 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2007/6/a/ii/solution

Parametrize the horizontal line as

z = t + ib

t \in R

. Then

e^{z} = e^{t} e^{ib}

, and

e^{t}

takes every positive real value exactly once. The image is

{r e^{ib} : r > 0} .

This is the ray with argument

b

2 π

, excluding its endpoint zero. As

t

increases, the image moves outward from arbitrarily small to arbitrarily large radius.

 New to topics? Read the docs here!