REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...
comment
Test and demonstration file for the Taylor expansion package,
by Rainer M. Schoepf. Works with version 2.1e (03-May-95);
%%% showtime;
on errcont;
% disable interruption on errors
comment Simple Taylor expansion;
xx := taylor (e**x, x, 0, 4);
1 2 1 3 1 4 5
xx := 1 + x + ---*x + ---*x + ----*x + O(x )
2 6 24
yy := taylor (e**y, y, 0, 4);
1 2 1 3 1 4 5
yy := 1 + y + ---*y + ---*y + ----*y + O(y )
2 6 24
comment Basic operations, i.e. addition, subtraction, multiplication,
and division are possible: this is not done automatically if
the switch TAYLORAUTOCOMBINE is OFF. In this case it is
necessary to use taylorcombine;
taylorcombine (xx**2);
2 4 3 2 4 5
1 + 2*x + 2*x + ---*x + ---*x + O(x )
3 3
taylorcombine (ws - xx);
3 2 7 3 5 4 5
x + ---*x + ---*x + ---*x + O(x )
2 6 8
taylorcombine (xx**3);
9 2 9 3 27 4 5
1 + 3*x + ---*x + ---*x + ----*x + O(x )
2 2 8
comment The result is again a Taylor kernel;
if taylorseriesp ws then write "OK";
OK
comment It is not possible to combine Taylor kernels that were
expanded with respect to different variables;
taylorcombine (xx**yy);
1 2 1 3 1 4 5
(1 + x + ---*x + ---*x + ----*x + O(x ))
2 6 24
1 2 1 3 1 4 5
**(1 + y + ---*y + ---*y + ----*y + O(y ))
2 6 24
comment But we can take the exponential or the logarithm
of a Taylor kernel;
taylorcombine (e**xx);
2 5*e 3 5*e 4 5
e + e*x + e*x + -----*x + -----*x + O(x )
6 8
taylorcombine log ws;
1 2 1 3 1 4 5
1 + x + ---*x + ---*x + ----*x + O(x )
2 6 24
comment A more complicated example;
hugo := taylor(log(1/(1-x)),x,0,5);
1 2 1 3 1 4 1 5 6
hugo := x + ---*x + ---*x + ---*x + ---*x + O(x )
2 3 4 5
taylorcombine(exp(hugo/(1+hugo)));
1 4 6
1 + x + ----*x + O(x )
12
comment We may try to expand about another point;
taylor (xx, x, 1, 2);
65 8 5 2 3
---- + ---*(x - 1) + ---*(x - 1) + O((x - 1) )
24 3 4
comment Arc tangent is one of the functions this package knows of;
xxa := taylorcombine atan ws;
65 1536 2933040 2 3
xxa := atan(----) + ------*(x - 1) - ----------*(x - 1) + O((x - 1) )
24 4801 23049601
comment The trigonometric functions;
taylor (tan x / x, x, 0, 2);
1 2 3
1 + ---*x + O(x )
3
taylorcombine sin ws;
cos(1) 2 3
sin(1) + --------*x + O(x )
3
taylor (cot x / x, x, 0, 4);
-2 1 1 2 2 4 5
x - --- - ----*x - -----*x + O(x )
3 45 945
comment The poles of these functions are correctly handled;
taylor(tan x,x,pi/2,0);
pi -1 pi
- (x - ----) + O(x - ----)
2 2
taylor(tan x,x,pi/2,3);
pi -1 1 pi 1 pi 3 pi 4
- (x - ----) + ---*(x - ----) + ----*(x - ----) + O((x - ----) )
2 3 2 45 2 2
comment Expansion with respect to more than one kernel is possible;
xy := taylor (e**(x+y), x, 0, 2, y, 0, 2);
1 2 3 3
xy := 1 + y + ---*y + x + y*x + (4 terms) + O(x ,y )
2
taylorcombine (ws**2);
2 3 3
1 + 2*y + 2*y + 2*x + 4*y*x + (4 terms) + O(x ,y )
comment We take the inverse and convert back to REDUCE's standard
representation;
taylorcombine (1/ws);
2 3 3
1 - 2*x + 2*x - 2*y + 4*y*x + (4 terms) + O(x ,y )
taylortostandard ws;
2 2 2 2 2 2
4*x *y - 4*x *y + 2*x - 4*x*y + 4*x*y - 2*x + 2*y - 2*y + 1
comment Some examples of Taylor kernel divsion;
xx1 := taylor (sin (x), x, 0, 4);
1 3 5
xx1 := x - ---*x + O(x )
6
taylorcombine (xx/xx1);
-1 2 1 2 3
x + 1 + ---*x + ---*x + O(x )
3 3
taylorcombine (1/xx1);
-1 1 3
x + ---*x + O(x )
6
tt1 := taylor (exp (x), x, 0, 3);
1 2 1 3 4
tt1 := 1 + x + ---*x + ---*x + O(x )
2 6
tt2 := taylor (sin (x), x, 0, 3);
1 3 4
tt2 := x - ---*x + O(x )
6
tt3 := taylor (1 + tt2, x, 0, 3);
1 3 4
tt3 := 1 + x - ---*x + O(x )
6
taylorcombine(tt1/tt2);
-1 2 2
x + 1 + ---*x + O(x )
3
taylorcombine(tt1/tt3);
1 2 1 3 4
1 + ---*x - ---*x + O(x )
2 6
taylorcombine(tt2/tt1);
2 1 3 4
x - x + ---*x + O(x )
3
taylorcombine(tt3/tt1);
1 2 1 3 4
1 - ---*x + ---*x + O(x )
2 6
