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last edited 9 years ago by test1 |
Edit detail for #418 series expansion, but not differentiable revision 2 of 4
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Editor: kratt6
Time: 2008/02/01 01:52:10 GMT-8 |
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| Note: | ||
added:
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) |
axiom
((-((-m*m)+1)**(1/2))+1)**(1/2)
![\label{eq3}\begin{array}{@{}l}
\displaystyle
{{1 \over{\sqrt{2}}}\ z}+{{1 \over{8 \ {\sqrt{2}}}}\ {z^3}}+{{7 \over{{128}\ {\sqrt{2}}}}\ {z^5}}+{{{33}\over{{1024}\ {\sqrt{2}}}}\ {z^7}}+
\
\
\displaystyle
{{{715}\over{{32768}\ {\sqrt{2}}}}\ {z^9}}+{O \left({z^{11}}\right)}](images/4534018470730410086-16.0px.png "\label{eq3}\begin{array}{@{}l}
\displaystyle
{{1 \over{\sqrt{2}}}\ z}+{{1 \over{8 \ {\sqrt{2}}}}\ {z^3}}+{{7 \over{{128}\ {\sqrt{2}}}}\ {z^5}}+{{{33}\over{{1024}\ {\sqrt{2}}}}\ {z^7}}+
\
\
\displaystyle
{{{715}\over{{32768}\ {\sqrt{2}}}}\ {z^9}}+{O \left({z^{11}}\right)}") | (3) |
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)