z

  
    Travel-Companion over the complex plane and along the imaginary axis

This journey requires some knowledge about vectors and You should be
able to add them, to multiply them with a scalar(number) and to build the
dot-product - and so equipped You travel through an euclidian vector-space,
as this the algebraic structure or the set of calculation rules is called. 
And You can do this in as many mathematical or physical dimensions as You 
like -of course there are not more than three of space in it. Here we restrict 
ourselves to just 2D, to a plane.
There are lots of reports about a "complex plane" or an "imaginary axis" - 
just translate it into "2D-plane" and "y-axis". I can assure You, that there 
is no mathematical difference between these.You can write one and the same 
vector in the form of coordinates (a,b)as well as an expression of 
two basis vectors (1,0)=1   and (0,1)=i  as a+i*b 
(the first form is called ordered pairs of real numbers, the second form 
linear combination of the basis elements).
In physics they use j instead of i for the vector (0,1)
A computer makes no difference, do You have to? See my calculator
For a beginner it was always very hard to grasp the difference - as there 
is none. And for all of those, who had to learn to do this brain-twist, it is 
even harder to disentangle their thoughts, but it is necessary, as You will 
see.

Once You got this, there's just one more difference to look at : the 
multiplication. It is defined as (a+i*b)*(u+i*v)=(a*u-b*v)+u*(a*v+b*u) or 
according to i*i=-1  and in coordinate-form
(u,v) * (a,b)  = (ua-vb,ub+va)  
Geometrically it means an addition of differences of direction and a 
multiplication of length.
This is an extension of the multiplication of real numbers, of 1D and it's 
different from the r.s.m. the multiplication of a vector by a real number and 
different from the dot-multiplication. And this multiplication is only 
possible in 2D. Again there is no justification, not even a historical one, 
to have a special "complex" world with this multiplication besides the normal 
2D-vectors. It's not worth to speculate, why  some like it so complicated.
This multiplication is included in the euclidian space, as you can build this
product by r.s.m., the addition and dot. Just as you can build with these three
let's say the subtraction or the metric(the distance between two points on the 
plane or in other words the length of the movement from the tip of one arrow 
to the tip of another,if they are tailed together).

Now You can start on Your journey.
In electronics You can learn to calculate circuits.Look for vibrations of 
ac-currents in a pointer-diagramm and You will find this in two forms, one 
without "complex-numbers"and one with them. The second one is easier, as here 
the multiplication is used. Compare both methods and see for Yourself: 
the pointer is pointing in the same directions, what is called the i-axis 
in one form is named y-axis in the other.

What is true for constant vectors will show up again, when You look at 
variables and functions.The most general form (of termdefined functions) in 
2D one can build is by means of 1D, just R: calculate from two independent 
variables x and y two functions u(x,y) and v(x,y), that is building two terms 
in R using x and y. Associate x and y with the two coordinates of the domain 
and let u and v do the same in the range.In this mathematical area you can 
find lot of useful material. What some might be unaware of is, that the 
functions of "complex variables" are part of this and what is said about the
first, is true for the second. And You can write the function also in the form 
f(x+i*y)=u(x,y)+i*v(x,y).
     Only for those who learned a bit about calculus:
         The extension of differentiation into 2D is the partial 
         differentiation, that is:
         (u_y) or delta u/delta y=(delta u/delta y) is differentiating u for y in the way You 
         learned, but treating the other variable x like any constant or number.
         You can get four terms and they are arranged in a matrix 
         the functionalmatrix, that is
         (deltamatrix)or(u_x&u_y/v_x&v_y)
         This is valid for both kind of functions.

Now comes the tricky part:
We restrict ourselves to the operations + and * and everything we can build with 
it, like the division. We will have no longer the dot, or the metric. We cant 
even go to polar coordinates.You might use a coordinate system, but even this 
is not essential. I just stick to the only definition of complex numbers one can 
find in literature. An important geometric conclusion is, that you cannot change 
orientation. The restriction to (R2,+,*) is not always followed in literature 
and that's why i proposed the name Bombelli instead of "C, the commutative field 
of complex numbers". (R2,+,*) is a commutative field, it's algebraic complete 
(that is: every polynomial has as many zeroes (roots), as its degree).
The same algebraic structure You get, when You look at some 2x2-matrices with 
their addition, their r.s.m. and their multiplication - restrict Yourself here 
to the matrices of this form (a,-b/b,a)
They have exactly the same possibilities:translations, rotations with 
shrinking/stretching.

Advance to variables. You must recognize, that here You cannot calculate the 
components of the independent variable differently, let's say x+3y as with the 
euclidian vectors. Build a term with x in R and replace x by z, or sligthly more 
expanded: build a term in 2D, but only use one independent variable z (instead of 
two independent variables, each of one dimension) - and only after this replace z 
by (a,b) or by a+i*b and 

calculate until You get two functions u(a,b) and v(a,b) combined into 2D as       
(u,v)=u+i*v.

It's the general form of any possible function here, build as a term in (R2,+,*) 
and using just z, - it's called holomorphic function, as You leave the variable 
whole, unsplit. Nearly every formula  You know already about functions of R1, 
which are termdefined You can directly transfer to these special 2D-functions 
(=in Bombelli termdefined). If You like flow charts: the angles and their 
orientation of any grid in the domain will be transferred by any such function 
exactly without change into the range (strict comform mapping). 
With vector fields.... ... ....lot's of results can be reached.
       Only for those who learned a bit about calculus:
           The functionalmatrix of these functions always takes the form
           (u_x,-v_x/v_x,u_x)
           or (u_x=v_y&u_y=-v_x)(Cauchy-Riemann equations)
           And as mentioned above, the matrices of this form are isomoph to
           2D-vectors: every matrix of this form can be replaced by
           (u_x,v_x) = (u_x+i*v_x)
           So the derivates of the holomorphic functions are 2D also!
           When You travel in the domain, the trace You leave will be 
           transferred to the range by the termdefined function. Is this a 
           holomorphic function, then the derivate is independent of 
           direction, that means the change of traveldirection  
           is only dependend on the point, You travel through, 
           but not on the way, You are going.

Have fun on Your trips around