Problem 3
Question
(a) For \(T(z)=\frac{z+1}{z-1}, T(0)=-1, T(1)=\infty,\) and \(T(\infty)=1\). (b) The circle \(|z|=1\) passes through the pole at \(z=1\) and so the image is a line. since \(T(-1)=0\) and \(T(i)=-i,\) the image is the line \(u=0 .\) If \(|z-1|=1\), $$|w-1|=\left|\frac{z+1}{z-1}-1\right|=\frac{2}{|z-1|}=2$$ and so the image is the circle \(|w-1|=2\) in the \(w\) -plane. (c) since \(T(0)=-1,\) the image of the disk \(|z| \leq 1\) is the half-plane \(u \leq 0\).
Step-by-Step Solution
Verified Answer
(a) Special point mappings: 0 to -1, 1 to ∞, ∞ to 1. (b) \(|z|=1\) maps to line \(u=0\), \(|z-1|=1\) maps to circle \(|w-1|=2\). (c) Disk \(|z| \leq 1\) maps to half-plane \(u \leq 0\).
1Step 1: Analyze the given function
The function is given by \( T(z) = \frac{z+1}{z-1} \). This is a Möbius transformation, which typically maps complex numbers and can have important mapping properties such as sending poles to infinity.
2Step 2: Determine mappings for special points
Evaluate \( T(z) \) at points of interest. For \( Z=0 \), \( T(0) = \frac{0+1}{0-1} = -1 \). For \( z=1 \), \( T(1) = \frac{1+1}{1-1} \) gives a boundary condition toward infinity (i.e., \( \infty \)). For \( z= \infty \), \( T(\infty)=1 \).
3Step 3: Describe the image of the unit circle \(|z|=1\)
The circle \(|z|=1\) encloses the pole at \(z=1\), thus it's mapped to a line in the \(w\)-plane. We identify this line based on additional points: \(T(-1)=0\) and \(T(i)=-i\), hence, the line is \(u=0\).
4Step 4: Analyze the circle \(|z-1|=1\)
This circle can be transformed using the given relationship \(|w-1|=|\frac{z+1}{z-1}-1|=\frac{2}{|z-1|}=2\). Therefore, it becomes a circle \(|w-1|=2\) in the \(w\)-plane.
5Step 5: Image of the disk \(|z| \leq 1\)
The disk \(|z| \leq 1\) when mapped using \(T\) has its boundary \(|z|=1\) mapped onto \(u=0\) (as derived earlier), implying that the interior of the disk maps onto \(u\leq 0\). Therefore, the image is the half-plane \( u \leq 0 \).
Key Concepts
Complex Function MappingConformal MappingImaginary and Real Components
Complex Function Mapping
In mathematics, a complex function mapping is a process where a function takes complex numbers as inputs and transforms them into another set of complex numbers. Consider the Möbius transformation, a specific type of complex function mapping, which takes the form \( T(z) = \frac{z+1}{z-1} \). This function maps the complex plane into itself and is characterized by its ability to transform lines and circles into other lines or circles. This distinctive feature makes Möbius transformations particularly useful in complex analysis and geometric interpretation of complex functions.
To explain further, this particular transformation has points of interest such as \( T(0) = -1 \), \( T(1) = \infty \), and \( T(\infty) = 1 \). These mappings illustrate how Möbius transformations can shift significant points to entirely new locations, including mapping finite points to infinity. Such transformations demonstrate the ability to manipulate complex numbers in a way that preserves angles and relative positions of figures on the complex plane. Understanding how these transformations work enhances the grasp of complex dynamics and the behavior of complex functions.
To explain further, this particular transformation has points of interest such as \( T(0) = -1 \), \( T(1) = \infty \), and \( T(\infty) = 1 \). These mappings illustrate how Möbius transformations can shift significant points to entirely new locations, including mapping finite points to infinity. Such transformations demonstrate the ability to manipulate complex numbers in a way that preserves angles and relative positions of figures on the complex plane. Understanding how these transformations work enhances the grasp of complex dynamics and the behavior of complex functions.