comment Here's what I call homogeneous expansion;
xx := taylor (e**(x*y), {x,y}, 0, 2);
3
xx := 1 + y*x + O({x,y} )
xx1 := taylor (sin (x+y), {x,y}, 0, 2);
3
xx1 := y + x + O({x,y} )
xx2 := taylor (cos (x+y), {x,y}, 0, 2);
1 2 1 2 3
xx2 := 1 - ---*y - y*x - ---*x + O({x,y} )
2 2
temp := taylorcombine (xx/xx2);
1 2 1 2 3
temp := 1 + ---*y + 2*y*x + ---*x + O({x,y} )
2 2
taylorcombine (ws*xx2);
3
1 + y*x + O({x,y} )
comment The following shows a principal difficulty:
since xx1 is symmetric in x and y but has no constant term
it is impossible to compute 1/xx1;
taylorcombine (1/xx1);
***** Not a unit in argument to invtaylor
comment Substitution in Taylor expressions is possible;
sub (x=z, xy);
1 2 3 3
1 + y + ---*y + z + y*z + (4 terms) + O(z ,y )
2
comment Expression dependency in substitution is detected;
sub (x=y, xy);
***** Invalid substitution in Taylor kernel: dependent variables y y
comment It is possible to replace a Taylor variable by a constant;
sub (x=4, xy);
13 2 3
13 + 13*y + ----*y + O(y )
2
sub (x=4, xx1);
3
4 + y + O(y )
sub (y=0, ws);
4
comment This package has three switches:
TAYLORKEEPORIGINAL, TAYLORAUTOEXPAND, and TAYLORAUTOCOMBINE;
on taylorkeeporiginal;
temp := taylor (e**(x+y), x, 0, 5);
y y y
y y e 2 e 3 e 4 6
temp := e + e *x + ----*x + ----*x + ----*x + (1 term) + O(x )
2 6 24
taylorcombine (log (temp));
6
y + x + O(x )
taylororiginal ws;
x + y
taylorcombine (temp * e**x);
y y y
x y y e 2 e 3 e 4 6
e *(e + e *x + ----*x + ----*x + ----*x + (1 term) + O(x ))
2 6 24
on taylorautoexpand;
taylorcombine ws;
y y
y y y 2 4*e 3 2*e 4 6
e + 2*e *x + 2*e *x + ------*x + ------*x + (1 term) + O(x )
3 3
taylororiginal ws;
2*x + y
e
taylorcombine (xx1 / x);
-1 2
y*x + 1 + O({x,y} )
on taylorautocombine;
xx / xx2;
1 2 1 2 3
1 + ---*y + 2*y*x + ---*x + O({x,y} )
2 2
ws * xx2;
3
1 + y*x + O({x,y} )
comment Another example that shows truncation if Taylor kernels
of different expansion order are combined;
comment First we increase the number of terms to be printed;
taylorprintterms := all;
taylorprintterms := all
p := taylor (x**2 + 2, x, 0, 10);
2 11
p := 2 + x + O(x )
p - x**2;
2 11 2
(2 + x + O(x )) - x
p - taylor (x**2, x, 0, 5);
6
2 + O(x )
taylor (p - x**2, x, 0, 6);
7
2 + O(x )
off taylorautocombine;
taylorcombine(p-x**2);
11
2 + O(x )
taylorcombine(p - taylor(x**2,x,0,5));
6
2 + O(x )
comment Switch back to finite number of terms;
taylorprintterms := 6;
taylorprintterms := 6
comment Some more examples;
taylor(1/(1+y^4+x^2*y^2+x^4),{x,y},0,6);
4 2 2 4 7
1 - y - y *x - x + O({x,y} )
taylor ((1 + x)**n, x, 0, 3);
2
n*(n - 1) 2 n*(n - 3*n + 2) 3 4
1 + n*x + -----------*x + ------------------*x + O(x )
2 6
taylor (e**(-a*t) * (1 + sin(t)), t, 0, 4);
3 2
a*(a - 2) 2 - a + 3*a - 1 3
1 + ( - a + 1)*t + -----------*t + ------------------*t
2 6
3 2
a*(a - 4*a + 4) 4 5
+ -------------------*t + O(t )
24
operator f;
taylor (1 + f(t), t, 0, 3);
sub(t=0,df(f(t),t,2)) 2
f(0) + 1 + sub(t=0,df(f(t),t))*t + -----------------------*t
2
sub(t=0,df(f(t),t,3)) 3 4
+ -----------------------*t + O(t )
6
taylor(f(sqrt(x^2+y^2)),x,x0,4,y,y0,4);
2 2 2 2
f(sqrt(x0 + y0 )) + sub(y=y0,df(f(sqrt(x0 + y )),y))*(y - y0)
2 2
sub(y=y0,df(f(sqrt(x0 + y )),y,2)) 2
+ -------------------------------------*(y - y0)
2
2 2
sub(y=y0,df(f(sqrt(x0 + y )),y,3)) 3
+ -------------------------------------*(y - y0)
6
2 2
sub(y=y0,df(f(sqrt(x0 + y )),y,4)) 4
+ -------------------------------------*(y - y0)
24
2 2
+ sub(x=x0,df(f(sqrt(x + y0 )),x))*(x - x0) + (19 terms)
5 5
+ O((x - x0) ,(y - y0) )
clear f;
taylor (sqrt(1 + a*x + sin(x)), x, 0, 3);
2 3 2
a + 1 - a - 2*a - 1 2 3*a + 9*a + 9*a - 1 3 4
1 + -------*x + -----------------*x + -----------------------*x + O(x )
2 8 48
taylorcombine (ws**2);
1 3 4
1 + (a + 1)*x - ---*x + O(x )
6
taylor (sqrt(1 + x), x, 0, 5);
