Mathematics- Complex Analysis-

Complex Numbers- Algebraic and Transcendental Complex Numbers, Real and Imaginary Complex Numbers, Pure Imaginary Numbers, Pure Real Numbers, Complex Conjugates, Roots of Unity, Primitive Roots of Unity, Complex Numbers in Polar Form.

Analytic Functions- Polynomial Functions, Exponential Functions, Trigonometric Functions, Logarithmic Functions, Complex Analytic Functions, Rational Functions, Hyperbolic Functions, Special Functions.

Complex Integration- Line Integral, Contour Integral, Closed Contour Integral, Path-Independent Integrals, Line Integrals over Paths in the Complex Plane, Improper Integrals, Parametric Integrals, Path Integral in Quantum Mechanics.

Residue Theory- Simple Residues, Higher Order Residues, Isolated Singularities, Integration Contours, Residue Theorem, Cauchy’s Residue Theorem, Computational Techniques.

Conformal Mapping- Linear Fractional Transformations (Möbius Transformations), Bilinear Transformations, Schwarz-Christoffel Transformation, Joukowsky Transformation, Riemann Mapping Theorem, Conformal Mappings of Special Regions, Conformal Mapping in Fluid Dynamics.

Power Series Representation- Taylor Series, Maclaurin Series, Binomial Series, Fourier Series.

Singularities- Removable Singularity, Pole, Essential Singularity, Branch Point, Isolated Singularity.

Analytic Continuation- Holomorphic Continuation, Meromorphic Continuation, Multiplicative Continuation, Integral Continuation, Analytic Continuation Along Paths, Pseudoanalytic Continuation, Continuation of Solutions to Differential Equations, Riemann Surfaces.

Special Functions- Trigonometric functions, Exponential and Logarithmic functions, Bessel functions, Legendre functions, Gamma function, Hypergeometric functions, Error functions, Elliptic functions, Zeta functions, Special orthogonal polynomials.