Sun Aug 18 18:30:54 2002 run on Windows
% Author: Alan Barnes <barnesa@aston.ac.uk>
psexplim 8;
6
% expand as far as 8th power (default is 6)
cos!-series:=ps(cos x,x,0);
1 2 1 4 1 6 1 8 9
cos-series := 1 - ---*x + ----*x - -----*x + -------*x + O(x )
2 24 720 40320
sin!-series:=ps(sin x,x,0);
1 3 1 5 1 7 9
sin-series := x - ---*x + -----*x - ------*x + O(x )
6 120 5040
atan!-series:=ps(atan x,x,0);
1 3 1 5 1 7 9
atan-series := x - ---*x + ---*x - ---*x + O(x )
3 5 7
tan!-series:=ps(tan x,x,0);
1 3 2 5 17 7 9
tan-series := x + ---*x + ----*x + -----*x + O(x )
3 15 315
cos!-series*tan!-series;
1 3 1 5 1 7 9
x - ---*x + -----*x - ------*x + O(x )
6 120 5040
% should series for sin(x)
df(cos!-series,x);
1 3 1 5 1 7 9
- x + ---*x - -----*x + ------*x + O(x )
6 120 5040
% series for sin(x) again
cos!-series/atan!-series;
-1 1 77 3 313 5 104539 7 9
x - ---*x - -----*x + ------*x - ---------*x + O(x )
6 360 3024 1814400
% should be expanded
tmp:=ps(1/(1+x^2),x,infinity);
1 1 1 1 1
tmp := ---- - ---- + ---- - ---- + O(----)
2 4 6 8 9
x x x x x
df(tmp,x);
1 1 1 1
- 2*---- + 4*---- - 6*---- + O(----)
3 5 7 9
x x x x
ps(df(1/(1+x^2),x),x,infinity);
1 1 1 1
- 2*---- + 4*---- - 6*---- + O(----)
3 5 7 9
x x x x
tmp*x;
1 1 1 1 1
(---- - ---- + ---- - ---- + O(----))*x
2 4 6 8 9
x x x x x
% not expanded as a single power series
ps(tmp*x,x,infinity);
1 1 1 1 1
--- - ---- + ---- - ---- + O(----)
x 3 5 7 9
x x x x
% now expanded
ps(1/(a*x-b*x^2),x,a/b);
2 3 4
1 a -1 b b a b a 2 b a 3
- ---*(x - ---) + ---- - ----*(x - ---) + ----*(x - ---) - ----*(x - ---)
a b 2 3 b 4 b 5 b
a a a a
5 6 7 8
b a 4 b a 5 b a 6 b a 7
+ ----*(x - ---) - ----*(x - ---) + ----*(x - ---) - ----*(x - ---)
6 b 7 b 8 b 9 b
a a a a
9
b a 8 a 9
+ -----*(x - ---) + O((x - ---) )
10 b b
a
% pole at expansion point
ps(cos!-series*x,x,2);
2
2*cos(2) + (cos(2) - 2*sin(2))*(x - 2) - (cos(2) + sin(2))*(x - 2)
- 3*cos(2) + 2*sin(2) 3 cos(2) + 2*sin(2) 4
+ ------------------------*(x - 2) + -------------------*(x - 2)
6 12
5*cos(2) - 2*sin(2) 5 - cos(2) - 3*sin(2) 6
+ ---------------------*(x - 2) + ----------------------*(x - 2)
120 360
- 7*cos(2) + 2*sin(2) 7 cos(2) + 4*sin(2) 8
+ ------------------------*(x - 2) + -------------------*(x - 2)
5040 20160
9
+ O((x - 2) )
tmp:=ps(x/atan!-series,x,0);
1 2 4 4 44 6 428 8 9
tmp := 1 + ---*x - ----*x + -----*x - -------*x + O(x )
3 45 945 14175
tmp1:=ps(atan!-series/x,x,0);
1 2 1 4 1 6 1 8 9
tmp1 := 1 - ---*x + ---*x - ---*x + ---*x + O(x )
3 5 7 9
tmp*tmp1;
1
% should be 1, of course
cos!-sin!-series:=ps(cos sin!-series,x,0);
1 2 5 4 37 6 457 8 9
cos-sin-series := 1 - ---*x + ----*x - -----*x + -------*x + O(x )
2 24 720 40320
% cos(sin(x))
tmp:=cos!-sin!-series^2;
2 2 4 14 6 37 8 9