1 1 2 1 3 5 4 7 5 6
1 + ---*x - ---*x + ----*x - -----*x + -----*x + O(x )
2 8 16 128 256
taylor ((cos(x) - sec(x))^3, x, 0, 5);
6
0 + O(x )
taylor ((cos(x) - sec(x))^-3, x, 0, 5);
-6 1 -4 11 -2 347 6767 2 15377 4 6
- x + ---*x + -----*x - ------- - --------*x - ---------*x + O(x )
2 120 15120 604800 7983360
taylor (sqrt(1 - k^2*sin(x)^2), x, 0, 6);
2 2 2 2 4 2
k 2 k *( - 3*k + 4) 4 k *( - 45*k + 60*k - 16) 6 7
1 - ----*x + ------------------*x + ----------------------------*x + O(x )
2 24 720
taylor (sin(x + y), x, 0, 3, y, 0, 3);
1 3 1 2 1 2 1 2 3 4 4
x - ---*x + y - ---*y*x - ---*y *x + ----*y *x + (2 terms) + O(x ,y )
6 2 2 12
taylor (e^x - 1 - x,x,0,6);
1 2 1 3 1 4 1 5 1 6 7
---*x + ---*x + ----*x + -----*x + -----*x + O(x )
2 6 24 120 720
taylorcombine sqrt ws;
1 1 2 1 3 1 4
---------*x + -----------*x + ------------*x + -------------*x
sqrt(2) 6*sqrt(2) 36*sqrt(2) 270*sqrt(2)
1 5 6
+ --------------*x + O(x )
2592*sqrt(2)
taylor(sin(x)/x,x,1,2);
- 2*cos(1) + sin(1) 2
sin(1) + (cos(1) - sin(1))*(x - 1) + ----------------------*(x - 1)
2
3
+ O((x - 1) )
taylor((sqrt(4+h)-2)/h,h,0,5);
1 1 1 2 5 3 7 4 21 5 6
--- - ----*h + -----*h - -------*h + --------*h - ---------*h + O(h )
4 64 512 16384 131072 2097152
taylor((sqrt(x)-2)/(4-x),x,4,2);
1 1 1 2 3
- --- + ----*(x - 4) - -----*(x - 4) + O((x - 4) )
4 64 512
taylor((sqrt(y+4)-2)/(-y),y,0,2);
1 1 1 2 3
- --- + ----*y - -----*y + O(y )
4 64 512
taylor(x*tanh(x)/(sqrt(1-x^2)-1),x,0,3);
7 2 4
- 2 + ---*x + O(x )
6
taylor((e^(5*x)-2*x)^(1/x),x,0,2);
3
3 3 73*e 2 3
e + 8*e *x + -------*x + O(x )
3
taylor(sin x/cos x,x,pi/2,3);
pi -1 1 pi 1 pi 3 pi 4
- (x - ----) + ---*(x - ----) + ----*(x - ----) + O((x - ----) )
2 3 2 45 2 2
taylor(log x*sin(x^2)/(x*sinh x),x,0,2);
1 2 3
log(x)*(1 - ---*x + O(x ))
6
taylor(1/x-1/sin x,x,0,2);
1 3
- ---*x + O(x )
6
taylor(tan x/log cos x,x,pi/2,2);
pi -1 pi
- (x - ----) + O(x - ----)
2 2
-------------------------------
log(pi - 2*x) - log(2)
taylor(log(x^2/(x^2-a)),x,0,3);
2
- x
taylor(log(--------),x,0,3)
2
a - x
comment Three more complicated examples contributed by Stan Kameny;
zz2 := (z*(z-2*pi*i)*(z-pi*i/2)^2)/(sinh z-i);
3 2 2 3
z*(2*i*pi - 12*i*pi*z - 9*pi *z + 4*z )
zz2 := -------------------------------------------
4*(sinh(z) - i)
dz2 := df(zz2,z);
3 3 2 2
dz2 := ( - 2*cosh(z)*i*pi *z + 12*cosh(z)*i*pi*z + 9*cosh(z)*pi *z
4 3 2
- 4*cosh(z)*z + 2*sinh(z)*i*pi - 36*sinh(z)*i*pi*z
2 3 2 3 3
- 18*sinh(z)*pi *z + 16*sinh(z)*z + 18*i*pi *z - 16*i*z + 2*pi
2 2
- 36*pi*z )/(4*(sinh(z) - 2*sinh(z)*i - 1))
z0 := pi*i/2;
i*pi
z0 := ------
2
taylor(dz2,z,z0,6);
2
i*(pi - 16) i*pi pi i*pi 2
- 2*pi + --------------*(z - ------) + ----*(z - ------)
4 2 2 2
2
i*( - 3*pi + 80) i*pi 3 pi i*pi 4
+ -------------------*(z - ------) - ----*(z - ------)
120 2 24 2
2
i*(5*pi - 168) i*pi 5 i*pi 7
+ -----------------*(z - ------) + (1 term) + O((z - ------) )
3360 2 2
zz3:=(z*(z-2*pi)*(z-pi/2)^2)/(sin z-1);
3 2 2 3
z*( - 2*pi + 9*pi *z - 12*pi*z + 4*z )
zz3 := ------------------------------------------
4*(sin(z) - 1)
dz3 := df(zz3,z);
3 2 2 3 4
dz3 := (2*cos(z)*pi *z - 9*cos(z)*pi *z + 12*cos(z)*pi*z - 4*cos(z)*z
3 2 2 3
- 2*sin(z)*pi + 18*sin(z)*pi *z - 36*sin(z)*pi*z + 16*sin(z)*z
3 2 2 3 2
+ 2*pi - 18*pi *z + 36*pi*z - 16*z )/(4*(sin(z) - 2*sin(z) + 1))
z1 := pi/2;
pi
z1 := ----
2
taylor(dz3,z,z1,6);
2 2
pi - 16 pi pi pi 2 3*pi - 80 pi 3
2*pi + ----------*(z - ----) + ----*(z - ----) + ------------*(z - ----)
4 2 2 2 120 2
2
pi pi 4 5*pi - 168 pi 5 pi 7
+ ----*(z - ----) + -------------*(z - ----) + (1 term) + O((z - ----) )
24 2 3360 2 2
taylor((sin tan x-tan sin x)/(asin atan x-atan asin x),x,0,6);
5 2 1313 4 2773 6 7
1 + ---*x + ------*x - -------*x + O(x )
3 1890 11907
comment If the expansion point is not constant, it has to be taken
