By John D. Paliouras

This awesome textual content for undergraduate scholars of technological know-how and engineering calls for just a commonplace path in simple calculus. Designed to supply a radical figuring out of basic strategies and create the foundation for higher-level classes, the remedy beneficial properties quite a few examples and vast workout sections of various trouble, plus solutions to chose exercises.

The two-part strategy starts with the improvement of the first thought of analytic functionality, advancing to the Cauchy fundamental idea, the sequence improvement of analytic features via evaluate of integrals by way of residues, and a few hassle-free functions of harmonic services. the second one half introduces a few of the deeper facets of advanced functionality conception: mapping homes of analytic capabilities, purposes to varied vector box issues of boundary stipulations, and a suite of extra theoretical results.

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**Extra info for Complex Variables for Scientists and Engineers: Second Edition**

**Example text**

26Prove that the unity of the complex number system is unique. 27Prove that the negative of any complex number z is unique. 28If z = x + iy is a nonzero complex number, derive z−l in terms of x and y and show that it is unique. SECTION 2 GEOMETRY OF COMPLEX NUMBERS The reader is familiar with certain correspondences between algebraic and geometric concepts that are described in analytic geometry. The real numbers correspond to the points on a number line. | a − b | corresponds to the distance between the numbers a and b.

The reader who is familiar with the concept of a vector space will recognize the complex number system C as a two-dimensional vector space over the field of real numbers. In this context a complex number (x, y) can be thought of as a vector with vector addition defined by (2). If one defines scalar multiplication by for any real number r and any complex number (x, y) it is an easy exercise to show that the postulates of a vector space are satisfied. Moreover, a basis for this vector space is given by the vectors This, of course, implies that any vector (x, y) in C can be written as a linear combination of the basis vectors; indeed, using (4), we find that Clearly, v1 is the unity of C.

Before we embark on a discussion of these and other items, we digress briefly to introduce certain basic concepts that are peripheral to the main development but indispensable nonetheless. We must also establish some terminology, which we will use for the remainder of this book. So here is a minimal dose of topological preliminaries. Absorb them; we will need them. In what follows we use the term set only for simple cases, such as a collection of points in the z-plane, or of integers. Although no knowledge of set theory on the part of the reader will be assumed, occasionally, we will use the conventional set-theoretic notation which we will read “the set of all x such that property P is true.