 2015 - 2016 | |||||||||||||||||||||||||||||
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| 0509-2844-12 | Complex Functions | ||||||||||||||||||||||||||||
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University credit hours: 1.0
Course description
Credit Points: 2.5
Prerequisites: Differential and Integral Calculus; Linear Algebra; Differential and
Integral Methods
The complex numbers system. Algebra of complex numbers. Geometrical interpretation and the complex plane. Polar representation. Complex conjugate. The logarithmic and exponential functions. Powers, roots, and their geometrical interpretations. Disk and ring regions in the complex plane. Functions of complex variables. Limit, continuity, derivative. Differentiation rules. Analytic functions. Cauchy-Riemann theorem and its applications. Elementary functions. Branch cuts and inverses. Integration in the complex plane. Contour integration. Simply connected domains. Independence of the path. Cauchy integral and its use to evaluate derivatives. Liouville theorem for entire functions. The fundamental theorem of algebra. Cauchy-Hadamard criteria for power series. Integration and differentiation of power series. Roots of analytic functions. Weierstrass M-test for uniform convergence of function series. Changing the order of summation and integration. Uniform convergence of function series. Isolated singular points of analytic functions. Residue theorem and its applications. The argument theorem. Branch cut integration. Conformal mapping.
Prerequisites: Differential and Integral Calculus; Linear Algebra; Differential and
Integral Methods
The complex numbers system. Algebra of complex numbers. Geometrical interpretation and the complex plane. Polar representation. Complex conjugate. The logarithmic and exponential functions. Powers, roots, and their geometrical interpretations. Disk and ring regions in the complex plane. Functions of complex variables. Limit, continuity, derivative. Differentiation rules. Analytic functions. Cauchy-Riemann theorem and its applications. Elementary functions. Branch cuts and inverses. Integration in the complex plane. Contour integration. Simply connected domains. Independence of the path. Cauchy integral and its use to evaluate derivatives. Liouville theorem for entire functions. The fundamental theorem of algebra. Cauchy-Hadamard criteria for power series. Integration and differentiation of power series. Roots of analytic functions. Weierstrass M-test for uniform convergence of function series. Changing the order of summation and integration. Uniform convergence of function series. Isolated singular points of analytic functions. Residue theorem and its applications. The argument theorem. Branch cut integration. Conformal mapping.