Understanding Complex Numbers as Parametric Functions

In summary, the conversation discusses how to sketch a parametric function in the complex plane, specifically for the given function \gamma. It is suggested to set x(t) = t^2 and y(t) = t for the interval [0,1] and x(t) = t and y(t) = 1 for the interval (1,2]. The conversation also discusses how to plot z^2 as a parametric function in the complex plane and whether it is correct to view complex numbers as parametric functions.
  • #1
Niles
1,866
0

Homework Statement


Hi all.

I am given the following parametric function in the complex plane C:

[tex]
\gamma = \left\{ {\begin{array}{*{20}c}
{t^2 + it\,\,\,\,\,\,\,\,{\rm{for }}\,\,t \in [0,1]} \\
{t + i\,\,\,\,\,\,\,\,\,\,\,\,{\rm{for }}\,\,t \in ]1,2]} \\
\end{array}} \right.
[/tex]

In order to sketch it for t in [0,1], will it be correct it I set x(t) = t2 and y(t) = t, and sketch it in the real plane?

Thanks in advance.Niles.
 
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  • #2
Niles said:

Homework Statement


Hi all.

I am given the following parametric function in the complex plane C:

[tex]
\gamma = \left\{ {\begin{array}{*{20}c}
{t^2 + it\,\,\,\,\,\,\,\,{\rm{for }}\,\,t \in [0,1]} \\
{t + i\,\,\,\,\,\,\,\,\,\,\,\,{\rm{for }}\,\,t \in ]1,2]} \\
\end{array}} \right.
[/tex]

In order to sketch it for t in [0,1], will it be correct it I set x(t) = t2 and y(t) = t, and sketch it in the real plane?

Thanks in advance.


Niles.
For [itex]0\le t\le 1[/itex], yes. For [itex]1< t\le 2[/itex], x= t, y= 1. Draw those two pieces.
 
  • #3
Thanks.

Lets look at e.g. w = z2 = r2ei2K = r2(cos(2K) + isin(2K)), where K is the argument of z and r is the modulus. If I wish to plot w = z2, then can I do this by plotting x(t) = r2cos(2K) and y(t) = r2sin(2K) as well?Niles.
 
  • #4
The reason why I am asking is that I seem to get confused when I look at complex numbers as mere parametric functions. Is it correct to look at them in this sense?
 
Last edited:

1. What is a complex parametric function?

A complex parametric function is a mathematical function that involves multiple variables, known as parameters, and produces complex values as its output. It is typically written in the form f(t) = x + iy, where x and y are real-valued functions of the parameter t and i is the imaginary unit.

2. How is a complex parametric function different from a regular function?

A regular function takes in one or more inputs and produces one output, while a complex parametric function takes in a parameter and produces a complex value as its output. The output of a regular function is a single value, whereas the output of a complex parametric function is a function itself.

3. What are some real-world applications of complex parametric functions?

Complex parametric functions have a wide range of applications in fields such as physics, engineering, and computer graphics. They are used to model and analyze phenomena such as fluid flow, electrical circuits, and motion of objects in 3D space. They are also commonly used in computer animation and graphics to create complex and realistic visual effects.

4. What are some common techniques for graphing complex parametric functions?

One common technique for graphing complex parametric functions is to plot the real and imaginary components separately on a 2D graph. Another approach is to use a parametric plot, where the x and y coordinates are determined by the real and imaginary components of the function, respectively. Computer software, such as graphing calculators and mathematical software programs, also offer tools for graphing complex parametric functions.

5. How can I manipulate or simplify a complex parametric function?

Manipulating and simplifying complex parametric functions can be done using standard techniques for manipulating algebraic expressions, such as combining like terms and factoring. Additionally, properties of complex numbers, such as the distributive property and rules for exponents, can be applied. It is also helpful to have a good understanding of the behavior of trigonometric and exponential functions, which are commonly used in complex parametric functions.

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