Describe the images of the following sets under the exponent

Describe the images of the following sets under the exponential function e^(z) :

(a) the line segment defined by z = iy, 0 y 2. Hint: Set w = e^(iy) and plot the values of w as a function of y.

(b) the line segment defined by z = 1 + iy, 0 y 2. Hint: Set w = e^(1+iy) = ee^(iy) .

(c) the rectangle {z = x + iy C : 0 x 1, 0 y 2}. Hint: Find the image of the boundary first.

Solution

MATLAB solution

close all
clear all
clc

%% part a

y=0:0.1:2*pi;
z=complex(0,y);
w=exp(z);
figure(1)
plot(y,w)

xlabel(\'y\')
ylabel(\'w\')

%% part b

y=0:0.1:2*pi;
z=exp(1)*complex(0,y);
w=exp(z);
figure(2)
plot(y,w)
xlabel(\'y\')
ylabel(\'w\')

%% part c

x=0:0.1:1;
y=0:(2*pi)/10:2*pi;
z=complex(x,y);
w3=exp(z);
figure(3)
plot(y,w3)
xlabel(\'y\')
ylabel(\'w\')