Introduction to Complex Numbers
1.1 ANALYTIC FUNCTION
A function is said to be analytic in a domain D if it is defined and differentiable at every point within that domain. This means that the function has a well-defined and continuous derivative at all points in D. Analytic functions exhibit smooth and well-behaved behavior within their domains.
Holomorphic FunctionsAnother term used to describe analytic functions in a domain D is "holomorphic." Holomorphic functions are those that possess complex differentiability throughout the entire domain. In simpler terms, a holomorphic function is analytic in D, indicating that it can be expanded into a Taylor series and exhibits properties like continuity and differentiability.
Entire FunctionsAn entire function refers to a function that is analytic and well-defined at all points in the complex plane. In other words, it is a function that is analytic throughout the entire complex domain. Polynomials of any degree are examples of entire functions, as their derivatives exist at every point. Entire functions have infinite radii of convergence for their power series expansions.
1.2. CAUCHY RIEMANN EQUATIONS
1.3. HARMONIC FUNCTION
1.4 CAUCHY’S INTEGRAL FORMULA
1.5 SINGULARITY
1.6 RESIDUES
Operations with Complex Numbers
Understanding the operations involving complex numbers is crucial. Addition, subtraction, multiplication, and division of complex numbers follow specific rules. For example, to add two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2, we simply add their real and imaginary parts: z1 + z2 = (a1 + a2) + i(b1 + b2). Similarly, subtraction, multiplication, and division can be performed following the respective rules.
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