By Beck M., Marchesi G., Pixton G.
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Remarks. 1. The third identity is a very special one and has no counterpart for the real exponential function. It says that the complex exponential function is periodic with period 2πi. This has many interesting consequences; one that may not seem too pleasant at first sight is the fact that the complex exponential function is not one-to-one. 2. The last identity is not only remarkable, but we invite the reader to meditate on its proof. 5 is reasonable. Finally, note that the last identity also says that exp is entire.
Then f has a primitive in G. Proof. Fix a point a ∈ G and let F (z) = f γz where γz is any smooth curve from a to z. We should make sure that F is well defined: Suppose δz is another smooth curve from a to z then γz − δz is closed and G-contractible, as G is simply connected. 5 0= f− f= γz −δz γz f δz which means we get the same integral no matter which path we take from a to z, so F is a well-defined function. It remains to show that F is a primitive of f : F (z + h) − F (z) 1 = lim h→0 h→0 h h f− F (z) = lim γz+h f .
Find the M¨ obius transformation f : (a) f maps 0 → 1, 1 → ∞, ∞ → 0. (b) f maps 1 → 1, −1 → i, −i → −1. (c) f maps x-axis to y = x, y-axis to y = −x, and the unit circle to itself. ˆ Show that z is on the circle passing through 16. Suppose z1 , z2 and z3 are distinct points in C. by z1 , z2 and z3 if and only if [z, z1 , z2 , z3 ] is real or infinite. 17. Describe the images of the following sets under the exponential function exp(z): (a) the line segment defined by z = iy, 0 ≤ y ≤ 2π. (b) the line segment defined by z = 1 + iy, 0 ≤ y ≤ 2π.