ*(r) + *φ*(s) for all r, s ∈ R_{1}. A ring isomorphism is a ring homomorphism that is one-to-one and onto. If G and G’ are two conformally equivalent domains in ℂ then there is an algebra isomorphism from *H*(G) to *H*(G’) as we saw in Chapter 5. An algebra isomorphism will be a ring isomorphism which additionally preserves scalar multiplication. It follows from Proposition 5.4 that *H*(G) and *H*(G’) are isomorphic as algebras if and only if G and G’ are conformally equivalent. But a ring isomorphism can exist without conformal equivalence.

## Sequences and Series of Holomorphic Functions

### Complex Analysis (2013-01-01) 245: 171-197 , January 01, 2013

We now turn from the study of a single holomorphic function to the study 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. 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 and results to obtain, as an illustrative example, a series expansion for the cotangent function. In the fourth section we characterize the compact subsets of the space of holomorphic functions on a domain. This powerful characterization is used in Sect. , to study approximations of holomorphic functions and, in particular, to prove Runge’s theorem, which describes conditions under which a holomorphic function can be approximated by rational functions with prescribed poles. The characterization will also be used in Chap. 8 to prove the Riemann mapping theorem.

## 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*.

## Laurent Series and Isolated Singularities

### Complex Analysis (2001-01-01): 165-194 , January 01, 2001

In Section 1 we derive the Laurent decomposition of a function that is analytic on an annulus, and in Section 2 we use the Laurent decomposition on a punctured disk to study isolated singularities of analytic functions. We classify these as removable singularities, essential singularities, or poles, and we characterize each type of singularity. In Section 3 we define isolated singularities at ∞, and in Section 4 we derive the partial fractions decomposition of a rational function. In Sections 5 and 6 we use the Laurent decomposition to study periodic functions and we relate Laurent series to Fourier series. Sections 5 and 6 can be omitted at first reading.

## Maximum-Modulus Theorems for Unbounded Domains

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

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*.

## Front Matter - Complex Analysis

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

## Twistor Theory (The Penrose Transform)

### Complex Analysis (1982-01-01) 950: 1-11 , January 01, 1982

Where does complex mathematics intervene in our real world? [5]

*Answer*: Twistor Theory! [19]

*Twistors* were introduced by Penrose [11, 13] in order to provide an alternative description of Minkowski-space which emphasizes the light rays rather than the points of space-time. Minkowski-space constructions must be replaced by corresponding constructions in *twistor-space*. The twistor programme [17] has met with much success:
(1)

The description of massless free fields (the *Penrose transform*)

The description of self-dual Einstein manifolds

(3)The description of self-dual Yang-Mills fields

(4)The description of elementary particles (rather tentative).

## Cauchy’s Theorem, First Part

### Complex Analysis (1999-01-01) 103: 86-132 , January 01, 1999

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}.

## Front Matter - Complex Analysis

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

## On some elementary Applications of the Reflection Principle to Schwarz-Christoffel Integrals

### Complex Analysis (1988-01-01): 101-106 , January 01, 1988

We shall consider here only *one-to-one conformal mappings*. In the application of the Schwarz-Christoffel formula for mapping the upper half-plane *G*_{z} onto a polygon *G*_{w}, the main difficulty is to find the (real) pre-images of the vertices of *G*_{w}. We want to emphasize here how the reflection principle may open an elementary path from a mapping *G*_{z} → *G*_{w} to a mapping *G*_{z} → Ĝ_{w}, where Ĝ_{w} is an extension of *G*_{w} by reflection across one or several boundary segments.

## Isolated Singularities of an Analytic Function

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

*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.”

## Front Matter - Complex Analysis

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

## Special Functions

### Complex Analysis (1998-01-01): 1-76 , January 01, 1998

This chapter introduces the classical special functions. It is certainly not intended to give an exhaustive treatment of them; rather we are interested in these functions primarily as examples of technique. The problem of determining the behavior of a definite integral, or of the solution to a differential equation, as a function of various parameters is one that arises frequently. If one is lucky, one finds that someone else has encountered the same function and that a reasonably complete theory exists. More often one is left to one’s own devices. For this reason it is much more important to understand the available techniques than to have memorized a list of formulae.

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

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

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

## Front Matter - Complex Analysis

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

## Angular Distribution of Meromorphic Functions in the Unit Disk

### Complex Analysis (1988-01-01): 239-245 , January 01, 1988

In 1959, W.K. Hayman [1] obtained a series of interesting results on Picard exceptional values of meromorphic functions. Among others, he proved.

*Theorem A*. *Let f(z) be a transcendental meromorphic function in the finite plane. If k is an integer not less than 5 and a is a finite non-zero complex value, then f’—af*^{k}*assumes every finite complex value infinitely often*.

