- The angle between two non-zero complex numbers z and \(w\) (taken in that order) is simply the oriented angle, in \(( - \pi , \pi ]\) , that is formed between the two vectors in \(\mathbb { R } ^ { 2 }\) corresponding to the points z and w. This oriented angle, say \(\alpha ,\) is uniquely determined by the two quantities
\[ \frac {(z , w)}{| z | | w |} \quad \mathrm{and} \quad \frac {(z , - i w)}{| z | | w |} \]
which are simply the cosine and sine of \(\alpha ,\) respectively. Here, the notation \(( \cdot , \cdot )\) corresponds to the usual Euclidian inner product in \(\mathbb { R } ^ { 2 }\) , which in terms of complex numbers takes the form \(( z , w ) = \operatorname { R e } ( z { \overline { { w } } } )\)
In particular, we may now consider two smooth curves \(\gamma : [ a , b ] \to \mathbb { C }\) and \(\eta :\) \([ a , b ] \to \mathbb { C }\) that intersect at \(z _ { 0 }\) , say \(\gamma ( t _ { 0 } ) = \eta ( t _ { 0 } ) = z _ { 0 }\) , for some \(t _ { 0 } \in ( a , b )\) . If the quantities \(\gamma ^ { \prime } ( t _ { 0 } )\) and \(\eta ^ { \prime } ( t _ { 0 } )\) are non-zero, then they represent the tangents to the curves \(\gamma\) and \(\eta\) at the point \(z _ { 0 }\) , and we say that the two curves intersect at \(z _ { 0 }\) at the angle formed by the two vectors \(\gamma ^ { \prime } ( t _ { 0 } )\) and \(\eta ^ { \prime } ( t _ { 0 } )\)
A holomorphic function \(f\) defined near \(z _ { 0 }\) is said to preserve angles at \(z _ { 0 }\) if for any two smooth curves \(\gamma\) and \(\eta\) intersecting at \(z _ { 0 }\) , the angle formed between the curves \(\gamma\) and \(\eta\) at \(z _ { \mathrm { 0 } }\) equals the angle formed between the curves \(f \circ \gamma\) and \(f \circ \eta\) at \(f ( z _ { 0 } )\) . (See Figure 12 for an illustration.) In particular, we assume that the tangents to the curves \(\gamma , \eta , f \circ \gamma\) , and \(f \circ \eta\) at the point \(z _ { 0 }\) and \(f ( z _ { 0 } )\) are all non-zero.
Figure 12. Preservation of angles at \(z _ { \mathrm { 0 } }\)
- Prove that if \(f : \Omega \to \mathbb { C }\) is holomorphic, and \(f ^ { \prime } ( z _ { 0 } ) \neq 0\) , then \(f\) preserves angles at \(z _ { \mathrm { 0 } }\) . [Hint: Observe that
\[ (f ^ {\prime} (z _ {0}) \gamma^ {\prime} (t _ {0}), f ^ {\prime} (z _ {0}) \eta^ {\prime} (t _ {0})) = | f ^ {\prime} (z _ {0}) | ^ {2} (\gamma^ {\prime} (t _ {0}), \eta^ {\prime} (t _ {0})). ] \]
- Conversely, prove the following: suppose \(f : \Omega \to \mathbb { C }\) is a complex-valued function, that is real-diferentiable at \(z _ { 0 } \in \Omega\) , and \(J _ { f } ( z _ { 0 } ) \ne 0\) . If f preserves angles at \(z _ { \mathrm { 0 } }\) , then \(f\) is holomorphic at \(z _ { \mathrm { 0 } }\) with \(f ^ { \prime } ( z _ { 0 } ) \neq 0\)
Solution
1. Part (a): Holomorphic functions with non-zero derivative preserve angles:
1.1. Let \(\gamma, \eta\) be smooth curves intersecting at \(z_0 = \gamma(t_0) = \eta(t_0)\) with non-zero tangent vectors \(v = \gamma'(t_0)\) and \(w = \eta'(t_0)\).The tangent vectors to the image curves \(\tilde{\gamma} = f \circ \gamma\) and \(\tilde{\eta} = f \circ \eta\) at \(f(z_0)\) are:\[ \tilde{v} = f'(z_0) v \quad \text{and} \quad \tilde{w} = f'(z_0) w. \]
chain rule for complex differentiation.
