1) (a) Prove that e^z1 = e^z2 implies z₁ - z₂ = i 2kΠ.

(b) Prove that the exponential mapping w = exp(z) is locally one-to-one; that is given a point z_o, there is an open neighborhood of z_o such that for points z₁, z₂ in this neighborhood the following holds: e^z1 = e^z2 only if z₁ = z₂ (Hint: you may want to try using part (a) ).

2) In each case, find all roots in rectangular coordinates and exhibit them as vertices of certain regular polygons:

(a) (-16)^1/4.

(b) (-8 - i8√3)^1/4.

(c) (-1)^1/3.

(d) (8)^1/6.

3) Find the four zeros of the polynomial z⁴ + 4 = 0.

4) Show if the following functions are analytic or not. In case they are analytic indicate the corresponding domain:

(a) f(z) = exp(z^-).

(b) f(z) = exp(z²).

5) Show the following inequalities:

(a) |exp(2z + i) + exp(iz²)| ≤ e^2x + e^-2xy.

(b) |exp(z²)| ≤ exp(|z|²)

6) Show that:

(a) Log[(1 + i)²]= 2Log(1 + i).

(b) Log[(-1 + i)²] ≠ 2Log(-1 + i).

7) Show that:

(a) The function f (z) = Log(z - i) is analytic everywhere except on the portion x ≤ 0 of the line y = 1.

(b) The function f(z) = Log(z + 4)/(z² + i) is analytic everywhere except at the points ± (1 - i)/√2 and on the portion x ≤ -4 of the real axis.

8) Mapping by the exponential w = exp(z). Sketch the following sets in the z-plane and their images under the exponential function in the w-plane. Indicate where the boundaries are mapped.

Here z = x + iy.

(a) Ω₁ = {z : x < 0, -Π < y ≤ Π}.

(b) Ω₂ = {z : x < 0 < ln(2), -Π < y ≤ Π}.

(c) Ω₃ = {z : x < 0, 0 < y < Π}.

(d) Ω₄ = {z : x ≥ 0, -Π < y ≤ Π}.

9) Show that: (-1)^1/Π = e^(2n+1)i n = 0, ±1, ±2,.......

10) Prove the following inequality |∫_c(z - z_o)^-1dz| ≤ 2Π, where C is the circle of radius r centered at z_o. Compare the inequality with the exact value of the integral.

11) Without evaluating the integral, show that

|∫_c dz/(z²-1) ≤ Π/3

where C is the arc of the circle |z|= 2 joining z_o = 2 and z₁ = 2i.

12) Let C denote the boundary of the triangle with vertices at the points 0, 3i and -4 oriented in the counterclockwise direction. Show that: |∫_c(e^z - z^-)dz ≤ 60.

13) Provide an example of a function f(z), defined on an open set Ω, that is analytic on Ω but such that f (z) is not the derivative F(z) of another function F(z) analytic throughout F'(z) of another function F(z) analytic throughout Ω.