Mapping and Rotation of Complex Curves

In summary, the question asks for the angle and scale factor of a curve rotated and scaled under a given mapping. Using the Taylor series expansion, the first term represents a translation while the higher powers of (z-z0) represent rotation and scaling. The coefficient of (z-z0) can be used to determine the desired values.
  • #1
elimenohpee
67
0

Homework Statement


Find the angle through which a curve drawn from the point z0 is rotated under the mapping w=f(z), and find the corresponding scale factor of the transformation.

z0 = -1, w=z^2

and

z0 = -1 + i, w = 1/z


Homework Equations



I honestly don't know how to begin

The Attempt at a Solution



I don't know where to start, these are 2 questions at the end of my complex variables book, and there is no example in the text. the answers for z0 = -1 are: arg w'(z0) = pi, |w'(z0)| = 2

and z0 = -1 + i are: arg w'(z0) = -pi/2, |w'(z0)| = 1/2
 
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  • #2
You want to expand f(z) into a taylor series around z=z0. Everything you want to know is then in the linear term. If f(z)=a*(z-z0) how can you find the scaling and rotation from the factor a?
 
  • #3
'a' would be the scaling factor would it not? Can you use the equation f(z) = a*(z-z0) to solve for the scaling factor somehow?

How would you find the arguement?
 
  • #4
I see if I just evaluate |w'(z0)|, I see that for the first question I get |-2| or 2, and 1/2 for the second.

I guess my question is really understanding what the question is asking. Like you stated, it looks as if its a taylor series expansion about a point, but what is the significance?
 
  • #5
elimenohpee said:
'a' would be the scaling factor would it not? Can you use the equation f(z) = a*(z-z0) to solve for the scaling factor somehow?

How would you find the arguement?

No, |a| would be the scaling. arg(a) would be the rotation, yes? Simplify it another notch. Just take f(z)=a*z. What's the rotation and what's the scaling? f(z)=i*z has scaling 1, it rotates by pi/2, right?
 
  • #6
elimenohpee said:
I see if I just evaluate |w'(z0)|, I see that for the first question I get |-2| or 2, and 1/2 for the second.

I guess my question is really understanding what the question is asking. Like you stated, it looks as if its a taylor series expansion about a point, but what is the significance?

The first term in the taylor series is just a translation, it doesn't scale or rotate. The powers of (z-z0) higher than 1 are negligible close to z0 compared with the first power. The coefficient of (z-z0) tells you everything.
 
  • #7
Ok perfect, makes sense now. Thanks for the help!
 

1. What are complex variables?

Complex variables are mathematical quantities that contain both real and imaginary components. They are represented by the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit. Complex variables are used in many branches of mathematics and science, including complex analysis, engineering, and physics.

2. Why are complex variables important?

Complex variables are important because they allow us to solve problems that cannot be solved using real numbers alone. They have many applications in fields such as engineering, physics, and economics. They also have a wide range of theoretical applications, including the study of complex functions and the behavior of physical systems.

3. What is the difference between a complex variable and a real variable?

The main difference between a complex variable and a real variable is that a complex variable contains an imaginary component, while a real variable does not. This means that complex variables have both real and imaginary parts, while real variables only have one real part. Additionally, complex variables have different properties and operations compared to real variables, such as the complex conjugate and complex multiplication.

4. How are complex variables used in physics?

Complex variables are used in physics to describe and analyze physical systems that involve oscillations or waves. This includes electromagnetism, quantum mechanics, and fluid dynamics. In these fields, complex variables are used to simplify equations and make calculations easier, as well as to represent physical quantities such as electric and magnetic fields.

5. What is the significance of the complex conjugate in complex variables?

The complex conjugate is an important concept in complex variables because it allows us to find the magnitude and phase angle of a complex number. It is also used in operations such as division and finding roots of complex numbers. In physics, the complex conjugate is used to represent the complex conjugate wave, which is an important concept in the study of electromagnetic waves.

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