 Solution

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

Parametrize the vertical line as

z = a + i t

t \in R

. Then

e^{z} = e^{a} (cos t + i sin t)

, whose modulus is

e^{a}

and whose argument takes every value. The image is

{w : ∣ w ∣ = e^{a}} .

Increasing

t

traverses the circle counterclockwise, repeating with period

2 π

. There is no injectivity on the full line.

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