Problem 14 Find and sketch or graph the ima... [FREE SOLUTION]

Chapter 17: Problem 14

Find and sketch or graph the images of the lines \(x=0\) \(\pm \pi / 6, \pm \pi / 3, \pm \pi / 2\) under the mapping \(w=\sin z\).

Short Answer

Expert verified

Lines map to horizontal strips in the real-imaginary plane with vertical oscillations.

Step by step solution

Identify the Mapping Function

The given mapping function is \( w = \sin z \). We need to find the images of lines parallel to the imaginary axis under this function.

Convert the Lines to Complex Form

The lines \( x = 0, \pm \pi/6, \pm \pi/3, \pm \pi/2 \) are vertical in the complex plane and can be expressed as \( z = c + iy \) where \( c \) is constant and equal to the respective line equation value.

Understanding Sine of a Complex Number

The sine of a complex number \( z = x + iy \) is defined as \( \sin z = \sin x \cosh y + i \cos x \sinh y \). For our case, the real part \( x \) is constant and the imaginary part varies.

Calculate the Image for Each Line

Using the formula for \( \sin z \), calculate the images for each line. E.g., for the line \( x = \pi/6 \), the mapping becomes \( \sin(\pi/6 + iy) = \frac{1}{2} \cosh y + i \frac{\sqrt{3}}{2} \sinh y \). Perform similar calculations for all lines.

Plot the Images of Each Line

Plot the images of each of these lines in the \( w \)-plane. You'll find that each vertical line maps to a horizontal range on the real-axis combined with vertical hyperbolic sine variations along the imaginary axis due to the \( \sinh \) and \( \cosh \) terms.

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Key Concepts

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

Complex Analysis

Complex analysis is a fascinating branch of mathematics that studies functions of complex numbers. Imagine a function that doesn't just take real numbers and output real numbers, but instead works entirely in the realm of complex numbers. These are numbers that have both a real part and an imaginary part, often written in the form \(z = x + iy\), where \(x\) is the real part, and \(y\) is the imaginary part.
Key properties of complex analysis include:

Holomorphic functions: These are complex functions that are complex differentiable in a neighborhood of each point in their domain.
Analytic functions: In the context of complex numbers, analytic functions are those representable by a power series.
Conformal mapping: A transformation that preserves angles and shapes on a small scale, which is a fascinating property used for creating mappings such as the sine function in the complex plane.

Understanding complex analysis helps in visualizing and solving complex mapping problems, like the one given in the exercise.

Sine Function

The sine function is more than just a trigonometric concept; in complex analysis, it takes on new dimensions. When dealing with complex numbers, the sine function needs to be defined in a way that captures both the real and imaginary parts. The formula used for a complex number \(z = x + iy\) is:
\[ \sin z = \sin x \cosh y + i \cos x \sinh y \]
Here鈥檚 a quick breakdown:

\(\sin x\) and \(\cos x\) are the regular real sine and cosine functions.
\(\cosh y\) and \(\sinh y\) represent hyperbolic functions, which are related to exponential functions. They determine the growth and decay among lines parallel to the imaginary axis.
The real and imaginary parts of \(\sin z\) help to map vertical lines in the \(z\)-plane to curves in the \(w\)-plane.

Understanding this formula helps students visualize how lines in the complex plane transform under the sine mapping.

Imaginary Axis

The imaginary axis plays a critical role in complex mappings. In the complex plane, this axis is where the real part of a condition is zero, meaning a point is purely imaginary; it is denoted as \(z = iy\).
When a function like \(w = \sin z\) acts on lines that are vertical (or parallel to the imaginary axis), it transforms them based on the behavior of both circular and hyperbolic trigonometric functions. For example:

Vertical lines such as \(x = 0, \pm \pi/6, \pm \pi/3, \pm \pi/2\) reflect symmetry alongside the imaginary axis because their real part is constant.
The imaginary component \(\sinh y\) in the sine function \(\sin z\) contributes to vertical stretching or compression, creating lines that shift into curves in the image plane.
Such transformations often yield unexpected yet beautiful patterns and help in grasping concepts like periodicity and extension in the complex plane.

The imaginary axis helps us visualize complex transformations, serving as a backbone for how points and lines get mapped into new, often more intricate shapes.

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91影视

Find and sketch or graph the images of the lines \(x=0\) \(\pm \pi / 6, \pm \pi / 3, \pm \pi / 2\) under the mapping \(w=\sin z\).

Short Answer

Step by step solution

Identify the Mapping Function

Convert the Lines to Complex Form

Understanding Sine of a Complex Number

Calculate the Image for Each Line

Plot the Images of Each Line

Key Concepts

Complex Analysis

Sine Function

Imaginary Axis

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