Question 7.1.3: Conformal Mappings. Find all points where the mapping f(z) =......
Conformal Mappings
Find all points where the mapping f(z) = sin z is conformal.
The function f(z) = sin z is entire, and from Section 4.3 we have that f^{\prime}(z) = cos z. In (21) of Section 4.3 we found that cos z = 0 if and only if z = (2n + 1)π/2, n = 0, ±1, ±2, . . . , and so each of these points is a critical point of f.
\cos z=0\ {\mathrm{if~and~only~if~}}z={\frac{(2n+1)\pi}{2}} for n = 0, ±1, ±2, . . . . (21) of Section 4.3
Therefore, by Theorem 7.1, w = sin z is a conformal mapping at z for all z = (2n + 1)π/2, n = 0, ±1, ±2, . . . . Furthermore, by Theorem 7.2, w = sin z is not a conformal mapping at z if z \neq (2n + 1)π/2, n = 0, ±1, ±2, . . . . Because f^{\prime \prime}(z) = −sin z = ±1 at the critical points of f, Theorem 7.2 also indicates that angles at these points are increased by a factor of 2.