## 12745 Reputation

8 years, 294 days

## f1...

```f1:= x -> local y:= x-floor(x/(2*Pi))*2*Pi; piecewise(y < (1/2)*Pi, 2*y, y < Pi, Pi, y <= (3/2)*Pi, 3*Pi-2*y);

```

Now f=f1.

I don't quite understand Q2.

## formal...

The hypergeom command uses analytic continuation, but sum does not (by default).

If you want it, just define:

```F:=z->sum(po(1,n)^2/po(1/2,n)^2*z^n/(n+1),n=1..infinity, formal);
```

Then:

F(4);
8*hypergeom([1, 2, 2, 2],[3/2, 3/2, 3],4)
F(4.);
-1.215795612 + 1.672715456 I

## size...

simplify(expr) is simpler in the sense that the result does not contain a sqrt any more.
Using simplify(expr, size), expr remains unchanged.

## value...

```p := pds:-value(output = procedurelist);
p(0, 0.1);
#        [x = 0., t = 0.1, u(x, t) = -0.587785252292473]
```

You may want to use

```f:=(x,t)->rhs(p(x,t)[-1]):
f(0.23,0.5)
#                       -0.991900750572342```

## proc...

```rem_d:=proc(n::integer, d::posint)
local q:=n, q1;
if (d=1 or n=0) then return infinity fi;
while irem(q,d,'q1')=0 do q:=q1 od;
q
end proc;```

rem_d(294912,8)  #  9

## Solution...

```f := -log[2](7/10*x)+log[3](3*x-1):
#plot(f, x=1/3 .. 40);
limit(f, x=infinity);
#                           -infinity

fsolve(f, x=0.34 .. 100, maxsols=4);
#                   0.3730125034, 16.60731835

a,b:=%;
#               a, b := 0.3730125034, 16.60731835

evalf(eval(f, x=(a+b)/2));  # >0
#                          0.339241451

#         answer := 0.3730125034 <= x, x <= 16.60731835
```

Exact solution possible but only in terms of RootOfs.

## Eigenvectors...

Of course. The right (or usual) eigenvectors of a matrix A are given by the command Eigenvectors(A). The left eigenvectors are given by the same command but for the transposed matrix.

## Bug...

Actually Maple can compute the limit.

## invfunc...

The expression of a inverse function is given in the table invfunc (and also Invfunc for multivalued inverses), see ?invfunc. When the user wants to use the inverse of one of his functions, he must define the entry in the table(s).

In Maple 2018 it works:

## Continued fractions...

Rational approximations are obtained via continued fractions. They are implemented in NumberTheory:-ContinuedFraction and the older `convert/confrac`.

For example:

```cf:=NumberTheory:-ContinuedFraction(Pi):
Convergent(cf, 1..5);
#    [22/7, 333/106, 355/113, 103993/33102, 104348/33215]
```

The actual computations are in `convert/real_rat`.

``<,>`(seq(`<|>`(seq(M[i,j],j=1..2)),i=1..2));`

or

`Matrix(4,4,(i,j)->M[iquo(i-1,2)+1,iquo(j-1,2)+1][irem(i-1,2)+1,irem(j-1,2)+1]);`

or

`Matrix(linalg[blockmatrix](2,2,[seq(M)]));`

## imbricated tables...

Replace sum with add in the definition of S.

## Careful with log branches...

 > ode := diff(y(x), x) - 2*(2*y(x) - x)/(x + y(x)) = 0; ic:=y(0)=2;
 (1)
 > sol:=dsolve(ode,'implicit');
 (2)
 > eq:=exp(lhs(sol))=1;
 (3)
 > EQ:=simplify(eq,exp);
 (4)
 > #plugin in y=2 at x=0 eval(%,[y(x)=2,x=0]);
 (5)
 > solve(%,_C1)
 (6)
 > EQ0:=eval(EQ, _C1=%);
 (7)
 > ysols:=solve(EQ0,y(x));
 (8)
 > simplify(limit~([ysols], x=0));
 (9)
 > ysol:=ysols[1];
 (10)
 > plot(ysol,x=-1..1);
 > odetest(y(x)=ysol,[ode,ic]);
 (11)
 >