3.52

Describe the image under exp of the line with equation \(y=x\).
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To do this you should find an equation (at least parametrically) for the image (you can start with the parametric form \(x=t,y=t\)), plot it reasonable carefully, and explain what happens in the limits as \(t \rightarrow \infty\) and \(t \rightarrow -\infty\).

Let \(z=x+iy\) be an arbitrary point on the line \(y=x\) plotted on the Complex plane. Parameterize \(y=x\) with \(x=t,y=t\). Then by definition 3.21, the image of \(z\) under \(exp\) is \(e^xe^iy=e^te^{it}\). This is the polar form of a complex number with magnitude of \(e^t\) and argument \(t\). Because \(t\) is arbitrary, every point on the line \(x=t,y=t\) corresponds to a point on \(e^te^{it}\). Now as \(t \rightarrow \infty\), the magnitude of \(z\) will grow exponentially to \(\infty\) and the argument approaches \(\infty\). As \(t \rightarrow -\infty\), the magnitude approaches \(0\). Thus, the image of the line \(y=x\) is a spiral outwards from the origin of the complex plane.