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Chapter 17: Problem 2

Find and sketch the image of the given region under \(w=e^{x}\). $$-1 \leqq x \leqq 0,0 \leq y \leq \pi / 2$$

Short Answer

Expert verified
The image is a circular sector in the first quadrant, radius \(\frac{1}{e}\leq r\leq 1\), angle \(0\) to \(\frac{\pi}{2}\).

Step by step solution

01

Understand the problem

We are given a region defined by \[-1 \leq x \leq 0,\quad 0 \leq y \leq \frac{\pi}{2}\]and need to find and sketch the image of this region under the transformation \[w = e^z = e^{(x + iy)}.\]
02

Apply the transformation

Using the complex number transformation, express it in magnitude and argument:\[w = e^{x+iy} = e^x e^{iy} = e^x (\cos y + i \sin y).\]This means the image is a region in the complex plane with radius controlled by \(e^x\) and a full angle range from \(0\) to \(\frac{\pi}{2}\) due to \(y\).
03

Evaluate the range for the real axis

Evaluate the interval \(-1 \leq x \leq 0\):- For \(x = -1\), \(e^x = e^{-1} = \frac{1}{e}.\)- For \(x = 0\), \(e^x = e^0 = 1.\) Therefore, the magnitude of \(w\) ranges from \(\frac{1}{e}\) to \(1\).
04

Evaluate the range for the imaginary axis

The argument of \(w\) is given by \(y\):- For \(y = 0\), the argument is \(0\).- For \(y = \frac{\pi}{2}\), the argument is \(\frac{\pi}{2}\).Thus, the region describes the first quadrant arc from angle \(0\) to \(\frac{\pi}{2}\) in the complex plane.
05

Sketch the image

The image of the region is a circular sector in the complex plane:- Radius: \(\frac{1}{e} \leq r \leq 1\).- Angle: from \(0\) to \(\frac{\pi}{2}\).Sketch this as a sector in the first quadrant, originating from the origin with an outer arc extending between the radius values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Transformation
Complex transformations change a complex number into another by applying a specified function. In our exercise, the transformation is given by the exponential function:
  • The transformation is expressed as \[ w = e^z = e^{(x + iy)} \]where \(z = x + iy\) is the complex variable.
  • In the exponential transformation, the complex number \(z\) is transformed into \(w\) by separating the real and imaginary components.
  • Here, \( e^z \) implies converting \( e^x \) as the magnitude and \( e^{iy} \) as the direction (angle).
  • The real part, \( e^x \), changes the distance from the origin, while the imaginary part provides the rotational angle, resulting in a new mapping of the original region.
Complex transformations can significantly alter shapes and structures when viewed in the complex plane, often creating curved and twisted images from simple rectangular regions.
Complex Plane
The complex plane, also known as the Argand plane, is a two-dimensional plane for representing complex numbers. Each complex number \(a + bi\) can be portrayed as a point or vector:
  • The horizontal axis (real axis) denotes the real part of a complex number.
  • The vertical axis (imaginary axis) stands for the imaginary part.
This plane offers a practical way to visualize complex numbers and their transformations. In the context of our exercise:
  • The transformation \( w = e^z \) reshapes the region defined within the complex plane.
  • Instead of treating complex numbers simply as a combination of real and imaginary parts, this plane allows them to be seen as geometric points which undergo dynamic transformations.
Understanding the complex plane is essential for picturing how regions transform under operations involving complex variables.
Circular Sector
A circular sector is a portion of a circle enclosed by two radii and an arc. It resembles a 'slice' of the circle. In complex analysis, transformations can map rectangular regions into unusual shapes like circular sectors, as seen in our example:
  • The arc is bounded by angles, ranging from \(0\) to \(\frac{\pi}{2}\) provided in the given transformation, which influences the sector's shape.
  • The radii are defined by the transformation's magnitude criteria: \( \frac{1}{e} \leq r \leq 1 \), which establishes the radial boundaries.
As such, the first quadrant of the standard rectangular region under the transformation becomes a circular sector. This sector can be visualized as a section of a circle defined in the complex plane, beautifully integrating the transformation properties with a geometric interpretation. This visual representation aids in simplifying the understanding of complex transformations in mathematical tasks.

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Most popular questions from this chapter

Find and sketch or graph the image of the given region under \(w=\cos z\). $$0 < x < \pi / 2,0 < y < 2$$

Show that substituting a linear fractional transformation ( \(\mathrm{LFT}\) ) into a LFT gives a LFT.

Find the LFT that maps the given three points onto the three given points in the respective order. $$0,1, \infty \text { onto } \infty, 1,0$$

Find the magnification ratio \(M\). Describe what it tells you about the mapping. Where is \(M\) equal to \(1 ?\) Find the Jacobian \(J\) $$w=z^{3}$$

Find and sketch or graph the image of the given region under the given mapping. $$x \geq 1, w=1 / z$$
See all solutions

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