Complex Functions of a complex variable

In summary: So you can see that z2 is just the sum of two lines and so the graph will look like this: In summary, the student is struggling to visualize complex functions of a complex variable. He needs help understanding what he is doing and how to sketch the function.
  • #1
Solidmozza
29
1
Hi,
Im doing a course in Differential Calculus and we are up to Euler's formula and representing complex functions of a complex variable. I'm finding it really difficult to visualise/sketch these types of functions: z|-->z^2 such that {z=x+2i}, or z|-->e^z = w such that {z=(-1+i)t | t e R, t>_0}
Could someone please explain to me how to sketch these sorts of things, or direct me to an explanation? For instance, in the first of these above equations, I've gone z^2 = x^2 - 4 + 4xi but then I get mixed up with the axes: arn't they meant to be real and imaginary axis, not x and y, so what can I do with this x value?
Any help muchly appreciated!
 
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  • #2
Yes, of course you are going to have trouble! every complex number, z= x+ iy, requires two dimension so a graph of f(z) would have to be in 4 dimensions- 2 for z itself and 2 for the value of f.
Do you really need to draw graphs in order to do the problems you are given?
 
  • #3
My fault - I didn't make myself clear. I realize fully that you would need 4 dimensions to draw it on one graph, however what I meant was that they want us to draw two complex planes - the z plane (inputs) and a w plane (outputs). For instance, if we chose a subset A of the complex numbers such that f:C-->C and A={z=x+iy|-pi<y_<pi} what are the set of output values of z maps to w, where w=e^z? Sorry if this doesn't make it clear either :S
 
  • #4
Ah, I've never really liked that method! The best you can do is take some simple curves or lines in the z= x+ iy plane and sketch the result in the f(z)= u+ iv plane.

For example to graph f(z)= z2, look at the vertical lines z= x. Then f(z)= x2 which is just u= x2 (constant for each vertical line and so a horizontal line- but not evenly spaced and above the real axis for both x and -x). The horizontal lines, z= iy, give z2= -y2, again horizontal lines but below the real axis.
 

What is a complex function?

A complex function is a mathematical function that takes a complex number as its input and produces a complex number as its output. It can be written as f(z) = u(x,y) + iv(x,y), where z = x + iy is a complex number, u(x,y) is the real part of the function, and v(x,y) is the imaginary part.

What is a complex variable?

A complex variable is a quantity that can take on complex values in addition to real values. It is typically denoted by z = x + iy, where x and y are real numbers and i = √(-1) is the imaginary unit.

What is the difference between a complex function and a real function?

The main difference between a complex function and a real function is that a complex function operates on complex numbers, while a real function operates on real numbers. This means that a complex function has both a real and an imaginary part, whereas a real function only has a real part.

What are some examples of complex functions?

Some examples of complex functions include polynomial functions, exponential functions, trigonometric functions, and logarithmic functions. These functions can be written in terms of complex variables and have both a real and an imaginary part.

What are the applications of complex functions?

Complex functions have many applications in mathematics, physics, engineering, and other fields. They are used to model and analyze a wide range of phenomena, including electrical circuits, fluid flow, quantum mechanics, and signal processing. They are also essential in the study of complex analysis, which has numerous applications in mathematics and physics.

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