care of in differentation, as the following examples show;
taylor(sin(x+a),x,a,8);
sin(2*a) 2 cos(2*a) 3
sin(2*a) + cos(2*a)*(x - a) - ----------*(x - a) - ----------*(x - a)
2 6
sin(2*a) 4 cos(2*a) 5 9
+ ----------*(x - a) + ----------*(x - a) + (3 terms) + O((x - a) )
24 120
df(ws,a);
cos(2*a) 2 sin(2*a) 3
cos(2*a) - sin(2*a)*(x - a) - ----------*(x - a) + ----------*(x - a)
2 6
cos(2*a) 4 sin(2*a) 5 8
+ ----------*(x - a) - ----------*(x - a) + (2 terms) + O((x - a) )
24 120
taylor(cos(x+a),x,a,7);
cos(2*a) 2 sin(2*a) 3
cos(2*a) - sin(2*a)*(x - a) - ----------*(x - a) + ----------*(x - a)
2 6
cos(2*a) 4 sin(2*a) 5 8
+ ----------*(x - a) - ----------*(x - a) + (2 terms) + O((x - a) )
24 120
comment A problem are non-analytical terms: rational powers and
logarithmic terms can be handled, but other types of essential
singularities cannot;
taylor(sqrt(x),x,0,2);
1/2 3
x + O(x )
taylor(asinh(1/x),x,0,5);
1 2 3 4 5 6 7
- log(x) + (log(2) + ---*x - ----*x + ----*x + O(x ))
4 32 96
taylor(e**(1/x),x,0,2);
1/x
taylor(e ,x,0,2)
comment Another example for non-integer powers;
sub (y = sqrt (x), yy);
1/2 1 1 3/2 1 2 5/2
1 + x + ---*x + ---*x + ----*x + O(x )
2 6 24
comment Expansion about infinity is possible in principle...;
taylor (e**(1/x), x, infinity, 5);
1 1 1 1 1 1 1 1 1 1
1 + --- + ---*---- + ---*---- + ----*---- + -----*---- + O(----)
x 2 2 6 3 24 4 120 5 6
x x x x x
xi := taylor (sin (1/x), x, infinity, 5);
1 1 1 1 1 1
xi := --- - ---*---- + -----*---- + O(----)
x 6 3 120 5 6
x x x
y1 := taylor(x/(x-1), x, infinity, 3);
1 1 1 1
y1 := 1 + --- + ---- + ---- + O(----)
x 2 3 4
x x x
z := df(y1, x);
1 1 1 1
z := - ---- - 2*---- - 3*---- + O(----)
2 3 4 5
x x x x
comment ...but far from being perfect;
taylor (1 / sin (x), x, infinity, 5);
1
taylor(--------,x,infinity,5)
sin(x)
clear z;
comment The template of a Taylor kernel can be extracted;
taylortemplate yy;
{{y,0,4}}
taylortemplate xxa;
{{x,1,2}}
taylortemplate xi;
{{x,infinity,5}}
taylortemplate xy;
{{x,0,2},{y,0,2}}
taylortemplate xx1;
{{{x,y},0,2}}
comment Here is a slightly less trivial example;
exp := (sin (x) * sin (y) / (x * y))**2;
2 2
sin(x) *sin(y)
exp := -----------------
2 2
x *y
taylor (exp, x, 0, 1, y, 0, 1);
2 2
1 + O(x ,y )
taylor (exp, x, 0, 2, y, 0, 2);
1 2 1 2 1 2 2 3 3
1 - ---*x - ---*y + ---*y *x + O(x ,y )
3 3 9
tt := taylor (exp, {x,y}, 0, 2);
1 2 1 2 3
tt := 1 - ---*y - ---*x + O({x,y} )
3 3
comment An example that uses factorization;
on factor;
ff := y**5 - 1;
4 3 2
ff := (y + y + y + y + 1)*(y - 1)
zz := sub (y = taylor(e**x, x, 0, 3), ff);
1 2 1 3 4 4 1 2 1 3 4 3
zz := ((1 + x + ---*x + ---*x + O(x )) + (1 + x + ---*x + ---*x + O(x ))
2 6 2 6
1 2 1 3 4 2 1 2 1 3 4
+ (1 + x + ---*x + ---*x + O(x )) + (1 + x + ---*x + ---*x + O(x ))
2 6 2 6
1 2 1 3 4
+ 1)*((1 + x + ---*x + ---*x + O(x )) - 1)
2 6
on exp;
zz;
1 2 1 3 4 5
(1 + x + ---*x + ---*x + O(x )) - 1
2 6
comment A simple example of Taylor kernel differentiation;
hugo := taylor(e^x,x,0,5);
1 2 1 3 1 4 1 5 6
hugo := 1 + x + ---*x + ---*x + ----*x + -----*x + O(x )
2 6 24 120
df(hugo^2,x);
2 8 3 4 4 5
2 + 4*x + 4*x + ---*x + ---*x + O(x )
3 3
comment The following shows the (limited) capabilities to integrate
Taylor kernels. Only simple cases are supported, otherwise
a warning is printed and the Taylor kernels are converted to
standard representation;
zz := taylor (sin x, x, 0, 5);
1 3 1 5 6
zz := x - ---*x + -----*x + O(x )
6 120
ww := taylor (cos y, y, 0, 5);
1 2 1 4 6
ww := 1 - ---*y + ----*y + O(y )
2 24
int (zz, x);
1 2 1 4 1 6 7
---*x - ----*x + -----*x + O(x )
2 24 720
int (ww, x);
x 2 x 4 6
x - ---*y + ----*y + O(y )
2 24
int (zz + ww, x);
1 2 1 4 1 6 7 x 2 x 4 6
(---*x - ----*x + -----*x + O(x )) + (x - ---*y + ----*y + O(y ))
2 24 720 2 24
comment And here we present Taylor series reversion.
