Functions of a Complex Variable

In summary, the conversation involved a request for help with a mapping problem. The mapping function was given as w = 1/(z-j) and the questions asked were to write z as a function of w, express X and Y in terms of u and v, and describe the image of the line x = 1 under this mapping. The solution involved basic algebraic manipulation to express z as a function of w and to find the image of the line x = 1.
  • #1
sunnybrarrd
2
0
Hi guys. Please help me with this.

Consider the Mapping:

w = 1/(z-j)

questions:(1) Write z as a function of w.

(2) Express X and Y in terms of u and v.

(3) Find and describe the image of the line x = 1 under this mapping.

Homework Equations

--

w = u+jv
z = x+jy

The Attempt at a Solution



I don't understand the first question and have no idea of how to have it as a function of w. However I do understand from then onwards as to express in terms u and v.
 
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  • #2
It's strange that you should be working with complex number and not understand something from basic algebra.
[tex]w= \frac{1}{z- j}[/tex]
[tex]z- j= \frac{1}{w}[/tex]
[tex]z= \frac{1}{w}+ j[/tex]
 
  • #3
Thank you.

That is really simple. Probably that is why i was confused as I did not thought it would be that simple.

Thanks alot!
 

FAQ: Functions of a Complex Variable

What are the basic properties of functions of a complex variable?

Functions of a complex variable are functions that take complex numbers as inputs and output complex numbers. They have similar properties to functions of a real variable, such as continuity and differentiability, but also have some unique properties, such as analyticity and the Cauchy-Riemann equations.

How are complex functions different from real functions?

Complex functions are different from real functions in that they take complex numbers as inputs and outputs, whereas real functions take real numbers as inputs and outputs. They also have different properties, such as analyticity and the Cauchy-Riemann equations, that are not present in real functions.

What is the significance of the Cauchy-Riemann equations in complex analysis?

The Cauchy-Riemann equations are a set of necessary and sufficient conditions for a complex function to be differentiable. They are important in complex analysis because they allow us to determine if a complex function is analytic, and if so, we can use them to find its derivative and other properties.

How are complex functions visualized?

Complex functions can be visualized in a number of ways, such as using contour plots, phase plots, or 3D plots. These visualizations can help us understand the behavior of the function and its properties, such as where it is analytic and where it has singularities.

What are some practical applications of functions of a complex variable?

Functions of a complex variable have many practical applications in fields such as physics, engineering, and economics. For example, they are used to model the behavior of electric and magnetic fields, fluid flow, and stock market trends. They are also used in signal processing and control systems.

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