Complex variables: a physical approach with applications and MATLAB
From the algebraic properties of a complete number field, to the analytic properties imposed by the Cauchy integral formula, to the geometric qualities originating from conformality, Complex Variables: A Physical Approach with Applications and MATLAB explores all facets of this subject, with particular emphasis on using theory in practice.The first five chapters encompass the core material of the book. These chapters cover fundamental concepts, holomorphic and harmonic functions, Cauchy theory and its applications, and isolated singularities. Subsequent chapters discuss the argument principle, geometric theory, and conformal mapping, followed by a more advanced discussion of harmonic functions. The author also presents a detailed glimpse of how complex variables are used in the real world, with chapters on Fourier and Laplace transforms as well as partial differential equations and boundary value problems. The final chapter explores computer tools, including Mathematica®, MapleTM, and MATLAB®, that can be employed to study complex variables. Each chapter contains physical applications drawing from the areas of physics and engineering.Offering new directions for further learning, this text provides modern students with a powerful toolkit for future work in the mathematical sciences.
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The Relationship of Holomorphic and Harmonic Functions
The Cauchy Theory
Applications of the Cauchy Theory
13 other sections not shown
accumulation point algebra annulus argument principle boundary bounded calculate Cauchy integral theorem Cauchy-Riemann equations closed curve coefficients complex derivative complex line integral complex number complex variables conformal mapping conformal self-map connected open set constant continuous function continuously differentiable contour defined differential equation Dirichlet problem domain entire function essential singularity Example Figure Fourier series Fourier transform function f(z harmonic function heat distribution holomorphic function infinite inverse Laurent expansion Laurent series Let U C C linear fractional transformation MatLab MatLab exercise MatLab routine maximum principle mean value property meromorphic function modulus multiplicity neighborhood number of zeros one-to-one open set plane pole of order polynomial power series expansion real number real-valued region removable singularity residue result roots Schwarz lemma Schwarz-Christoffel Section sequence simply connected solution Suppose unit disc upper half-plane vanishes Verify Write a MatLab zero of order zero set