Which of the following properties hold(s) for every complex function $f(z) = u(x, y) + i v(x, y)$ that is analytic in an open domain $D$?
Options:
(A) The Cauchy–Riemann equations $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ hold throughout $D$
(B) Both real and imaginary parts $u(x, y)$ and $v(x, y)$ are harmonic: $\nabla^2 u = 0$ and $\nabla^2 v = 0$
(C) The family of curves $u(x, y) = c_1$ and $v(x, y) = c_2$ intersect orthogonally at every point where $f'(z) \neq 0$
(D) $f(z)$ possesses derivatives of all orders throughout $D$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
For an analytic function $f(z) = u + iv$:
1. **Cauchy–Riemann Equations (Option A)**: Differentiability with respect to $z$ requires $u_x = v_y$ and $u_y = -v_x$ (True).
2. **Harmonicity (Option B)**: Differentiating C-R equations gives $u_{xx} + u_{yy} = v_{yx} - v_{xy} = 0$, so both $u$ and $v$ satisfy Laplace's equation (True).
3. **Orthogonality of Level Curves (Option C)**: $\nabla u \cdot \nabla v = u_x v_x + u_y v_y = (v_y)(-u_y) + u_y(u_x) = 0$, so gradients are orthogonal, meaning level contours intersect orthogonally (True).
4. **Infinite Differentiability (Option D)**: Complex analyticity in an open domain implies existence of derivatives of all orders (True).