We start with the example given by Knuth for the algorithm;
taylor (t - t**2, t, 0, 5);
2 6
t - t + O(t )
taylorrevert (ws, t, x);
2 3 4 5 6
x + x + 2*x + 5*x + 14*x + O(x )
tan!-series := taylor (tan x, x, 0, 5);
1 3 2 5 6
tan-series := x + ---*x + ----*x + O(x )
3 15
taylorrevert (tan!-series, x, y);
1 3 1 5 6
y - ---*y + ---*y + O(y )
3 5
atan!-series:=taylor (atan y, y, 0, 5);
1 3 1 5 6
atan-series := y - ---*y + ---*y + O(y )
3 5
tmp := taylor (e**x, x, 0, 5);
1 2 1 3 1 4 1 5 6
tmp := 1 + x + ---*x + ---*x + ----*x + -----*x + O(x )
2 6 24 120
taylorrevert (tmp, x, y);
1 2 1 3 1 4 1 5 6
y - 1 - ---*(y - 1) + ---*(y - 1) - ---*(y - 1) + ---*(y - 1) + O((y - 1) )
2 3 4 5
taylor (log y, y, 1, 5);
1 2 1 3 1 4 1 5 6
y - 1 - ---*(y - 1) + ---*(y - 1) - ---*(y - 1) + ---*(y - 1) + O((y - 1) )
2 3 4 5
comment The following example calculates the perturbation expansion
of the root x = 20 of the following polynomial in terms of
EPS, in ROUNDED mode;
poly := for r := 1 : 20 product (x - r);
20 19 18 17 16 15
poly := x - 210*x + 20615*x - 1256850*x + 53327946*x - 1672280820*x
14 13 12
+ 40171771630*x - 756111184500*x + 11310276995381*x
11 10 9
- 135585182899530*x + 1307535010540395*x - 10142299865511450*x
8 7 6
+ 63030812099294896*x - 311333643161390640*x + 1206647803780373360*x
5 4
- 3599979517947607200*x + 8037811822645051776*x
3 2
- 12870931245150988800*x + 13803759753640704000*x
- 8752948036761600000*x + 2432902008176640000
on rounded;
tpoly := taylor (poly, x, 20, 4);
2
tpoly := 1.21649393692e+17*(x - 20) + 4.31564847287e+17*(x - 20)
3 4
+ 6.68609351672e+17*(x - 20) + 6.10115975015e+17*(x - 20)
5
+ O((x - 20) )
taylorrevert (tpoly, x, eps);
2 3
20 + 8.22034512178e-18*eps - 2.39726594662e-34*eps + 1.09290580232e-50*eps
4 5
- 5.97114159465e-67*eps + O(eps )
comment Some more examples using rounded mode;
taylor(sin x/x,x,0,4);
2 4 5
1 - 0.166666666667*x + 0.00833333333333*x + O(x )
taylor(sin x,x,pi/2,4);
2
1 + 6.12303176911e-17*(x - 1.57079632679) - 0.5*(x - 1.57079632679)
3 4
- 1.02050529485e-17*(x - 1.57079632679) + 0.0416666666667*(x - 1.57079632679)
5
+ O((x - 1.57079632679) )
taylor(tan x,x,pi/2,4);
-1
- (x - 1.57079632679) + 0.333333333333*(x - 1.57079632679)
3 5
+ 0.0222222222222*(x - 1.57079632679) + O((x - 1.57079632679) )
off rounded;
comment An example that involves computing limits of type 0/0 if
expansion is done via differentiation;
taylor(sqrt((e^x - 1)/x),x,0,15);
1 5 2 1 3 79 4 3 5 16
1 + ---*x + ----*x + -----*x + -------*x + -------*x + (10 terms) + O(x )
4 96 128 92160 40960
comment An example that involves intermediate non-analytical terms
which cancel entirely;
taylor(x^(5/2)/(log(x+1)*tan(x^(3/2))),x,0,5);
1 1 2 7 3 139 4 67 5 6
1 + ---*x - ----*x - ----*x - -----*x + ------*x + O(x )
2 12 24 720 1440
comment Other examples involving non-analytical terms;
taylor(log(e^x-1),x,0,5);
1 1 2 1 4 5
log(x) + (---*x + ----*x - ------*x + O(x ))
2 24 2880
taylor(e^(1/x)*(e^x-1),x,0,5);
1/x 1 2 1 3 1 4 1 5 6
e *(x + ---*x + ---*x + ----*x + -----*x + O(x ))
2 6 24 120
taylor(log(x)*x^10,x,0,5);
10 11
log(x)*(x + O(x ))
taylor(log(x)*x^10,x,0,11);
10 12
log(x)*(x + O(x ))
taylor(log(x-a)/((a-b)*(a-c)) + log(2(x-b))/((b-c)*(b-a))
+ log(x-c)/((c-a)*(c-b)),x,infinity,2);
log(2) 1 1 1
- ---------------------- - ---*---- + O(----)
2 2 2 3
a*b - a*c - b + b*c x x
ss := (sqrt(x^(2/5) +1) - x^(1/3)-1)/x^(1/3);
2/5 1/3
sqrt(x + 1) - x - 1
ss := ---------------------------
1/3
x
taylor(exp ss,x,0,2);
1 1 1/15 1 2/15 1 1/5 1 4/15 1 1/3
--- + -----*x + -----*x + ------*x + -------*x + --------*x
e 2*e 8*e 48*e 384*e 3840*e
31/15
+ (25 terms) + O(x )
taylor(exp sub(x=x^15,ss),x,0,2);
1 1 1 2 3
--- + -----*x + -----*x + O(x )
e 2*e 8*e
taylor(dilog(x),x,0,4);
1 2 1 3 1 4 5
log(x)*(x + ---*x + ---*x + ---*x + O(x ))
2 3 4
2
pi 1 2 1 3 1 4 5
+ (----- - x - ---*x - ---*x - ----*x + O(x ))
6 4 9 16
taylor(ei(x),x,0,4);
1 2 1 3 1 4 5
log(x) - psi(1) + (x + ---*x + ----*x + ----*x + O(x ))
4 18 96
comment In the following we demonstrate the possibiblity to compute the
expansion of a function which is given by a simple first order
differential equation: the function myexp(x) is exp(-x^2);
operator myexp,myerf;
let {df(myexp(~x),~x) => -2*x*myexp(x), myexp(0) => 1,
df(myerf(~x),~x) => 2/sqrt(pi)*myexp(x), myerf(0) => 0};
taylor(myexp(x),x,0,5);
2 1 4 6
1 - x + ---*x + O(x )
2
taylor(myerf(x),x,0,5);
2*sqrt(pi) 2*sqrt(pi) 3 sqrt(pi) 5 6
------------*x - ------------*x + ----------*x + O(x )
