What are complex functions and how can they be graphed?

[madah12]

I know that a complex number can be written in form of a+bi and r(cos(theta) + isin(theta))
but I don't understand the the representation of it as r*e^(i * theta) also

 

[zhermes]

I'm not exactly sure what you're looking for, but this is a good article:
http://en.wikipedia.org/wiki/Euler's_formula
With a section on proofs of the relation.

 

[madah12]

uhm so since I like things by examples tell me if I got it right
5+2i
in polar form its
5.4 (isin(tan^-1(2/5)+.cos(tan^-1(2/5)
and it could be written as 5.4e^(i*tan^-1(2/5))?

 

[zhermes]

Yeah! that's exactly right.

If you imagine that complex numbers are a position in the x-y plane, where the x-axis is real numbers and the y-axis is imaginary numbers; then a+ib is just a standard rectilinear (Cartesian) way of describing a point [e.g. 5+2i = 5x + 2y = (5,2) ]; when you use r*e^{i\theta}, its like writing it in polar coordinates, r is the magnitude, and theta the angle with the x-axis.

 

[madah12]

Yeah! that's exactly right.

If you imagine that complex numbers are a position in the x-y plane, where the x-axis is real numbers and the y-axis is imaginary numbers; then a+ib is just a standard rectilinear (Cartesian) way of describing a point [e.g. 5+2i = 5x + 2y = (5,2) ]; when you use r*e^{i\theta}, its like writing it in polar coordinates, r is the magnitude, and theta the angle with the x-axis.

is it possible with the rectangular coordinate to graph a complex function? I searched the net but couldn't figure out the right key words I mean like f(x)=2x+2ix and you input (2-i) and get 6+2i or does no such thing exist in mathematics?

 

[zhermes]

I mean like f(x)=2x+2ix and you input (2-i) and get 6+2i or does no such thing exist in mathematics?

"Complex functions" are the general term for functions which operate on (or yield) complex numbers. But note, you have to input two scalars (the equivalent of a complex number)
$$z = f(x+iy)<br /> [\tex]<br /> You, therefore, can't graph such functions in 2 dimensions, because you have 2 input dimensions (e.g. x and y) and then output dimensions (1 if your result is a real number, and 2 if your result is a complex number).<br /> <br /> If you have a function which takes a complex number and gives a real number, you could plot it as a surface in 3 dimensions.$$