@acer Examing this function arctan as real and complex for his definition is to study

--------------------------------------------------------------------------------------

For real arguments x, y, the two-argument function arctan(y, x), computes the principal value of the argument of the complex number   x + yI , so 
          -Pi < arctan(y, x) and arctan(y, x) <= Pi

. This function is extended to complex arguments by the formula
      arctan(y, x) = -I*ln((x + y*I)/sqrt(x^2 + y^2))

--------------------------------------------------------

Numerical_evaluation_post_acer_verder_bekijken.mw

@vv 

---------------------------------------------

@janhardo You should use the symbolic option only in the rare cases when you don't care about the branch of the log.
Check here e.g. z=-1, a=-1.

-------------------------------------

What is a?.. -> ln ( a+bI) , z= a+bI 

plot_compare_en_functionadvisorvoor_onderzoek_complexe_functies_.mw

@Carl Love 

Thanks

Indeed a extensive help page

In complex analysis, there are several classes of singularities. These include the isolated singularities, the nonisolated singularities and the branch points.

Branch points are generally the result of a multi-valued function, such as  or 

Multivariate functions are studied using Riemann surfaces. The formal definition of branching point even uses the concept of a Riemann surface.

Riemann surfaces can be thought of as deformed versions of the complex plane:

The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.

When discussing functions of a complex variable it is often convenient to think of a cut in the complex plane.

The wiki can help : Holomorphic function - Wikipedia

Try to get a idea what this all abouts 

@Carl Love Thanks
I checked it and it "works"
I will give attention to plotcompare command.
Seems to me that it easier to study both plots side by side 

The plotcompare command can be used for one function to start with and see branchpoints/cuts?

@janhardo 

2d input

This 2d input is converted to maple to 1d input

=I/(2*b)*ln((b*I + z)/(b*I - z));

Seems to be impossible in Maple to get this 2d input as answer, via 1d maple input like in my textbook ( the simplification done by @acer is correct (good job), but it is not in the wanted form)

Ok you can say why bother further about this, although its right?
Its so satifsying to see in Maple exact the same as in a textbook. 

@Carl Love 

Thanks

Probably  i was tired after trying another code getting working and doing figuring out Maple math for some hours. I took it over from a example.
Yes, this what  meant, it is the plotttingrange from x=-1..1 for all 6 plots

The idea was only to show two plots besides eachother for a complex plot with a real and imaginair part. 

@janhardo 

Something like this

restart;
with(plots):
a:=Array(1..2,1..3);

a[1,1]:=plot(sin(Pi*x), x=1..1, color=black):
a[1,2]:=plot(sin(2*Pi*x), x=1..1, color=black):
a[1,3]:=plot(sin(3*Pi*x), x=1..1, color=black):

a[2,1]:=plot(cos(Pi*x), x=1..1, color=black):
a[2,2]:=plot(cos(2*Pi*x), x=1..1, color=black):
a[2,3]:=plot(cos(3*Pi*x), x=1..1, color=black):
display (a, tickmarks= [[],[]],labels=["",""]);

Its only not working yet.

@Ronan Thanks 
I can use it 
But i do need the plots placed in a table structure (  like in Word text program) 
Don't know if there is another way too  

@Carl Love

 Thanks, looks great those plots

The distiction between a conformal plot and the spacial complex plot(  u-v form :)  must be yet understood by me ?
z = r*exp(I*theta) = r*cos(theta) + I*r*sin(theta)

This equation is the complex number written in two of his three forms and the (formula of Euler)  can be derived from this by cancelling out the r.  

Yes, this plotting representation can be used to show two to or more plots side by side 
Its only that there is no direct relation by formula is between the graphs in my case 
Its here for 2D , but now for 3D.

I do want use it for for compare plots

with(plots)c# verder uitzoeken
plotcompare(ln(x), x - x^3/6, x = -3 .. 3); Can this be used?

For the FunctionAdvisor ( ln(z) give lots of interesting information , but for plot it gives only a plot. 
I like to see the plotting code underneath the plot as i saw with a other plot example.

@vv 

Thanks

Unfortenaly  must first study this subject of branche(s) in order to get a idea about it 

@acer 

Thanks for searching to more easier commands.

Looking at both commands then is the combine command a much easier to use then the nested simplify command

But there was  third command used lost in the delete post what uses a e power  in his nested simplify command => no it is done by @acer

So three  rather simple ways to for transforming the expression: expr

a third ? ..no ln(exp(-I*expr))*I is done = expr

@acer
Thanks 

The most direct approach by you is something difficult to master for me, or with more explenation how it excactly all works.
For commercial use it cannot be short enough , because time is money
But for me it is only educational  use ,so when i get the same result with more easier command then i am also satisfied. 

Indeed i made mistake by over take the expression, it shows again how to be accurate is important.

@janhardo 

Download post_acer_extra_na_verwijdering_nm_en_kitonum.mw

@janhardo 

Ok, i posted a double post ..it slipped in and the post of @nm 6131 and @kitonum are removed

The question was there was how to rearrange this  complex ln expression 

As you can see i did it by hand too, but don't know if it is right? :

If i remember the answer from nm 6131 it was like this :
expr : = I*ln(-z*I + b) - ln(z*I + b); 
I*simplify(log(exp(1)*(expr)) ;   ?        (in general : ln x^a = a*ln x)  or  take ln^a             

Note: should be nice  to  have the answer of @nm 6131 and @kitonum back again

@Kitonum 

Thanks

Your calculation brings  me back to the direction of the start formula