By Steven G. Krantz

From the algebraic houses of an entire quantity box, to the analytic houses imposed by way of the Cauchy crucial formulation, to the geometric features originating from conformality, advanced Variables: A actual process with purposes and MATLAB explores all features of this topic, with specific emphasis on utilizing thought in practice.

The first 5 chapters surround the middle fabric of the publication. those chapters conceal basic suggestions, holomorphic and harmonic services, Cauchy idea and its purposes, and remoted singularities. next chapters speak about the argument precept, geometric conception, and conformal mapping, by means of a extra complex dialogue of harmonic features. the writer additionally offers a close glimpse of the way advanced 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 laptop instruments, together with Mathematica®, Maple™, and MATLAB®, that may be hired to check complicated variables. every one bankruptcy comprises actual purposes drawing from the parts of physics and engineering.

Offering new instructions for extra studying, this article offers glossy scholars with a strong toolkit for destiny paintings within the mathematical sciences.

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Extra resources for Complex Variables: A Physical Approach with Applications and MATLAB

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2. Find complex numbers z, w such that |z| = 5 , |w| = 7, |z + w| = 9. 3. Find complex numbers z, w such that |z| = 1, |w| = 1, and z/w = i3. 4. Let z = 4 − 6i, w = 2 + 7i. Calculate z/w, w/z, and 1/w. 5. Sketch these discs on the same set of axes: D(2 + 3i, 4), D(1 − 2i, 2), D(i, 5), D(6 − 2i, 5). 6. Which of these sets is open? Which is closed? Why or why not? (a) {x + iy ∈ C : x2 + 4y 2 ≤ 4} (b) {x + iy ∈ C : x < y} (c) {x + iy ∈ C : 2 ≤ x + y < 5} (d) {x + iy ∈ C : 4 < (e) {x + iy ∈ C : 5 ≤ x2 + 3y 2 } x4 + 2y 6} 7.

Z ∂z ∂z ∂z 17. The function f(z) = z 2 − z 3 is holomorphic. Why? It has real part u that describes a steady state flow of heat on the unit disc. Calculate this real part. Verify that u satisfies the partial differential equation ∂ ∂ u(z) ≡ 0 . ∂z ∂z This is the Laplace equation. We shall study it in greater detail as the book progresses. 18. Do the last exercise with “real part” u replaced by “imaginary part” v. 2. 1 45 The Relationship of Holomorphic and Harmonic Functions Harmonic Functions A C 2 (twice continuously differentiable) function u is said to be harmonic if it satisfies the equation ∂2 ∂2 + ∂x2 ∂y 2 u = 0.

Prove that, for any C 2 function f, △(f ◦ ρ) = (△f) ◦ ρ . 6. Let a ∈ R2 and let λa be the operator λa (x, y) = (x, y) + a. This is translation by a. Verify that, for any C 2 function f, △(f ◦ λa ) = (△f) ◦ λa .

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Complex Variables: A Physical Approach with Applications and by Steven G. Krantz
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