Problem 9
Question
(a) \(\psi(x, y)=\operatorname{Im}\left(z^{4}\right)=4 x y\left(x^{2}-y^{2}\right)\) and so \(\psi(x, y)=0\) when \(y=x\) and \(y=0\) (b) \(\mathbf{V}=\overline{G^{\prime}(z)}=\overline{4 z^{3}}=4\left(x^{3}-3 x y^{2}, y^{3}-3 x^{2} y\right)\) (c) In polar coordinates \(r^{4} \sin 4 \theta=c\) or \(r=(c \csc 4 \theta)^{1 / 4},\) for \(0 < \theta<\pi / 4\), are the streamlines. See the figure.
Step-by-Step Solution
Verified Answer
The streamlines in polar coordinates are described by \(r = (c \csc 4 \theta)^{1/4}\) for \(0 < \theta < \pi/4\).
1Step 1: Understand the Given Problem
The problem involves three parts related to functions of a complex variable. The first part deals with the imaginary part of a complex function, the second part involves the derivative of a function, and the third part looks at streamlines in polar coordinates.
2Step 2: Analyzing Part A
Part A states that \(\psi(x, y) = \operatorname{Im}(z^4) = 4xy(x^2 - y^2)\). This is the imaginary component of the function when expressed in terms of \( x \) and \( y \). It equals zero when \(y = x\) and \(y = 0\), as substituting these values will zero out the expression \(4xy(x^2 - y^2)\).
3Step 3: Analyzing Part B
Part B gives \(\mathbf{V} = \overline{G'(z)} = \overline{4z^3}\), which translates to the complex conjugate of the derivative of the function \( G(z) = z^4 \). The expansion \( 4(x^3 - 3xy^2, y^3 - 3x^2y) \) shows the components of the vector field \( \mathbf{V} \) derived from taking the real and imaginary parts of \(\overline{4z^3}\).
4Step 4: Analyzing Part C
In Part C, the streamlines of the vector field in polar coordinates are described by the equation \(r^4 \sin 4\theta = c\). This can be rearranged to solve for \(r\) as \(r = (c\csc 4 \theta)^{1/4}\). The condition for \(0 < \theta < \pi/4\) implies the region of validity for these streamlines in the given range of angles.
Key Concepts
Imaginary Part of Complex FunctionsComplex ConjugatePolar CoordinatesStreamlines
Imaginary Part of Complex Functions
In complex analysis, complex numbers are represented in the form of \( z = x + yi \), where \( x \) is the real part and \( y \) the imaginary part, with \( i \) being the imaginary unit satisfying \( i^2 = -1 \).
When dealing with complex functions, it's important to understand how to extract the imaginary component. For example, if you consider the complex function \( z^4 = (x+yi)^4 \), you expand it to find its real and imaginary parts.
The imaginary part in this expansion is given by \( \operatorname{Im}(z^4) = 4xy(x^2 - y^2) \). This represents a specific relationship in the complex function, revealing where the imaginary part equals zero.
This occurs when \( y = x \) or \( y = 0 \), highlighting symmetry and specific null points on a plane. Understanding these points is crucial for analyzing the behavior of the function across complex fields.
When dealing with complex functions, it's important to understand how to extract the imaginary component. For example, if you consider the complex function \( z^4 = (x+yi)^4 \), you expand it to find its real and imaginary parts.
The imaginary part in this expansion is given by \( \operatorname{Im}(z^4) = 4xy(x^2 - y^2) \). This represents a specific relationship in the complex function, revealing where the imaginary part equals zero.
This occurs when \( y = x \) or \( y = 0 \), highlighting symmetry and specific null points on a plane. Understanding these points is crucial for analyzing the behavior of the function across complex fields.
Complex Conjugate
The complex conjugate of a complex number \( z = x + yi \) is \( \overline{z} = x - yi \). This reflection across the real axis is fundamental in complex analysis.
For complex functions, taking the conjugate helps in interpreting operations and visualizing transformations. Part of the exercise involved understanding \( \overline{G'(z)} \), where \( G(z) = z^4 \) and \( G'(z) = 4z^3 \).
Thus, \( \overline{4z^3} \) results in a new vector field expression \( 4(x^3 - 3xy^2, y^3 - 3x^2y) \).
This expression is utility-driven in both real and imaginary parts, displaying how a complex conjugate simplifies or alters vector field representations. It plays a significant role in the symmetry and behavior of complex functions.
Using complex conjugates assists in simplifying computations and understanding deeper properties of complex mappings.