pi 3*pi 5*pi
clear {df(myexp(~x),~x) => -2*x*myexp(x), myexp(0) => 1,
df(myerf(~x),~x) => 2/sqrt(pi)*myexp(x), myerf(0) => 0};
clear myexp,erf;
%%% showtime;
comment There are two special operators, implicit_taylor and
inverse_taylor, to compute the Taylor expansion of implicit
or inverse functions;
implicit_taylor(x^2 + y^2 - 1,x,y,0,1,5);
1 2 1 4 6
1 - ---*x - ---*x + O(x )
2 8
implicit_taylor(x^2 + y^2 - 1,x,y,0,1,20);
1 2 1 4 1 6 5 8 7 10 21
1 - ---*x - ---*x - ----*x - -----*x - -----*x + (5 terms) + O(x )
2 8 16 128 256
implicit_taylor(x+y^3-y,x,y,0,0,8);
3 5 7 9
x + x + 3*x + 12*x + O(x )
implicit_taylor(x+y^3-y,x,y,0,1,5);
1 3 2 1 3 105 4 3 5 6
1 - ---*x - ---*x - ---*x - -----*x - ---*x + O(x )
2 8 2 128 2
implicit_taylor(x+y^3-y,x,y,0,-1,5);
1 3 2 1 3 105 4 3 5 6
- 1 - ---*x + ---*x - ---*x + -----*x - ---*x + O(x )
2 8 2 128 2
implicit_taylor(y*e^y-x,x,y,0,0,5);
2 3 3 8 4 125 5 6
x - x + ---*x - ---*x + -----*x + O(x )
2 3 24
comment This is the function exp(-1/x^2), which has an essential
singularity at the point 0;
implicit_taylor(x^2*log y+1,x,y,0,0,3);
***** Computation of Taylor series of implicit function failed
Input expression non-zero at given point
inverse_taylor(exp(x)-1,x,y,0,8);
1 2 1 3 1 4 1 5 1 6 9
y - ---*y + ---*y - ---*y + ---*y - ---*y + (2 terms) + O(y )
2 3 4 5 6
inverse_taylor(exp(x),x,y,0,5);
1 2 1 3 1 4 1 5 6
y - 1 - ---*(y - 1) + ---*(y - 1) - ---*(y - 1) + ---*(y - 1) + O((y - 1) )
2 3 4 5
inverse_taylor(sqrt(x),x,y,0,5);
2 6
y + O(y )
inverse_taylor(log(1+x),x,y,0,5);
1 2 1 3 1 4 1 5 6
y + ---*y + ---*y + ----*y + -----*y + O(y )
2 6 24 120
inverse_taylor((e^x-e^(-x))/2,x,y,0,5);
1 3 3 5 6
y - ---*y + ----*y + O(y )
6 40
comment In the next two cases the inverse functions have a branch
point, therefore the computation fails;
inverse_taylor((e^x+e^(-x))/2,x,y,0,5);
***** Computation of Taylor series of inverse function failed
inverse_taylor(exp(x^2-1),x,y,0,5);
***** Computation of Taylor series of inverse function failed
inverse_taylor(exp(sqrt(x))-1,x,y,0,5);
2 3 11 4 5 5 6
y - y + ----*y - ---*y + O(y )
12 6
inverse_taylor(x*exp(x),x,y,0,5);
2 3 3 8 4 125 5 6
y - y + ---*y - ---*y + -----*y + O(y )
2 3 24
%%% showtime;
comment An application is the problem posed by Prof. Stanley:
we prove that the finite difference expression below
corresponds to the given derivative expression;
operator diff,a,f,gg;
% We use gg to avoid conflict with high energy
% physics operator.
let diff(~f,~arg) => df(f,arg);
derivative_expression :=
diff(a(x,y)*diff(gg(x,y),x)*diff(gg(x,y),y)*diff(f(x,y),y),x) +
diff(a(x,y)*diff(gg(x,y),x)*diff(gg(x,y),y)*diff(f(x,y),x),y) ;
derivative_expression := 2*a(x,y)*df(f(x,y),x,y)*df(gg(x,y),x)*df(gg(x,y),y)
+ a(x,y)*df(f(x,y),x)*df(gg(x,y),x,y)*df(gg(x,y),y)
+ a(x,y)*df(f(x,y),x)*df(gg(x,y),x)*df(gg(x,y),y,2)
+ a(x,y)*df(f(x,y),y)*df(gg(x,y),x,y)*df(gg(x,y),x)
+ a(x,y)*df(f(x,y),y)*df(gg(x,y),x,2)*df(gg(x,y),y)
+ df(a(x,y),x)*df(f(x,y),y)*df(gg(x,y),x)*df(gg(x,y),y)
+ df(a(x,y),y)*df(f(x,y),x)*df(gg(x,y),x)*df(gg(x,y),y)
finite_difference_expression :=
+a(x+dx,y+dy)*f(x+dx,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2)
+a(x+dx,y)*f(x+dx,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2)
+a(x,y+dy)*f(x+dx,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2)
+a(x,y)*f(x+dx,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x+dx,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x+dx,y)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x,y+dy)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2)
-a(x,y)*f(x,y)*gg(x+dx,y+dy)^2/(32*dx^2*dy^2)
-gg(x,y)*a(x+dx,y+dy)*f(x+dx,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2)
-gg(x,y)*a(x+dx,y)*f(x+dx,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2)
-gg(x,y)*a(x,y+dy)*f(x+dx,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2)
-a(x,y)*gg(x,y)*f(x+dx,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x+dx,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x+dx,y)*gg(x+dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x,y+dy)*gg(x+dx,y+dy)/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*gg(x,y)*gg(x+dx,y+dy)/(16*dx^2*dy^2)
-gg(x+dx,y)^2*a(x+dx,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2)
+gg(x,y+dy)*gg(x+dx,y)*a(x+dx,y+dy)*f(x+dx,y+dy)/(16*dx^2*dy^2)
-gg(x,y+dy)^2*a(x+dx,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x+dx,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2)
