allvalues(RootOf(_Z^2+alpha+_Z, label = _L3));

                               1/2                       1/2
                  (1 - 4 alpha)             (1 - 4 alpha)
          - 1/2 + ----------------, - 1/2 - ----------------
                         2                         2

I'll take a look at the procedure.

Alec

Subtracting the 2nd equation from the 1st, and the 4th from the 3rd, and using diff(P,y)-diff(Q,y) = diff(P-Q,y), we can find P-Q = C(x)*exp(-y). That, together with either 1st, or 2nd equation can be used to find either P and Q (after substituting R or S from the 3rd or the 4th equation), but as Robert Israel said above, the results look complicated.

Alec

Another option may be using Maplets:-Examples:-ShowTable. Something like

L:=[[a,0.5,0.6,0.7,0.8,0.9],
    [r,0.5000,0.5999,0.7003,0.8004,0.9079],
    [alpha,0.3315,0.3455,0.2890,0.2311,-0.6636]]:
Maplets:-Examples:-ShowTable(L);

That doesn't save the table in the worksheet though, and alpha is displayed as a word.

Alec

Maplets:-Examples:-GetFile();

Alec

In the first expression, all terms in the sum are negative with these conditions, so their sum is also negative. What other proof do you need?

Alec

I would do that either in Python, or C# - but your mileage may vary.

Alec

For linear transformations (i.e. matrices) the answer is the n-th power of the matrix. For fractional-linear as well - in addition to your example, another example, say is q:=x-> (2*x+3)/(x+2). It's matrix is

A:=<2,3;1,2>;

                                 [2    3]
                            A := [      ]
                                 [1    2]

And it's n-th power describes the n-times composition while rsolve doesn't produce an answer (it should, but it is not implemented).

On the other hand, rsolve handles some non-linear and non-fractional-linear cases, such as q:=x->x^2, for example.

Alec

On the other hand, that would be very easy to do in Sage.

Alec

So, it is not allowed to use any built-in procedure, but it is allowed to submit the answer given here as your homework?

Alec

Theoretically speaking, in Windows one can insert an object into a worksheet (using Insert menu). In particular, if there is a nice table in some Excel (.xls) file, then it can be inserted - and it inserts in the center and looks nice.

Also, there is such a package, ExcelTools, that allows exporting data into an .xls file.

Practically, however, when I tried to produce the table on p. 167 that way, I got a few problems leading to series of both Maple and Excel crashes.

I guess, the manual copying and pasting is the best (if not the only) way to creating such a table in a worksheet.

Alec

Looking at the plot,

plot({x,2*x,1/x,2/x},x=0.5..1.5,y=0.9..2.1);

the area can be calculated as

int(2*x-1/x,x=sqrt(2)/2..1)+int(2/x-x,x=1..sqrt(2));

                              1/2 ln(2)

as well as

int(y-1/y,y=1..sqrt(2))+int(2/y-y/2,y=sqrt(2)..2);

                              1/2 ln(2)

Alec

Beside other well-known tools, Reinteract produces decent plots

from numpy import *
from replot import plot
x=linspace(-10,10,10000)
plot(x,cos(2*x**2))

Alec

Well, now there is a good way to check your calculations - using Wolfram Alpha. Enter either cos(2x^2) there, or

cos(2x^2) from -10 to 10

Alec

Great minds think alike...

Alec

Another solution can be obtained using rsolve,

rsolve({a(n)=a(n-1)+1,a(0)=x},a(n));

                                x + n

Alec