Theorem, Proof, Definition | Analytic functions
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
The transformation w = az + b/ cz + d’ – bc ≠ 0 where a, b, c, d are complex numbers, is called a bilinear transformation.
Problems with Answer | Analytic functions
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
Probability and complex function: Unit III: Analytic functions : Exercise 3.4
Solved Example Problems | Analytic functions
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
The transformation w = C + z, where C is a complex constant, represents a translation.
Solved Example Problems | Analytic functions
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
Consider the transformation w = f (z), where f (z) is a single valued function of z, under this transformation, a point z0 and any two curves C1 and C2 passing through z0 in the z plane, will be mapped onto a point w0 and two curves C1' and C2' in the w plane.
Problems with Answer
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
Probability and complex function: Unit III: Analytic functions : Exercise 3.3
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
Probability and complex function: Unit III: Analytic functions : Problems based on construction of analytic functions
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
There are three methods to find f (z).
Analytic functions
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
Probability and complex function: Unit III: Analytic functions : Problems based on harmonic conjugate
Analytic functions
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
As u and v are harmonic, their first partial derivatives are continuous. They satisfy C-R conditions also, by the definition of harmonic conjugate.
Analytic functions
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
A real function of two real variables x and y that possesses continuous second order partial derivatives and that satisfies Laplace equation is called a harmonic function.
Problems with Answer
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
Probability and complex function: Unit III: Analytic functions : Exercise 3.1
Solved Example Problems
Subject and UNIT: Probability and complex function: Unit III: Analytic functions
Probability and complex function: Unit III: Analytic functions : Examples