By Steven G. Krantz

From the algebraic houses of a whole quantity box, to the analytic homes imposed through the Cauchy indispensable formulation, to the geometric traits originating from conformality, complicated Variables: A actual procedure with functions and MATLAB explores all aspects of this topic, with specific emphasis on utilizing idea in perform. the 1st 5 chapters surround the center fabric of the publication. those chapters conceal basic options, holomorphic and harmonic features, Cauchy idea and its functions, and remoted singularities. next chapters speak about the argument precept, geometric idea, and conformal mapping, by means of a extra complex dialogue of harmonic services. the writer additionally offers an in depth glimpse of ways complicated variables are utilized in the true global, with chapters on Fourier and Laplace transforms in addition to partial differential equations and boundary worth difficulties. the ultimate bankruptcy explores machine instruments, together with Mathematica®, Maple™, and MATLAB®, that may be hired to check complicated variables. every one bankruptcy comprises actual functions drawing from the parts of physics and engineering. providing new instructions for additional studying, this article presents sleek scholars with a strong toolkit for destiny paintings within the mathematical sciences.

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**Example text**

For if k is an integer, then eiθ = cos θ + i sin θ = cos(θ + 2kπ) + i sin(θ + 2kπ) = ei(θ+2kπ) . 23) Remark: Of course the inverse of the exponential function is the (complex) logarithm. 5. Exercises 1. 2. Calculate (with your answer in the form a + ib) the values of eπi , e(π/3)i, 5e−i(π/4), 2ei , 7e−3i . √ √ 3i, 3 − i, Write these complex numbers in polar form: 2 + 2i, 1 + √ √ 2 − i 2, i, −1 − i. 3. If ez = 2 − 2i then what can you say about z? ] 4. If w5 = z and |z| = 3 then what can you say about |w|?

In fact this power series property is a complete characterization of holomorphic functions; we shall discuss it in detail below. The use of “differentiable” derives from properties related to the complex derivative. These pieces of terminology and their significance will all be sorted out as the book develops. 3) and other physical phenomena. 1 and Chapter 7. We shall treat physical applications of conformality in Chapter 8. Exercises 1. Verify that each of these functions is holomorphic whereever it is defined: (a) f(z) = sin z − 1 z2 z+1 A more classical formulation of the result is this.

One may calculate directly, just differentiating the power series term-by-term, that ∂ z e = ez . ∂z In addition, ∂ z e = 0, ∂z so the exponential function is holomorphic. Of course we know, and we have already noted, that ex+iy = ex (cos y + i sin y) . When x = 0 this gives Euler’s famous formula eiy = cos y + i sin y . It follows immediately that cos y = eiy + e−iy 2 and eiy − e−iy . 2i We explore other derivations of Euler’s formula in the exercises. In analogy with these basic formulas from calculus, we now define complexanalytic versions of the trigonometric functions: sin y = eiz + e−iz cos z = 2 and eiy − e−iy .