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Complex Functions Plot By Hand On Paper

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Moderator: galactus


Post Mon Jul 14, 2014 8:09 am

Posts: 7
Hi,
consider a function $z=x+iy$ and $w=z^2$.
I have problem imagining it's plot, I know there are lots of techniques to show this plots and I know we can't plot such function because we need 4 dimensions due to the fact that we've got for variables ( $x$,$y$, $u$, $v$) - $u$ and $v$ are in the $w$ plane. ($w=u+iv$ , $u=x^2-y^2$, $v=2xy$)
Now, I want to ask you how did you get along with such issue if you had one?
I think my problem is due to the fact that I've gotten used to functions which can be plotted by hand like Real functions and I have confusion in imagining complex functions and my mind doesn't have particular imagination or some kind of intuition from it.
I'm looking forward to hear your suggestions to clarify this problem.
Thanks in advance

zaidalyafey Global Moderator
Global Moderator

Posts: 357
Usually we dont look at plotting complex valued functions rather we look at transformations. For example, what is the mapping of lines and circles under the transformation $w=z^2$ this will help you a lot understand how such mappings work. You can also look at mobius transformations, Riemann surfaces and stereographics projections.
Wanna learn what we discuss , see Book


Posts: 7
zaidalyafey wrote:
Usually we dont look at plotting complex valued functions rather we look at transformations. For example, what is the mapping of lines and circles under the transformation $w=z^2$ this will help you a lot understand how such mappings work. You can also look at mobius transformations, Riemann surfaces and stereographics projections.

Thanks for your reply, but understanding Riemann surfaces and Mobius transformations needs decent knowledge in complex analysis.
I'm stuck at early stages and some questions are seriously bothering me and one of them was the one I asked.
I wish it was a problem at which caring about transformations would be handy so I could understand the importance of the transformations in the Complex functions.

zaidalyafey Global Moderator
Global Moderator

Posts: 357
I understand your confusion and it is really something that bothers many at early stages in complex analysis. I say you forget about it at the moment and focus more on the introductory topics. There is a book by Zill which I consider easy to understand. He introduces transformations in an easy and more intuitive approach than other books.
Wanna learn what we discuss , see Book


Posts: 7
zaidalyafey wrote:
I understand your confusion and it is really something that bothers many at early stages in complex analysis. I say you forget about it at the moment and focus more on the introductory topics. There is a book by Zill which I consider easy to understand. He introduces transformations in an easy and more intuitive approach than other books.

Most of the books I've read on the introduction to the Complex Analysis, DO NOT talk about the general picture of it and its usefulness.They immediately dive into the detail and impose the learner to read the stuff before having an clear insight about what is going to happen.
I have got some general questions about the Complex analysis and the world of complex numbers which I haven't been satisfied by the answers given by books yet.
What I have figured out of the subject is, mathematicians defined the square root of minus one to have the power to play with the answers a little bit more, it's kind of like inventing a way to bypass the restrictions.
What I haven't figured out yet is, what we're supposed to do with the extra variable we've got if we only need the manipulation of one of them.For example, consider the $z=x+iy$, if we only need $x$ to solve a particular problem then can we consider the real part of the answer? and don't take care of the manipulation of imaginary part?
The serious question is, is it the nature of the complex algebra to solve two problems simultaneously?
And the other problem is, I haven't got any intuition about the limit of the complex functions, for example we have intuition of the limit on the graph of the real functions, is it any similar thing for the complex functions?

zaidalyafey Global Moderator
Global Moderator

Posts: 357
NoVice wrote:
Most of the books I've read on the introduction to the Complex Analysis, DO NOT talk about the general picture of it and its usefulness.They immediately dive into the detail and impose the learner to read the stuff before having an clear insight about what is going to happen.


The most interesting part of complex analysis is not the part where they talk about " limits and analtyicity". Unfortunatley to dive into the the interesting part of applications such as (solving real integrals, series and mobius transformations) you have to go through all the introductory parts. Dont worry, at first you see yourself blind at the concepts and you gotta read more and more to get a slight intuition of what is happening.

What I have figured out of the subject is, mathematicians defined the square root of minus one to have the power to play with the answers a little bit more, it's kind of like inventing a way to bypass the restrictions.
What I haven't figured out yet is, what we're supposed to do with the extra variable we've got if we only need the manipulation of one of them.For example, consider the $z=x+iy$, if we only need $x$ to solve a particular problem then can we consider the real part of the answer? and don't take care of the manipulation of imaginary part?


Sometimes when you are on the process of solving a certain problem you are only interested on the real part or in the imaginary part hence converting the problem to the real calculus.

The serious question is, is it the nature of the complex algebra to solve two problems simultaneously?
And the other problem is, I haven't got any intuition about the limit of the complex functions, for example we have intuition of the limit on the graph of the real functions, is it any similar thing for the complex functions?


Most definitions in complex analysis are abstract and don't have a geometrical meaning as of slope and area under a certain curve like in single variable calculus.
Wanna learn what we discuss , see Book


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