Plotting e^(a+ix) for Beginners

[silent_hunter]

I was just wondering how can I plot e^(a+ix) and e^a[cos(x) + i*sin(x)] (=e^(a+ix)) in cartesian coordinate. (a is constant,x is independent variable & i is imaginary number).
This is my first post,so please forgive for any mistakes :) Thanks in advance.

 

[Edgardo]

Welcome to PF silent_hunter,

if a and b are constants, then e^(a+ib) is just a number. Do you mean something different?

 

[silent_hunter]

Welcome to PF silent_hunter,

if a and b are constants, then e^(a+ib) is just a number. Do you mean something different?

Thanks for replaying ,I made a mistake. b is not constant, I'm editing my post .Sorry for my stupidity.

Actually I want to know how to plot the above complex function in cartesian coordinate.I mean in which axis do I consider domain and which one will be range?
By the way somewhere I heard that 4 dimensions are needed to plot complex functions.

 

[Edgardo]

The problem with complex functions such as exp(a+ib) is that their domain is two dimensional and their range can be two dimensional as well. How do you visualize a two dimensional range?

One way is to create two different pictures, one for domain and one for range:
http://www-math.mit.edu/daimp/ComplexExponential.html

In the link above you have the function exp(z) with z = a+ib.
The left picture represents the domain and the right picture the function exp(z).

 

[silent_hunter]

Thanks bro, now I understand.

The problem with complex functions such as exp(a+ib) is that their domain is two dimensional and their range can be two dimensional as well. How do you visualize a two dimensional range?

I thought that it could be placed in same plane,but it seems it still can be done.

 

[meldraft]

You can also use 3D Cartesian coordinates and use colour as your 4th dimension. Two examples from wikipedia:

complex exponential:

http://upload.wikimedia.org/wikipedia/commons/d/de/ExponentialAbs.png

complex logarithm:

http://upload.wikimedia.org/wikipedia/commons/4/41/Riemann_surface_log.jpg

 

[burochokkotti]

If we plot z=cos(x) and z= i sin(y) then it looks like as the following attachments.

 

[burochokkotti]

and if you treat i as another coordinate, then I guess the fn behaves like z= cos(x) + sin(y) (sorry if I've made any mistakes) and the graphs are as follows!