For complex functions, taking the conjugate helps in interpreting operations and visualizing transformations. Part of the exercise involved understanding \( \overline{G'(z)} \), where \( G(z) = z^4 \) and \( G'(z) = 4z^3 \).
Thus, \( \overline{4z^3} \) results in a new vector field expression \( 4(x^3 - 3xy^2, y^3 - 3x^2y) \).
This expression is utility-driven in both real and imaginary parts, displaying how a complex conjugate simplifies or alters vector field representations. It plays a significant role in the symmetry and behavior of complex functions.
Using complex conjugates assists in simplifying computations and understanding deeper properties of complex mappings.
Polar Coordinates
Polar coordinates allow us to map points on a plane using a radius and angle \( (r, \theta) \), offering a different view over Cartesian coordinates.
This method is particularly useful in complex analysis as it often simplifies mathematical relationships and solutions. In the exercise, the relation \( r^4 \sin 4\theta = c \) represents a transformation in polar coordinates.
Transforming into polar coordinates requires converting the Cartesian expressions \( x = r\cos\theta \) and \( y = r\sin\theta \), integrating seamlessly with the radial symmetry often seen in complex functions.
This can lead to simplified expression or insight, like determining shapes or behaviors of functions as circular or spiral patterns. In complex dynamics, polar coordinates simplify trigonometric identities and streamline the process of integration and differentiation over complex planes.
Understanding how to transition between coordinate systems is vital for solving complex problems efficiently.
This method is particularly useful in complex analysis as it often simplifies mathematical relationships and solutions. In the exercise, the relation \( r^4 \sin 4\theta = c \) represents a transformation in polar coordinates.
Transforming into polar coordinates requires converting the Cartesian expressions \( x = r\cos\theta \) and \( y = r\sin\theta \), integrating seamlessly with the radial symmetry often seen in complex functions.
This can lead to simplified expression or insight, like determining shapes or behaviors of functions as circular or spiral patterns. In complex dynamics, polar coordinates simplify trigonometric identities and streamline the process of integration and differentiation over complex planes.
Understanding how to transition between coordinate systems is vital for solving complex problems efficiently.
Streamlines
Streamlines describe paths that a massless particle would follow in a fluid flow represented by a vector field, helpful in both physics and complex analysis.
In the exercise, streamlines arise from the equation \( r^4 \sin 4\theta = c \), demonstrating paths based on angle and radius in polar coordinates. This equation shows how the flow lines are affected by the underlying polar transformations.
The condition \( 0 < \theta < \pi/4 \) restricts the streamline view to a specific range of angles, simulating how particles move within those constraints.
Streamlines allow visualization of vector fields and understand behaviors, such as flow stability or the existence of vortices.
In complex analysis, analyzing streamlines informs us about fluid flows, electromagnetic fields, and can also be applied to finance and economics for understanding various flow dynamics invisibly dictating system behaviors.
In the exercise, streamlines arise from the equation \( r^4 \sin 4\theta = c \), demonstrating paths based on angle and radius in polar coordinates. This equation shows how the flow lines are affected by the underlying polar transformations.
The condition \( 0 < \theta < \pi/4 \) restricts the streamline view to a specific range of angles, simulating how particles move within those constraints.
Streamlines allow visualization of vector fields and understand behaviors, such as flow stability or the existence of vortices.
In complex analysis, analyzing streamlines informs us about fluid flows, electromagnetic fields, and can also be applied to finance and economics for understanding various flow dynamics invisibly dictating system behaviors.
Other exercises in this chapter
Problem 7
\(S^{-1}(T(z))=\frac{a z+b}{c z+d}\) where $$\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\operatorname{adj}\left(\left[\begin{array}{ll} 1 & -2 \\
View solution Problem 8
If \(r=2\) and \(0 \leq \theta \leq \frac{\pi}{2}, w=\sqrt{2} e^{i \theta / 2} .\) Therefore \(|w|=\sqrt{2}\) and \(0 \leq \operatorname{Arg} w \leq \pi / 4 .\)
View solution Problem 10
Since \(\alpha_{1}=3 \pi / 2=\alpha_{2}, f^{\prime}(z)=A(z+1)^{1 / 2}(z-1)^{1 / 2}=A\left(z^{2}-1\right)^{1 / 2} .\) Therefore, $$f(z)=A\left[\frac{z\left(z^{2}
View solution Problem 10
(a) \(\operatorname{since} G\left(r e^{i \theta}\right)=r^{2 / 3} e^{i 2 \theta / 3}, \psi(r, \theta)=\operatorname{Im}\left(G\left(r e^{i \theta}\right)\right)
View solution