Analisis Complejo – Lars. Ahlfors – [PDF Document]. – Lars Valerian Ahlfors ( April – 11 October. ) was a Finnish mathematician. Lars Ahlfors Complex Analysis Third Edition file PDF Book only if you are registered here. Analisis Complejo Lars Ahlfors PDF Document. – COMPLEX. Ahlfors, L. V.. Complex analysis: an introduction to the theory of Boas Análisis real y complejo. Sansone, Giovanni. Lectures on the theory of functions of a.

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Published on Dec View Download Copyright by McGraw-Hill, Inc. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without analisjs prior written permission of the publisher.

International series in pure and applied mathematics Includes index. There is, never-theless, need for a new edition, partly because of changes in current mathe-matical terminology, partly because of differences in student preparedness and aims.

Analisis Complejo – Lars Ahlfors

There are no radical innovations in the new edition. The author still believes strongly in a geometric approach to the basics, and for this reason the introductory chapters are virtually unchanged. In a few places, throughout the book, it was desirable to clarify certain points that ex-perience has shown to have been a source of possible misunderstanding or difficulties.

Misprints and minor errors that have come to my attention have been corrected. Otherwise, the main differences between the second and third editions can be summarized as follows: Notations and terminology have been modernized, but it did not seem necessary to change the style in any significant way.

In Chapter 2 a brief section on the change of length and area under conformal mapping has been added. To some degree this infringes on the otherwise self-contained exposition, for it forces the reader to fall back on calculus for the definition and manipulation of double integrals.

The disadvantage is minor. In Chapter 4 there complsjo a new and simpler proof of the general form of Cauchy’s theorem. It is due to A. Beardon, who has kindly permitted me to reproduce it. It complements but does not replace the old proof, which has been retained and improved.

A short section on the Riemann zeta function has been included. Large parts ahfors Chapter 8 have been completely rewritten. The main purpose was to introduce the reader to the terminology of germs and sheaves while emphasizing all the classical concepts.

It goes without saying that nothing beyond the basic notions of sheaf theory would have been compatible with the elementary nature of the book. The author has successfully resisted the temptation to include Riemann surfaces as one-dimensional complex manifolds.

The book would lose much of its usefulness if it went beyond its purpose of being no more than an introduction to the basic methods and results of complex function theory in the plane.

It is my pleasant duty to thank the many who have helped me by pointing out misprints, weaknesses, and errors in the second edition. I am particularly grateful to my colleague Lynn Loomis, who kindly let me share student reaction to a recent course based on my book. We begin our study of complex func-tion theory by stressing and implementing this analogy.


If the imaginary unit is combined with two real num-bers a,: Zero is the only number which is at once real and purely imaginary. Two complex numbers are equal if and only if they have the same real part and the same imaginary part. Addition and multiplication do not lead out from the system of complex numbers.

It is less obvious that division is also possible. We have thus the result 3 Once the existence of the quotient has been proved, its value can be found in a simpler way. They correspond to values of n which divided by 4 leave the remainders 0, 1, 2, 3. We shall now show that the square root of a complex number can be found explicitly. Observe that these quantities are positive or zero regardless of the sign of a. The equations 5 yield, in general, two opposite values for x and two for y.

But these values cannot be combined arbitrarily, for the second equation 4 is not a consequence of 5. Our discussion of the real-number system is incomplete inasmuch as we have not proved the existence and uniqueness up to isomorphisms of a system R with the postulated properties. Denote a solution by i. The subset Cis a subfield of F. What is more, the structure of Cis independent of F. The word is used quite generally to indicate a corre-spondence which is one to one and preserves all relations that are considered important in a given connection.

It is thus demonstrated that C and C’ are isomorphic. We now define the field of complex numbers to be the sub field C of an arbitrarily given F. We have just seen that the choice of F makes no difference, but we have not yet shown that there exists a field F with the required properties.

There are many ways in which such a field can be constructed. The simplest and most direct method is the following: Show that the system of all matrices of the special form combined by matrix addition and matrix multiplication, is isomorphic to the field of complex numbers. The real and imaginary part of a complex number a will also be denoted by Re a, Im a.

Since -i has the same property, all rules must remain valid if i is everywhere replaced by -i. Direct verification shows that this is indeed so.

The conjugate of a is denoted by a.

Complex Analysis, 3rd ed. by Lars Ahlfors | eBay

A number is real if and only if it is equal to its conjugate. The conjugation is analieis involutory transformation: By systematic use of the notations a and a it is hence possible to dispense with the use of separate letters for the real and imaginary part. It is more convenient, though, to make free use of both notations. The corresponding property for quotients is a consequence: More generally, let R a,b,c, As an application, consider the equation If!: Its nonnega-tive square root is called the modulus or absolute value of the complex num-ber a; it is denoted by Ia!.

For the absolute value of a product we obtain and hence lab! The absolute value of a product is equal to the product of the absohtte values of the factors.

Analisis Complejo – Lars Ahlfors

It is clear that this property extends to arbitrary finite products: What exception must be made if Ia! Prove Lagrange’s identity in the complex form 1. We shall now prove some important inequalities which will be of constant use.


It is perhaps well to point out that there is no order relation in the complex-number system, and hence all inequali-ties must be between real numbers. From the definition of the absolute value we deduce the inequalities 9 -Ia!

This is called the triangle inequality for reasons which will emerge later. By induction it can be extended to arbitrary sums: The absolute value of a sum is at most equal to the sum of the absolute values of the terms. The reader is well aware of the importance of the estimate 11 in the real case, and we shall find it no less important in the theory of complex numbers.

Let us determine all cases of equality in But the numbering of the terms is arbitrary; thus the ratio of any two nonzero terms must be positive. Suppose conversely that this condition is fulfilled. Many other inequalities whose proof is less immediate are also of fre-quent use.

It seems pointless to ban its use. We can choose for if the denominator should vanish there is nothing to prove.

This choice is not arbitrary, but it is dictated by the desire to make the expression 15 as small as possible. Naalisis in 15 we find, after simplifications, which is equivalent to From 15 we conclude further that the sign of equality holds in 14 if and only if the a; are proportional to the b. Cauchy’s inequality can also be proved by means of Lagrange’s identity Sec. An easy argument shows that this distinction is independent of the parametric representation. Write the equation of an ellipse, ahlvors, parabola in complex form.

Prove that the diagonals of a parallelogram bisect each other and that the diagonals of a rhombus are orthogonal. Prove analytically that the midpoints of parallel chords to a circle lie on a diameter perpendicular to the chords. For many purposes it is useful to extend the system C of complex numbers by introduction of a symbol co to represent infinity. In the plane there is no room for a point vomplejo to co, but we can of course introduce an “ideal” point which we call the point at infinity.

The points in analiis plane together with the point at infinity form the extended complex plane. Ahlfoes agree that every straight line shall pass through the point at infinity. By contrast, no half plane shall contain the ideal point.

It is desirable to introduce a geometric model in which all points of rhe extended plane have a concrete representative. The correspondence can be completed by letting the point at infinity correspond to 0,0,1and we can thus regard the sphere as a repre-sentation of the extended plane or of qhlfors extended number system.

In function theory the sphere S is referred to as the Riemann sphere. If the complex plane is identified with the x1,x2 -plane with the xr and x 2-axis corresponding to the real and imaginary axis, respectively, the transformation 24 takes on a simple geometric meaning. Hence the correspondence is a central projection from the center 0,0,1 as shown in Fig. It is called a stereographic projection.