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in-depth study guide for both IGNOU's MMT005 Complex Analysis and MMT007 Differential Equations and Numerical Solutions:MMT005 Complex Analysis:Fundamental Concepts: Begin by understanding the basic concepts of complex numbers, including arithmetic operations, polar form, and complex conjugates.Analytic Functions: Study the definition and properties of analytic functions, including Cauchy-Riemann equations, harmonic functions, and the concept of holomorphic functions.Complex Integration: Explore complex integration techniques, such as contour integration, Cauchy's integral theorem, and Cauchy's integral formula. Understand the residue theorem and its applications in evaluating complex integrals.Power Series and Laurent Series: Learn about power series and Laurent series representations of complex functions. Understand their convergence properties and applications in solving differential equations and evaluating complex integrals.Conformal Mapping: Delve into the concept of conformal mapping and its significance in preserving angles and shapes. Study common conformal mappings, such as linear transformations, exponential functions, and fractional linear transformations.Analytic Continuation: Understand the concept of analytic continuation and its role in extending the domain of analytic functions beyond their original region of definition.Applications: Explore practical applications of complex analysis in various fields, such as physics, engineering, and mathematical finance. Understand how complex analysis techniques are used to solve problems in fluid dynamics, electromagnetism, and signal processing.Additional Topics: Explore advanced topics in complex analysis, such as singularities, branch points, Riemann surfaces, and the theory of entire functions.