tmp := 1 - x + ---*x - ----*x + -----*x + O(x )
3 45 315
tmp1:=ps((sin(sin!-series))^2,x,0);
2 2 4 14 6 37 8 9
tmp1 := x - ---*x + ----*x - -----*x + O(x )
3 45 315
tmp+tmp1;
9
1 + O(x )
% sin^2 + cos^2
psfunction tmp1;
2
sin(sin(x))
% function represented by power series tmp1
tmp:=tan!-series^2;
2 2 4 17 6 62 8 9
tmp := x + ---*x + ----*x + -----*x + O(x )
3 45 315
psdepvar tmp;
x
% in case we have forgotten the dependent variable
psexpansionpt tmp;
0
% .... or the expansion point
psterm(tmp,6);
17
----
45
% select 6th term
psterm(tmp,10);
1382
-------
14175
% select 10th term (series extended automtically)
tmp1:=ps(1/(cos x)^2,x,0);
2 2 4 17 6 62 8 9
tmp1 := 1 + x + ---*x + ----*x + -----*x + O(x )
3 45 315
tmp1-tmp;
9
1 + O(x )
% sec^2-tan^2
ps(int(e^(x^2),x),x,0);
1 3 1 5 1 7 9
x + ---*x + ----*x + ----*x + O(x )
3 10 42
% integrator not called
tmp:=ps(1/(y+x),x,0);
1 1 1 2 1 3 1 4 1 5 1 6 1 7
tmp := --- - ----*x + ----*x - ----*x + ----*x - ----*x + ----*x - ----*x
y 2 3 4 5 6 7 8
y y y y y y y
1 8 9
+ ----*x + O(x )
9
y
ps(int(tmp,y),x,0);
1 1 2 1 3 1 4 1 5 1 6
log(y) + ---*x - ------*x + ------*x - ------*x + ------*x - ------*x
y 2 3 4 5 6
2*y 3*y 4*y 5*y 6*y
1 7 1 8 9
+ ------*x - ------*x + O(x )
7 8
7*y 8*y
% integrator called on each coefficient
pscompose(cos!-series,sin!-series);
1 2 5 4 37 6 457 8 9
1 - ---*x + ----*x - -----*x + -------*x + O(x )
2 24 720 40320
% power series composition cos(sin(x)) again
cos!-sin!-series;
1 2 5 4 37 6 457 8 9
1 - ---*x + ----*x - -----*x + -------*x + O(x )
2 24 720 40320
% should be same as previous result
psfunction cos!-sin!-series;
cos(sin(x))
tmp:=ps(log x,x,1);
1 2 1 3 1 4 1 5
tmp := x - 1 - ---*(x - 1) + ---*(x - 1) - ---*(x - 1) + ---*(x - 1)
2 3 4 5
1 6 1 7 1 8 9
- ---*(x - 1) + ---*(x - 1) - ---*(x - 1) + O((x - 1) )
6 7 8
tmp1:=pscompose(tmp, cos!-series);
1 2 1 4 1 6 17 8 9
tmp1 := - ---*x - ----*x - ----*x - ------*x + O(x )
2 12 45 2520
% power series composition of log(cos(x))
df(tmp1,x);
1 3 2 5 17 7 9
- x - ---*x - ----*x - -----*x + O(x )
3 15 315
% series for -tan x
psreverse tan!-series;
1 3 1 5 1 7 9
x - ---*x + ---*x - ---*x + O(x )
3 5 7
% should be series for atan x
atan!-series;
1 3 1 5 1 7 9
x - ---*x + ---*x - ---*x + O(x )
3 5 7
tmp:=ps(e^x,x,0);
1 2 1 3 1 4 1 5 1 6 1 7
tmp := 1 + x + ---*x + ---*x + ----*x + -----*x + -----*x + ------*x
2 6 24 120 720 5040
1 8 9
+ -------*x + O(x )
40320
psreverse tmp;
1 2 1 3 1 4 1 5 1 6
x - 1 - ---*(x - 1) + ---*(x - 1) - ---*(x - 1) + ---*(x - 1) - ---*(x - 1)
2 3 4 5 6
1 7 1 8 9
+ ---*(x - 1) - ---*(x - 1) + O((x - 1) )
7 8
% NB expansion of log x in powers of (x-1)
pschangevar(tan!-series,y);
1 3 2 5 17 7 9
y + ---*y + ----*y + -----*y + O(y )
3 15 315
end;
Time for test: 1123 ms, plus GC time: 120 ms