Question

Cauchy Riemann Equations for an Analytic function are

• A. \( u_x = v_y; u_y = -v_x \)
• B. \( u_x = v_x; u_y = v_y \)
• C. \( u_x = v_y; u_y = v_x \)
• D. \( u_x = -v_x; u_y = -v_y \)

Hint:

Cauchy-Riemann Equations. Necessary conditions for a complex function \(f(z)=u+iv\) to be analytic: \(u_x = v_y\) and \(u_y = -v_x\).

Answer 1

The Correct Option is A

Let \( f(z) = u(x, y) + i v(x, y) \) be a complex function, where \( z = x + iy \), and \( u \) and \( v \) are real-valued functions of \( x \) and \( y \)
For \( f(z) \) to be analytic (differentiable) in a region, its real part \( u \) and imaginary part \( v \) must satisfy the Cauchy-Riemann equations throughout that region
These equations relate the partial derivatives of \( u \) and \( v \) with respect to \( x \) and \( y \): $$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad (\text{or } u_x = v_y) $$ $$ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \quad (\text{or } u_y = -v_x) $$ This matches option (1)