#### 6.75 asympt ignores order on ﬁrst use (10.1.00)

Here is a quirk discovered by a colleague:

asympt does not seem to honour the Order setting the ﬁrst time it is called for a particular
expression:

> restart: R:=sqrt(r^2-a^2);
2 2
R := sqrt(r - a )

Series expansion for large r converted to an algebraic expression. Note that Order is
ignored.

I think this is well described as a ’quirk’; Maple seems to need to do one series expansion of
the expression before it starts to honour the value of Order. If you skip/execute the a0 line
below then the a1 line ignores/honours the Order option. If you cange the - in the a0 line to
+, it works diﬀerently, so it seems to be something to do with remembering the series for
that particular expression.

> a0:=asympt(1/sqrt(x^2-a^2),x);
2 4 6
a a a 1
a0 := 1/x + 1/2 ---- + 3/8 ---- + 5/16 ---- + O(----)
3 5 7 9
x x x x
> a1:=asympt(1/R,r,4);
2
a 1
a1 := 1/r + 1/2 ---- + O(----)
3 5
r r
> a2:=asympt(1/R,r,4);
2
a 1
a2 := 1/r + 1/2 ---- + O(----)
3 5
r r
> a3:=asympt(1/R,r);
2 4
a a 1
a3 := 1/r + 1/2 ---- + 3/8 ---- + O(----)
3 5 7
r r r

Without executing the a0 line we get:

> restart: R:=sqrt(r^2-a^2);
> a1:=asympt(1/R,r,4);
2 4
a a 1
a1 := 1/r + 1/2 ---- + 3/8 ---- + O(----)
3 5 7
r r r
> a2:=asympt(1/R,r,4);
2
a 1
a2 := 1/r + 1/2 ---- + O(----)
3 5
r r
> a3:=asympt(1/R,r);
2 4 6
a a a 1
a3 := 1/r + 1/2 ---- + 3/8 ---- + 5/16 ---- + O(----)
3 5 7 9
r r r r

This is not a great problem, but I am intrigued as to why it happens, and if there is a better
anser than ’do it twice’

This is actually a bug in "series" (which "asympt" calls). I don’t have any advice other than
to call it twice. The reason it works the second time, I think, is that "series" consults its
remember table to see if it has computed the same series before to this or higher order; if it
has done so, it truncates the series appropriately.