help links subscribe changes refresh edit

	IssueTracker
#417 slow integration ←	↑	→ #419 bug in type inference for anonymous functions

#418 series expansion, but not differentiable

Edit detail for #418 series expansion, but not differentiable revision 1 of 4

	1 2 3 4
Editor: kratt6 Time: 2008/02/01 01:50:46 GMT-8
Note:

changed:
-
Consider
\begin{axiom}
ex := ((-((-z*z)+1)**(1/2))+1)**(1/2)
\end{axiom}
which is certainly continuous, but not differentiable at zero.  Still, axiom gives
\begin{axiom}
m:=z::UTS(AN, z, 0)
((-((-m*m)+1)**(1/2))+1)**(1/2)
\end{axiom}
which is correct for positive values, but not for negative values of z.  Of course, there are simpler functions that are developed like that, abs for example.

I'm not sure what's the correct way to deal with this.  MuPAD is very careful here::
    >> ex := ((-((-z*z)+1)^(1/2))+1)^(1/2)
    ex := ((-((-z*z)+1)^(1/2))+1)^(1/2)

                                          2 1/2 1/2
                               (1 - (1 - z )   )
    >> series(ex, z=0, 5)
    series(ex, z=0, 5)

                       /  2 \1/2        /  2 \1/2         /  2 \1/2
                     2 | z  |         4 | z  |          6 | z  |
        /  2 \1/2   z  | -- |      7 z  | -- |      33 z  | -- |
        | z  |         \  2 /           \  2 /            \  2 /         8
        | -- |    + ------------ + -------------- + --------------- + O(z )
        \  2 /            8              128              1024
    >> series(abs(z), z=0, 5)
    series(abs(z), z=0, 5)

                                     z         6
                                  ------- + O(z )
                                  sign(z)

Consider

axiom
ex := ((-((-z*z)+1)**(1/2))+1)**(1/2)

(1)

Type: Expression Integer

which is certainly continuous, but not differentiable at zero. Still, axiom gives

axiom
m:=z::UTS(AN, z, 0)

(2)

Type: UnivariateTaylorSeries?(AlgebraicNumber?,z,0)

axiom
((-((-m*m)+1)**(1/2))+1)**(1/2)

(3)

Type: UnivariateTaylorSeries?(AlgebraicNumber?,z,0)

which is correct for positive values, but not for negative values of z. Of course, there are simpler functions that are developed like that, abs for example.

I'm not sure what's the correct way to deal with this. MuPAD? is very careful here:: >> ex := ((-((-zz)+1)^(1/2))+1)^(1/2) ex := ((-((-zz)+1)^(1/2))+1)^(1/2)

1/2 1/2 (1 - (1 - z ) ) >> series(ex, z=0, 5) series(ex, z=0, 5)

/ 2 \1/2 / 2 \1/2 / 2 \1/2 2 | z | 4 | z | 6 | z | / 2 \1/2 z | -- | 7 z | -- | 33 z | -- | | z | \ 2 / \ 2 / \ 2 / 8 | -- | + ------------ + -------------- + --------------- + O(z ) \ 2 / 8 128 1024 >> series(abs(z), z=0, 5) series(abs(z), z=0, 5)

z 6 ------- + O(z ) sign(z)