1.2. The Euclidean inner product on \(\mathbb{R}^2 \cong \mathbb{C}\) is \((z_1, z_2) = \operatorname{Re}(z_1 \bar{z}_2)\).Compute the inner product of \(\tilde{v}\) and \(\tilde{w}\):\[ (\tilde{v}, \tilde{w}) = \operatorname{Re}\big((f'(z_0) v) \overline{(f'(z_0) w)}\big) = \operatorname{Re}\big(|f'(z_0)|^2 v \bar{w}\big) = |f'(z_0)|^2 \operatorname{Re}(v \bar{w}) = |f'(z_0)|^2 (v, w). \]
algebraic properties of complex conjugation and modulus.
1.3. The norms are \(|\tilde{v}| = |f'(z_0)| |v|\) and \(|\tilde{w}| = |f'(z_0)| |w|\).Thus the cosine of the angle \(\tilde{\alpha}\) between \(\tilde{v}\) and \(\tilde{w}\) satisfies:\[ \cos \tilde{\alpha} = \frac{(\tilde{v}, \tilde{w})}{|\tilde{v}| |\tilde{w}|} = \frac{|f'(z_0)|^2 (v, w)}{|f'(z_0)|^2 |v| |w|} = \frac{(v, w)}{|v| |w|} = \cos \alpha. \]
2 and norm scaling.
1.4. Similarly, compute the sine of the angle \(\tilde{\alpha}\):\[ (\tilde{v}, -i\tilde{w}) = \operatorname{Re}\big((f'(z_0) v) \overline{(-i f'(z_0) w)}\big) = |f'(z_0)|^2 \operatorname{Re}(v \cdot i \bar{w}) = |f'(z_0)|^2 (v, -iw). \]Dividing by \(|\tilde{v}| |\tilde{w}|\) yields:\[ \sin \tilde{\alpha} = \frac{(\tilde{v}, -i\tilde{w})}{|\tilde{v}| |\tilde{w}|} = \frac{(v, -iw)}{|v| |w|} = \sin \alpha. \]
definition of oriented sine.
1.5. Since \(\cos \tilde{\alpha} = \cos \alpha\) and \(\sin \tilde{\alpha} = \sin \alpha\), the oriented angle is preserved: \(\tilde{\alpha} = \alpha\).
angle is uniquely determined in \((-\pi, \pi]\) by its sine and cosine.
2. Part (b): Preservation of oriented angles implies holomorphicity:
2.1. Let \(T = Df(z_0): \mathbb{R}^2 \to \mathbb{R}^2\) be the real total derivative, represented by the Jacobian matrix:\[ J = \begin{pmatrix} u_x & u_y \\ v_x & v_y \end{pmatrix}, \quad \text{with } \det(J) \neq 0. \]
real differentiability of \(f = u + iv\).
2.2. Since \(f\) preserves oriented angles at \(z_0\), the linear map \(T\) preserves oriented angles between all pairs of non-zero vectors in \(\mathbb{R}^2\).
tangent mapping of smooth curves.
2.3. Consider the standard basis \(e_1 = (1, 0)^T\) and \(e_2 = (0, 1)^T\).\(e_1\) and \(e_2\) are orthogonal of equal length with \(\det(e_1, e_2) = 1 > 0\).Thus their images \(T e_1 = (u_x, v_x)^T\) and \(T e_2 = (u_y, v_y)^T\) must be orthogonal, have equal non-zero length \(\lambda > 0\), and satisfy \(\det(T e_1, T e_2) > 0\):\[ u_x u_y + v_x v_y = 0, \qquad u_x^2 + v_x^2 = u_y^2 + v_y^2 = \lambda^2 > 0, \qquad u_x v_y - u_y v_x > 0. \]
preservation of right angles and orientation.
2.4. Any \(2 \times 2\) matrix satisfying these conditions has the form of a positive scalar times a rotation matrix:\[ J = \begin{pmatrix} a & -b \\ b & a \end{pmatrix} \quad (a, b \in \mathbb{R}, \, a^2 + b^2 = \lambda^2 > 0). \]Therefore:\[ u_x = v_y = a, \qquad u_y = -v_x = -b. \]
classification of orientation-preserving conformal linear maps on \(\mathbb{R}^2\).
2.5. The equations in 4 are the Cauchy–Riemann equations for \(f\) at \(z_0\).Since \(f\) is real-differentiable at \(z_0\) and satisfies the Cauchy–Riemann equations, \(f\) is complex-differentiable (holomorphic) at \(z_0\) with:\[ f'(z_0) = u_x(z_0) + i v_x(z_0) = a + ib \neq 0. \]
complex differentiability characterization via Cauchy–Riemann equations and real differentiability.
3. Conclusion:\(f\) preserves angles if and only if \(f\) is holomorphic with \(f'(z_0) \neq 0\). Q.E.D.
1 and 2.