Answered: If l is a line through (0,0), the point…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

If l is a line through (0,0), the point (0,0) divides the line l into two rays. Also, a complex-valued function g on a set S is said to be bounded if there is some finite M such that |g| < M on the set S. So we need to find out for which rays through the origin does e^z satisfy this property.

Determine which rays starting at the origin are such that e² is bounded
on the ray.

Expert Solution

Step 1: Step 1:

To determine for which rays starting at the origin $e^{z}$ is bounded, we need to analyze the behavior of the complex exponential function $e^{z}$ along these rays.

First, let's consider the complex exponential function $e^{z}$ , where $z = x + i y$ for real numbers x and y.

$e to the power of z equals e to the power of x plus i y end exponent space space space space space equals e to the power of x cross times e to the power of i y end exponent$

The magnitude of $e^{z}$ is given by:

$vertical line e to the power of z vertical line equals vertical line e to the power of x cross times e to the power of i y end exponent vertical line space space space space space space space equals vertical line e to the power of x vertical line cross times vertical line e to the power of i y end exponent vertical line$

Now, $vertical line e to the power of i y end exponent vertical line equals 1$ , as $e to the power of i y end exponent$ traces a unit circle in the complex plane, and its magnitude is always 1.

So, $vertical line e to the power of z vertical line equals vertical line e to the power of x vertical line cross times 1 space space space space space space space equals vertical line e to the power of x vertical line$