Complex Analysis (MC2061)

Statement: The function $$f\left(z\right)=\overline{z}$$ is nowhere differentiable. Choose the correct reason.

Choose the correct statement(s)

(i) If $$f\left(z\right)=u+iv$$ is differentiable at a point, then $$u$$ and $$v$$ satisfies the C.R equation.

(ii) If $$u$$ and $$v$$ does not satisfies the C.R equation at any point, then $$f\left(z\right)=u+iv$$ is not differentiable.

(iii) If $$u$$ and $$v$$ satisfies the C.R equation at a point, then $$f\left(z\right)=u+iv$$ is differentiable at that point.

(iv) If the first order partial derivatives of $$u$$ and $$v$$ are continuous at a point, then $$f\left(z\right)=u+iv$$ is differentiable at that point.

Say True or False: If a function is differentiable at a point, then it is continuous at that point.

Which of the following is/ are nowhere differentiable?

A function which is analytic at every point of the complex plane is called .............. function

Say True or False: An analytic function with its derivative zero at every point of the domain is constant

Which of the following is/are correct?

(i) An analytic function in a region with constant modulus is constant.

(ii) An analytic function with arg $$f\left(z\right)$$ constant is constant.

(iii) An analytic function whose real part is constant is constant.

(iv) If $$f\left(z\right)$$ and $$\overline{f\left(z\right)}$$ are analytic, then $$f\left(z\right)$$ is constant.

If $$v$$ is a harmonic conjugate of $$u$$ , then what is the harmonic conjugate of $$-v$$ ?

Which of the following given below is/are elementary transformations?

Choose the correct answers