Problem 1 Find and sketch the image of the... [FREE SOLUTION]

Chapter 17: Problem 1

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

Short Answer

Expert verified

The image is an annular region in the complex plane with inner radius 1 and outer radius $e^2$.

Step by step solution

Understand the transformation

The transformation given is of the form $ w = e^z $, where $ z = x + iy $. Hence, $ w = e^x \cdot e^{iy} $. This means the transformation maps points $ (x,y) $ in the Cartesian plane to points in the complex plane.

Analyze the real part transformation

In $ w = e^{x+iy} $, the modulus $ |w| = e^x $. We see that $ x $ controls the magnitude of $ w $. Given the range $ 0 \leq x \leq 2 $, this maps to the magnitude range $ 1 \leq |w| \leq e^2 $.

Analyze the imaginary part transformation

The argument of $ w = e^{iy} $ gives us the angle in the complex plane. Since $ -\pi \leq y \leq \pi $, the angle range of $ w $ is from $-\pi $ to $\pi$, covering the entire semicircle in the complex plane.

Determine the shape of the image

Combining both analyses, the image under the transformation $ w = e^z $ is an annular region, with an inner radius of 1 and an outer radius of $ e^2 $. It extends over the angles $ -\pi $ to $ \pi $, covering a full circle from negative to positive real axis.

Sketch the image

To sketch, draw a circle centered at the origin with a radius ranging from 1 to $ e^2 $. This defines an annular region. The region spans the full circle because $ y $ covers all arguments between $-\pi$ and $\pi$.

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Key Concepts

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

Complex Transformation

A complex transformation is a process that turns one shape or region in the complex plane into another. When we talk about a transformation like $ w = e^z $, we are mapping complex numbers $ z = x + iy $ to another complex number $ w $. This particular transformation involves both the real part $ x $ and the imaginary part $ y $.
Transformation functions are powerful,

providing a way to manipulate regions in the complex domain.
They often have applications in complex analysis, a field critical to engineering and physics.
For instance, in our exercise, we're using the exponential function to transform a rectangular region into an annular region in the complex plane.

Understanding complex transformations helps unravel many complex problems by changing the domain in which we work. Just remember, each transformation has its own set of rules and effects on the complex plane geometries.

Conformal Mapping

Conformal mapping is a concept where transformations preserve angles. Although the shape of a region can change under a conformal map, angles between intersecting curves remain unchanged.
When applying the transformation $ w = e^z $, the mapping is conformal except at points where the function derivative is zero, which is not common in our specific scenario.
This preservation of angles

allows complex functions to be useful in fields such as fluid dynamics and electromagnetism.
Ensures that local geometrical structures within a region are maintained post-mapping.
Provides a consistent way to understand transformations across different complex scenarios.

Conformal mappings, like the one in this exercise, enable us to move easily from the complicated real-world problems to simpler geometries for analysis, while still maintaining essential characteristics like angles.

Complex Plane

The complex plane, also known as the Argand Plane, is a way of visualizing complex numbers geometrically. Each complex number $ z = x + iy $ is represented as a point, $ (x, y) $, with the real part $ x $ and the imaginary part $ y $. It's like a regular xy-coordinate plane, but with an added twist 鈥� literally!
Consider the exercise's region defined by $ 0 \leq x \leq 2 $ and $-\pi \leq y \leq \pi $. This specifies a rectangular region in the complex plane.
The transformation $ w = e^z $

morphs this rectangle into an annular region.
The real part determines the "distance" from the origin, as we transform $ |w| = e^x $.
The imaginary part rotates these distances, covering angles from $-\pi$ to $\pi$.

Visualizing these transformations on the complex plane helps us better understand novel representations and the vast possibilities of complex number manipulations.

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91影视

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

Short Answer

Step by step solution

Understand the transformation

Analyze the real part transformation

Analyze the imaginary part transformation

Determine the shape of the image

Sketch the image

Key Concepts

Complex Transformation

Conformal Mapping

Complex Plane

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