Which of the following statements regarding complex variable mappings and bilinear (Möbius) transformations $w = \frac{a z + b}{c z + d}$ ($ad - bc \neq 0$) are **CORRECT**?

(A) A bilinear transformation maps circles and straight lines in the $z$-plane into circles and straight lines in the $w$-plane

(B) The cross-ratio of any four distinct points $(z_1, z_2, z_3, z_4)$ is invariant under a bilinear transformation

(C) A mapping $w = f(z)$ is conformal at every point $z_0$ where $f(z)$ is analytic and $f'(z_0) \neq 0$

(D) The Joukowski transformation $w = z + \frac{c^2}{z}$ is conformal everywhere in the complex plane, including at the critical points $z = \pm c$

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles 1. **Circle-Preserving Property:** Bilinear transformations map the extended complex plane onto itself. Generalized circles (circles and straight lines, where a line is a circle of infinite radius) always map to generalized circles. 2. **Invariance of Cross-Ratio:** For any four points, the cross-ratio $\frac{(w_1 - w_4)(w_3 - w_2)}{(w_1 - w_2)(w_3 - w_4)} = \frac{(z_1 - z_4)(z_3 - z_2)}{(z_1 - z_2)(z_3 - z_4)}$ is strictly invariant under all Möbius maps. 3. **Conformality:** A mapping is conformal (preserves angles in magnitude and sense) at all points where $f(z)$ is analytic and its derivative does not vanish ($f'(z) \neq 0$). 4. **Joukowski Transformation:** $w = z + \frac{c^2}{z} \implies \frac{dw}{dz} = 1 - \frac{c^2}{z^2} = \frac{z^2 - c^2}{z^2}$. At $z = \pm c$, $\frac{dw}{dz} = 0$ (critical points). Hence, conformality breaks down at $z = \pm c$ (angles are doubled, which allows forming the sharp trailing edge of an airfoil cusped or finite-angle wedge). ### 2. Statement Evaluation - **Statement A:** **CORRECT.** Generalized circles map to generalized circles. - **Statement B:** **CORRECT.** Cross-ratio is invariant. - **Statement C:** **CORRECT.** Standard definition of conformality. - **Statement D:** **INCORRECT.** Conformality fails at $z = \pm c$ because $f'(z) = 0$. ### 3. Exam Trap & Standard Reference - **Trap Alert:** Critical points where $f'(z) = 0$ are precisely where conformality fails. - **Standard Reference:** Churchill, R.V. & Brown, J.W., *Complex Variables and Applications*, Chapter 8 & 9.

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