Reference no: EM13887043
1) (a) Prove that ez1 = ez2 implies z1 - z2 = i 2kΠ.
(b) Prove that the exponential mapping w = exp(z) is locally one-to-one; that is given a point zo, there is an open neighborhood of zo such that for points z1, z2 in this neighborhood the following holds: ez1 = ez2 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)| ≤ e2x + 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 - zo)-1dz| ≤ 2Π, where C is the circle of radius r centered at zo. 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 zo = 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(ez - 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.