@Axel Vogt 

"Is it a bug if posts are in languages other than English? That is actually an incredible attitude."

Please calm down. I thought it is a bug as what happened when the Maple web site was showing the web site in non-English due to geolocation bug. Since Mapleprime forum is full of bugs I thought maybe the same thing happens to some entries.

I visit many technical forums, and never seen a technical and international forum no-less, have discussion or a single question in it in something other than in English (except this).  Try to post a question at Mathematica or Matlab forums in something other than English and see what happens. Try that at stackexchange and see.

Unless the forum is local to a country, or private and not an international forum like this that is open to everyone, then there should be one common language used. period.

btw, English is not my birth language myself. And I would never think of asking a question here in my own language myself. It will be rude to do so since many (if not all) would not understand what I am saying.

why not post plain text code? Do you expect someone to type all of this?

Example

Diff(x1,t) = x2^2,
Diff(x2,t) = u,
y = x1

result in

Diff(y,t$2)-4*u^2*Diff(y,t) = 0
2*u*x2 - Diff(y,t^2) = 0
x1 - y = 0

Which version are using? Using Maple 17 I do not get the output you show, I get this:

--------------------
restart;
Diff(x1,t) = x2^2,
Diff(x2,t) = u,
y = x1;

      Diff(x1, t) = x2^2, Diff(x2, t) = u, y = x1

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

I do not even see where the `4` could have come from.

@Alejandro Jakubi 

Let me just step back one step. First I think we agree there can be different regions of expansion, and hence there is a need to tell laurent() (or series() the region, btw, Mathematica does not have Laurent[] function, but Series does both, Taylor and Laurent). 

How this API should look like, I am not sure. I am only a user. All what I am saying is that, from a user point of view, I want to ask a CAS to expand a complex function in some region, and currently this is not possible directly (unless one can do some smart tricks as shown above, and not all users would know how to do this).

To make this more concrete, here is a HW problem I had to solve (the one I asked about)

"Obtain two distinct Laurent expansions for f(z)=((3z+1)/(z^2-1)) around z=1 and tell where each converges"

Currently, neither Maple nor Mathematica as well supports specifying a region. Only an expansion point, and what seems to happen is that the disk around this point up to the first pole is used by default.

As far as implementation inside Maple, I do not know. I know I solved it by hand, by making a diagram and looking and seeing where to use Laurent vs. Taylor depending on direction. How to program this in a CAS, that is up to the math experts.

For an API, it can be as simple as supporting regions specificed as a disk or an annulus. So any mathematical way deemed to be the right way to specify these regions to use in the API would be fine.

Again, I am answering as a user, saying there is a need for this. I will leave how to do it for those experts in CAS programming :)

What does (T(z))^r ® 0 as r ® ∞?  mean?  what is circle with R inside it? Is this some kind of worksheet coding?

@Preben Alsholm 

Thanks for the nice idea. It worked, it showed the same solution I did by hand:

I still feel there should be a standard API into laurent series to tell it the region to select.  

why not post plain text code so one can copy it and use it? Do you really expect someone to type all this looking at the screen?

I do not understand this forum. This sort of thing is never allowed, for someone to paste an image for the code on the Mathematica forum. I do not understand the rules of this forum that will allow this, I see it so often here and no one seems to complain about it.

@Carl Love 

"Whew! I am sick and tired of cut-and-pasting worksheets line by line."

If I can suggest, you can, using the mouse after pasting the image here, make it little smaller by resizing it. This makes it easier to read the whole post. Right now the images are too large.

To paste more than one equation here, since copy/paste does not work with more than one equation, what I do is this: I take snapshot of the screen, put in paint (on windows), then select using the mouse all the content I want pasted here, then paste here.This way I can paste large content as one image.

At the mathematica stackexchange forum there is a very nice tool to use to paste directly from notebook itself to posts at the forum. One can paste anything from the notebook. Simply select the cell(s), and with one click on a button the tool copies it then one just past it into the post with the mouse. Here is a link to it if you like to see how it looks like. May be someone at Mapleprime can make such tool for this forum.

@Carl Love 

Thanks, I mixed it up with dolittle. No problem, simple transpose is all what is needed. Will fix now

@ANANDMUNAGALA 

updated answer for Crout

What does

    Pi= infinity Σ n=0

mean?

Is this Maple code?

@Carl Love 

Thank you, I did try that also. but it was giving me a strange result.

restart;
F:= w-> cos(w*L)*cos(w*x)/(w^6+w^2*k);
with(inttrans);
simplify(fouriercos(F(w),w,x));

@Axel Vogt 

Thanks. But it will really help to give an answer using a self contained code using the function I gave. I still do not see what you are doing.

This is what I typed.

-------------------------------
restart;
F:= w-> cos(w*L)*cos(w*x)/(w^6+w^2*k);
with(inttrans);
fouriercos(F(x),x,w);
---------------------------------

It does not evaluate as you can see. I do not know what the convert(%,Int) is for, but here it is:

convert(%,Int);

Mathematica has Inverse and Forward Fourier Transform functions for both Sin/Cos. I do not understand why Maple does not have these.

@Alejandro Jakubi 

thanks. I tried it. It works but the final result is hard to read. Needs lots of manually formating, and I am not going to waste my time formating code by hand so I can read it.

I prefer plain normal text based programming. All this 2D stuff and document mode and menus and click this and click that and such is making Maple hard to use for me.

@digerdiga 

I'd like to help, but I can't stand document mode. If you can make a version using worksheet that one can open and use that will help. I have not figured out how to open a file like yours, written in document mode as a worksheet. If I save the file as SAVE AS->Maple classical worksheet, then many things get mangled up.

May be there is a easy way, but I have not figured it out.