Complex Analysis Notes for BS/MSc

Complex Analysis Notes for BS/MSc

Complex analysis also known as the theory of functions of a complex variable, is a branch of mathematics that deals with complex numbers and the functions of a complex variable. It explores the properties, functions, and calculus of complex numbers and their extensions.

Complex analysis is a branch of mathematics that explores the properties and behavior of functions defined on the complex plane, involving complex numbers. It combines elements of calculus and complex numbers to study functions of a complex variable.

1. Complex Numbers and Functions:

• Complex numbers as (a + bi), where (a) and (b) are real numbers and (i) is the imaginary unit ((i^2 = -1)).
• Complex functions (f(z)) of a complex variable (z = x + iy).
• Analytic functions, Cauchy-Riemann equations.

1. Complex Integration:

• Contour integrals and paths in the complex plane.
• Cauchy’s integral theorem and integral formula.
• Residue theorem and applications.

1. Power Series and Taylor Series:

• Representation of complex functions as power series.
• Convergence of power series.
• Taylor series expansion and Laurent series.

1. Singularities:

• Classification of singularities (poles, removable singularities, essential singularities).
• Residue at a singularity.

1. Conformal Mapping:

• Definition and properties of conformal mappings.
• Examples of conformal mappings.
• Riemann mapping theorem.

Complex Differentiation:

• Derivatives of complex functions.
• Entire functions and meromorphic functions.
• Liouville’s theorem.

1. Complex Integration Techniques:

• Contour integration techniques (e.g., residues, partial fractions).
• Evaluation of real integrals using complex analysis.

1. Analytic Continuation:

• Definition and examples.
• Monodromy theorem.

1. Harmonic Functions:

• Definition and properties of harmonic functions.
• Poisson’s formula.

1. Applications:
   • Applications to physics and engineering (e.g., potential theory, fluid dynamics).
   • Analytic functions in electrical engineering.
2. Advanced Topics:
   • Schwarz reflection principle.
   • Mittag-Leffler theorem.
   • Special functions (e.g., Gamma function, Zeta function).

Complex analysis has applications in various scientific and engineering fields, providing powerful tools to understand and manipulate functions involving complex variables. The subject also lays the groundwork for more advanced topics in mathematics, such as Riemann surfaces and algebraic geometry. The study of complex analysis enhances problem-solving skills and contributes to a deeper understanding of mathematical structures.

Complex Analysis Notes for BS/MSc

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