Complex Analysis (MC2061) 2024-2025 - Quiz I

• No, the function is only differentiable at z=0.
• Yes, the function is differentiable at all real numbers only.
• The function is not differentiable anywhere in the complex plane.
• Yes, the function is differentiable at all points in the complex plane.

Which function property implies another?

Differentiable, Continuous, Analytic

• Continuous implies Differentiable
• Analytic implies Differentiable

• Differentiable implies Analytic
• Differentiable implies Continuous

What is Cauchy Riemann Equation?

• The Cauchy-Riemann equations are ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x.
• The Cauchy-Riemann equations are ∂u/∂x = ∂v/∂y and ∂u/∂y = ∂v/∂x.

• The Cauchy-Riemann equations state that ∂u/∂x = ∂v/∂x and ∂u/∂y = ∂v/∂y.
• The Cauchy-Riemann equations state that ∂u/∂x = ∂v/∂x and ∂u/∂y = -∂v/∂y.

• f'(z) = ∂u/∂x + i(∂v/∂x)

• f'(z) = ∂u/∂y + i(∂v/∂y)
• f'(z) = ∂u/∂x - i(∂v/∂x)
• f'(z) = ∂v/∂y - i(∂u/∂y)

.......................... provide a necessary condition for a function to be differentiable at a point.

• Continuity at that point.
• Continuity in the entire domain.

• Existence of a limit at that point.
• Cauchy-Riemann Equations

Write True or False: If a function satisfies the Cauchy-Riemann equations, it is necessarily differentiable.

Choose the correct answer.

• (i). Differentiable implies C.R. equation satisfied

  (ii). Satisfies C.R. equation implies not differentiable

  (iii). Satisfies C.R. equation and first order partial derivatives are continuous implies Differentiable

• (ii). Satisfies C.R. equation implies not differentiable

  (iii). Satisfies C.R. equation and first order partial derivatives are continuous implies Differentiable

• (i). Satisfies C.R. equation implies Differentiable

  (ii). Satisfies C.R. equation and first order partial derivatives are continuous implies Differentiable

• None of these

Complex form of C.R. Equation

Cauchy-Riemann Equation in complex form