Conformal Mapping
Conformal mapping involves using functions to preserve angles while transforming shapes. An essential aspect of conformal mapping is its ability to maintain the local angles at which curves intersect, though the shapes and sizes of objects may change. This mapping is crucial in complex analysis and engineering fields, such as fluid dynamics and electromagnetic theory.
In the context of the Möbius transformation \( T(z) = \frac{z+1}{z-1} \), conformal mapping is visualized when the unit circle \(|z|=1\) is transformed into a straight line \(u=0\) in the \(w\)-plane. This transformation maintains the angles and curvature properties but modifies the object's form from a circle to a line.
The transformation also affects circles not centered at the origin, such as \(|z-1|=1\), which is transformed into a circle \(|w-1|=2\). The ability of conformal mappings to so drastically change the appearance of shapes while maintaining angles makes them a powerful mathematical tool for analyzing and visualizing complex functions.
In the context of the Möbius transformation \( T(z) = \frac{z+1}{z-1} \), conformal mapping is visualized when the unit circle \(|z|=1\) is transformed into a straight line \(u=0\) in the \(w\)-plane. This transformation maintains the angles and curvature properties but modifies the object's form from a circle to a line.
The transformation also affects circles not centered at the origin, such as \(|z-1|=1\), which is transformed into a circle \(|w-1|=2\). The ability of conformal mappings to so drastically change the appearance of shapes while maintaining angles makes them a powerful mathematical tool for analyzing and visualizing complex functions.
Imaginary and Real Components
When dealing with complex functions, understanding imaginary and real components is crucial. Complex numbers can be expressed in the form \( z = x + yi \) where \( x \) is the real part and \( y \) is the imaginary part. For complex functions, both these components can change with transformation depending on the mapping function.
In the Möbius transformation \( T(z) = \frac{z+1}{z-1} \), various concepts relate to real and imaginary parts. For example, the line \( u=0 \) in the \( w\)-plane results from mapping the unit circle \(|z|=1\), where \( u \) denotes the real component of the transformed position \( w \). This demonstrates the complex interplay between real and imaginary parts in transformations.
By transforming a circle \(|z| \leq 1\) into the half-plane \( u \leq 0 \), the boundary and area of the original shape reflect movements of both real and imaginary components. To understand complex function mappings, one must conceptualize how these mappings affect both parts and visualize the impact on geometrical shapes.
In the Möbius transformation \( T(z) = \frac{z+1}{z-1} \), various concepts relate to real and imaginary parts. For example, the line \( u=0 \) in the \( w\)-plane results from mapping the unit circle \(|z|=1\), where \( u \) denotes the real component of the transformed position \( w \). This demonstrates the complex interplay between real and imaginary parts in transformations.
By transforming a circle \(|z| \leq 1\) into the half-plane \( u \leq 0 \), the boundary and area of the original shape reflect movements of both real and imaginary components. To understand complex function mappings, one must conceptualize how these mappings affect both parts and visualize the impact on geometrical shapes.
Other exercises in this chapter
Problem 1
For \(w=\frac{1}{z}, u=\frac{x}{x^{2}+y^{2}}\) and \(v=\frac{-y}{x^{2}+y^{2}} .\) If \(y=x, u=\frac{1}{2} \frac{1}{x}, v=-\frac{1}{2} \frac{1}{x},\) and so \(v=
View solution Problem 2
Using (3) with \(x_{0}=-2, x_{1}=0, x_{2}=1\) and \(u_{1}=5\) and \(u_{2}=1\) \\[ u=\frac{5}{\pi} \operatorname{Arg}\left(\frac{z}{z+2}\right)+\frac{1}{\pi} \op
View solution Problem 3
For \(w=z^{2}, u=x^{2}-y^{2}\) and \(v=2 x y .\) If \(x y=1, v=2\) and so the hyperbola \(x y=1\) is mapped onto the line \(v=2.\)
View solution Problem 4
\(f(z)=z+\operatorname{Ln} z+1\) is analytic for all \(z\) except \(z=y \leq 0 .\) But \(f^{\prime}(z)=1+1 / z\) and \(1+1 / z=0\) for \(z=-1\) Therefore \(f(z)
View solution