-a(x+dx,y)*gg(x+dx,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
-a(x,y+dy)*gg(x+dx,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
-a(x,y)*gg(x+dx,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
+gg(x,y+dy)*a(x+dx,y)*gg(x+dx,y)*f(x+dx,y+dy)/(16*dx^2*dy^2)
+a(x,y+dy)*gg(x,y+dy)*gg(x+dx,y)*f(x+dx,y+dy)/(16*dx^2*dy^2)
+a(x,y)*gg(x,y+dy)*gg(x+dx,y)*f(x+dx,y+dy)/(16*dx^2*dy^2)
-gg(x,y+dy)^2*a(x+dx,y)*f(x+dx,y+dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x+dx,y)*f(x+dx,y+dy)/(32*dx^2*dy^2)
-a(x,y+dy)*gg(x,y+dy)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
-a(x,y)*gg(x,y+dy)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x,y+dy)*f(x+dx,y+dy)/(32*dx^2*dy^2)
+a(x,y)*gg(x,y)^2*f(x+dx,y+dy)/(32*dx^2*dy^2)
+f(x,y)*gg(x+dx,y)^2*a(x+dx,y+dy)/(32*dx^2*dy^2)
-f(x,y)*gg(x,y+dy)*gg(x+dx,y)*a(x+dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y+dy)^2*a(x+dx,y+dy)/(32*dx^2*dy^2)
-f(x,y)*gg(x,y)^2*a(x+dx,y+dy)/(32*dx^2*dy^2)
+a(x+dx,y-dy)*f(x+dx,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2)
+a(x+dx,y)*f(x+dx,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2)
+a(x,y-dy)*f(x+dx,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2)
+a(x,y)*f(x+dx,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x+dx,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x+dx,y)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x,y-dy)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2)
-a(x,y)*f(x,y)*gg(x+dx,y-dy)^2/(32*dx^2*dy^2)
-gg(x,y)*a(x+dx,y-dy)*f(x+dx,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2)
-gg(x,y)*a(x+dx,y)*f(x+dx,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2)
-gg(x,y)*a(x,y-dy)*f(x+dx,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2)
-a(x,y)*gg(x,y)*f(x+dx,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x+dx,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x+dx,y)*gg(x+dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x,y-dy)*gg(x+dx,y-dy)/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*gg(x,y)*gg(x+dx,y-dy)/(16*dx^2*dy^2)
-gg(x+dx,y)^2*a(x+dx,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2)
+gg(x,y-dy)*gg(x+dx,y)*a(x+dx,y-dy)*f(x+dx,y-dy)/(16*dx^2*dy^2)
-gg(x,y-dy)^2*a(x+dx,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x+dx,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2)
-a(x+dx,y)*gg(x+dx,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
-a(x,y-dy)*gg(x+dx,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
-a(x,y)*gg(x+dx,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
+gg(x,y-dy)*a(x+dx,y)*gg(x+dx,y)*f(x+dx,y-dy)/(16*dx^2*dy^2)
+a(x,y-dy)*gg(x,y-dy)*gg(x+dx,y)*f(x+dx,y-dy)/(16*dx^2*dy^2)
+a(x,y)*gg(x,y-dy)*gg(x+dx,y)*f(x+dx,y-dy)/(16*dx^2*dy^2)
-gg(x,y-dy)^2*a(x+dx,y)*f(x+dx,y-dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x+dx,y)*f(x+dx,y-dy)/(32*dx^2*dy^2)
-a(x,y-dy)*gg(x,y-dy)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
-a(x,y)*gg(x,y-dy)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x,y-dy)*f(x+dx,y-dy)/(32*dx^2*dy^2)
+a(x,y)*gg(x,y)^2*f(x+dx,y-dy)/(32*dx^2*dy^2)
+f(x,y)*gg(x+dx,y)^2*a(x+dx,y-dy)/(32*dx^2*dy^2)
-f(x,y)*gg(x,y-dy)*gg(x+dx,y)*a(x+dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y-dy)^2*a(x+dx,y-dy)/(32*dx^2*dy^2)
-f(x,y)*gg(x,y)^2*a(x+dx,y-dy)/(32*dx^2*dy^2)
+f(x,y)*a(x+dx,y)*gg(x+dx,y)^2/(16*dx^2*dy^2)
+f(x,y)*a(x,y+dy)*gg(x+dx,y)^2/(32*dx^2*dy^2)
+f(x,y)*a(x,y-dy)*gg(x+dx,y)^2/(32*dx^2*dy^2)
+a(x,y)*f(x,y)*gg(x+dx,y)^2/(16*dx^2*dy^2)
-f(x,y)*gg(x,y+dy)*a(x+dx,y)*gg(x+dx,y)/(16*dx^2*dy^2)
-f(x,y)*gg(x,y-dy)*a(x+dx,y)*gg(x+dx,y)/(16*dx^2*dy^2)
-f(x,y)*a(x,y+dy)*gg(x,y+dy)*gg(x+dx,y)/(16*dx^2*dy^2)
-a(x,y)*f(x,y)*gg(x,y+dy)*gg(x+dx,y)/(16*dx^2*dy^2)
-f(x,y)*a(x,y-dy)*gg(x,y-dy)*gg(x+dx,y)/(16*dx^2*dy^2)
-a(x,y)*f(x,y)*gg(x,y-dy)*gg(x+dx,y)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y+dy)^2*a(x+dx,y)/(32*dx^2*dy^2)
+f(x,y)*gg(x,y-dy)^2*a(x+dx,y)/(32*dx^2*dy^2)
-f(x,y)*gg(x,y)^2*a(x+dx,y)/(16*dx^2*dy^2)
+a(x-dx,y+dy)*f(x-dx,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2)
+a(x-dx,y)*f(x-dx,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2)
+a(x,y+dy)*f(x-dx,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2)
+a(x,y)*f(x-dx,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x-dx,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x-dx,y)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x,y+dy)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2)
