A First Course in Complex Analysis · Problem 38

Integration — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · complex-trig-functions

Problem

Determine the image of the strip

\{z \in \mathbb{C} : -\frac{\pi}{2} < \operatorname{Re} z < \frac{\pi}{2}\}

under the sine function. Hint. Exercise 3.6.30 makes it easy to convert parametric equations for horizontal or vertical lines to parametric equations for their images. Note that the equations

x = A\sin t

and

y = B\cos t

represent an ellipse and the equations

x = A\cosh t

and

y = B\sinh t

represent a hyperbola. Start by finding the images of the boundary lines of the strip, and then find the images of a few horizontal segments and vertical lines in the strip.

Related theory

Complex Numbers Arithmetic And The Complex Plane

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