## 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.

## 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.

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

### Complex Analysis (1999-01-01) 103: 339-371 , January 01, 1999

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

## On Fixed Points of Conformal Automorphisms of Riemann Surfaces

### Complex Analysis (1988-01-01): 207-210 , January 01, 1988

*0.* Let *R* be a compact Riemann surface of genus *g*, and let *f* denote a conformal automorphism of *R* that is not the identity mapping. By a classical theorem the number of fixed points of *f* is at most 2*g* + 2. This result has in recent years been generalized to non-compact Riemann surfaces (cf. [P/L] for *g* = 0 and [M], [S] for *g* ≥ 0). Our paper contains new proofs of that fact.

## On Circulants

### Complex Analysis (1988-01-01): 121-130 , January 01, 1988

Circulants were introduced as determinants of matrices of the form

1 $$ Q= \begin{bmatrix} c_0& c_1& .& .& .& c_{n-1}\\ c_{n-1}& c_0& c_1& .& .& c_{n-2}\\ .& .& .& .& .&. \\ .& .& .& .& .&.\\ c_1& c_2& .& .& .&c_0 \end{bmatrix} = (q_{ij}) $$

Thus *q*_{ij} = *c*_{mod(j−i, n)}. One now refers to *circulant matrices*. Closely related to *Q* is the *skew-circulant* matrix *Q’* in which each entry in *Q* above the principal diagonal is multiplied by −1.

## 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

## 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*.

## Front Matter - Complex Analysis

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

## The Fundamental Theorem in Complex Function Theory

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

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.

## Back Matter - Complex Analysis

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

## Calculus of Residues

### Complex Analysis (1999-01-01) 103: 173-207 , January 01, 1999

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.

## Power Series

### Complex Analysis (1985-01-01) 103: 38-86 , January 01, 1985

So far, we have given only rational functions as examples of holomorphic functions. We shall study other ways of defining such functions. One of the principal ways will be by means of power series. Thus we shall see that the series
$$1 + z + \frac{{{z^2}}}{{2!}} + \frac{{{z^3}}}{{3!}} + ...$$
converges for all *z* to define a function which is equal to *e*^{z}. Similarly, we shall extend the values of sin *z* and cos *z* by their usual series to complex valued functions of a complex variable, and we shall see that they have similar properties to the functions of a real variable which you already know.

## 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.

## A Quick Review of Real Analysis

### Complex Analysis (1998-01-01): 201-220 , January 01, 1998

The purpose of this appendix is to provide the reader with exactly enough real analysis to read the remainder of the text. It is intended as a reference rather than as a substitute for a real analysis course. No attempt has been made at either elegance or generality.

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

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

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.

## Plurisubharmonic functions on ring domains

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

## Front Matter - Complex Analysis

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

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

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

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.

## 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.

## Compact Families of Meromorphic Functions

### Complex Analysis (2001-01-01): 315-341 , January 01, 2001

In Sections 1 and 2 we treat normal families of meromorphic functions. These are families that are sequentially compact when regarded as functions with values in the extended complex plane. We give two characterizations of normal families, Marty’s theorem in Section 1 and the Zalcman lemma in Section 2. From the latter characterization we deduce Montel’s theorem on compactness of families of meromorphic functions that omit three points, and we also prove the Picard theorems. Sections 3 and 4 constitute an introduction to iteration theory and Julia sets. In Section 3 we proceed far enough into the theory to see how Montel’s theorem enters the picture and to indicate the fractal nature (self-similarity) of Julia sets. In Section 4 we relate the connectedness of Julia sets to the orbits of critical points. In Section 5 we introduce the Mandelbrot set, which has been called the “most fascinating and complicated subset of the complex plane.”

## Proper mappings between balls in Cn

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

## 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.

## 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

## The Cauchy Theory–A Fundamental Theorem

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

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.

## Dual Space Topologies

### Complex Analysis (1984-01-01): 136-150 , January 01, 1984

This chapter is intended as a prerequisite for later chapters. In it we introduce a topology on the dual of a topological vector space. We present some of the standard results in the theory of Fréchet spaces and some additional results on topological vector spaces in general.

## 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.

## Back Matter - Complex Analysis

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

## The points of maximum modulus of a univalent function

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

## Harmonic Functions

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

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

## Values shared by an entire function and its derivative

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

## Integral Calculus in the Complex Plane ℂ

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

## Applications of Cauchy’s Integral Formula

### Complex Analysis (1985-01-01) 103: 144-164 , January 01, 1985

In the present chapter we show how a holomorphic function and the derivative of a holomorphic function can be expressed as an integral in terms of the function. We then give applications of this, getting the expansion of a holomorphic function into a power series, and studying the possible singularities which may arise when a function is analytic near a point, but may not be holomorphic at the point itself.

## 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.

## 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.