PYQs Based Test - 22 : Analytic Function and C-R Equations

An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + i v(x, y), where i = √-1. If u(x, y) = x² – y², then expression for v(x, y) in terms of x, y and a general constant c would be

An analytic function of a complex variable $z=x+iy\left(i=\sqrt{-1}\right)$ is defined as

f(z) = x²- y² + iψ (x, y),

where ψ(x,y) is a real function. The value of the imaginary part of f(Z) at z = (1 + i) is _______ (round off to 2 decimal places).

A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x² − 2y² + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is

Given $f\left(z\right)=g\left(z\right)+h\left(z\right)$, where f, g, h are complex valued functions of a complex variable z. which one of the following statements is TRUE?

Let 𝑓₁(𝑧) = 𝑧² and 𝑓₂(𝑧) = z̅ be two complex variable functions. Here z̅ is the complex conjugate of z. Choose the correct answer.