-a(x,y)*f(x,y)*gg(x-dx,y+dy)^2/(32*dx^2*dy^2)
-gg(x,y)*a(x-dx,y+dy)*f(x-dx,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2)
-gg(x,y)*a(x-dx,y)*f(x-dx,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2)
-gg(x,y)*a(x,y+dy)*f(x-dx,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2)
-a(x,y)*gg(x,y)*f(x-dx,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x-dx,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x-dx,y)*gg(x-dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x,y+dy)*gg(x-dx,y+dy)/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*gg(x,y)*gg(x-dx,y+dy)/(16*dx^2*dy^2)
-gg(x-dx,y)^2*a(x-dx,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2)
+gg(x,y+dy)*gg(x-dx,y)*a(x-dx,y+dy)*f(x-dx,y+dy)/(16*dx^2*dy^2)
-gg(x,y+dy)^2*a(x-dx,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x-dx,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2)
-a(x-dx,y)*gg(x-dx,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
-a(x,y+dy)*gg(x-dx,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
-a(x,y)*gg(x-dx,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
+gg(x,y+dy)*a(x-dx,y)*gg(x-dx,y)*f(x-dx,y+dy)/(16*dx^2*dy^2)
+a(x,y+dy)*gg(x,y+dy)*gg(x-dx,y)*f(x-dx,y+dy)/(16*dx^2*dy^2)
+a(x,y)*gg(x,y+dy)*gg(x-dx,y)*f(x-dx,y+dy)/(16*dx^2*dy^2)
-gg(x,y+dy)^2*a(x-dx,y)*f(x-dx,y+dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x-dx,y)*f(x-dx,y+dy)/(32*dx^2*dy^2)
-a(x,y+dy)*gg(x,y+dy)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
-a(x,y)*gg(x,y+dy)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x,y+dy)*f(x-dx,y+dy)/(32*dx^2*dy^2)
+a(x,y)*gg(x,y)^2*f(x-dx,y+dy)/(32*dx^2*dy^2)
+f(x,y)*gg(x-dx,y)^2*a(x-dx,y+dy)/(32*dx^2*dy^2)
-f(x,y)*gg(x,y+dy)*gg(x-dx,y)*a(x-dx,y+dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y+dy)^2*a(x-dx,y+dy)/(32*dx^2*dy^2)
-f(x,y)*gg(x,y)^2*a(x-dx,y+dy)/(32*dx^2*dy^2)
+a(x-dx,y-dy)*f(x-dx,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2)
+a(x-dx,y)*f(x-dx,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2)
+a(x,y-dy)*f(x-dx,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2)
+a(x,y)*f(x-dx,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x-dx,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x-dx,y)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2)
-f(x,y)*a(x,y-dy)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2)
-a(x,y)*f(x,y)*gg(x-dx,y-dy)^2/(32*dx^2*dy^2)
-gg(x,y)*a(x-dx,y-dy)*f(x-dx,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2)
-gg(x,y)*a(x-dx,y)*f(x-dx,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2)
-gg(x,y)*a(x,y-dy)*f(x-dx,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2)
-a(x,y)*gg(x,y)*f(x-dx,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x-dx,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x-dx,y)*gg(x-dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y)*a(x,y-dy)*gg(x-dx,y-dy)/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*gg(x,y)*gg(x-dx,y-dy)/(16*dx^2*dy^2)
-gg(x-dx,y)^2*a(x-dx,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2)
+gg(x,y-dy)*gg(x-dx,y)*a(x-dx,y-dy)*f(x-dx,y-dy)/(16*dx^2*dy^2)
-gg(x,y-dy)^2*a(x-dx,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x-dx,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2)
-a(x-dx,y)*gg(x-dx,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
-a(x,y-dy)*gg(x-dx,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
-a(x,y)*gg(x-dx,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
+gg(x,y-dy)*a(x-dx,y)*gg(x-dx,y)*f(x-dx,y-dy)/(16*dx^2*dy^2)
+a(x,y-dy)*gg(x,y-dy)*gg(x-dx,y)*f(x-dx,y-dy)/(16*dx^2*dy^2)
+a(x,y)*gg(x,y-dy)*gg(x-dx,y)*f(x-dx,y-dy)/(16*dx^2*dy^2)
-gg(x,y-dy)^2*a(x-dx,y)*f(x-dx,y-dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x-dx,y)*f(x-dx,y-dy)/(32*dx^2*dy^2)
-a(x,y-dy)*gg(x,y-dy)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
-a(x,y)*gg(x,y-dy)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
+gg(x,y)^2*a(x,y-dy)*f(x-dx,y-dy)/(32*dx^2*dy^2)
+a(x,y)*gg(x,y)^2*f(x-dx,y-dy)/(32*dx^2*dy^2)
+f(x,y)*gg(x-dx,y)^2*a(x-dx,y-dy)/(32*dx^2*dy^2)
-f(x,y)*gg(x,y-dy)*gg(x-dx,y)*a(x-dx,y-dy)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y-dy)^2*a(x-dx,y-dy)/(32*dx^2*dy^2)
-f(x,y)*gg(x,y)^2*a(x-dx,y-dy)/(32*dx^2*dy^2)
+f(x,y)*a(x-dx,y)*gg(x-dx,y)^2/(16*dx^2*dy^2)
+f(x,y)*a(x,y+dy)*gg(x-dx,y)^2/(32*dx^2*dy^2)
+f(x,y)*a(x,y-dy)*gg(x-dx,y)^2/(32*dx^2*dy^2)
+a(x,y)*f(x,y)*gg(x-dx,y)^2/(16*dx^2*dy^2)
-f(x,y)*gg(x,y+dy)*a(x-dx,y)*gg(x-dx,y)/(16*dx^2*dy^2)
-f(x,y)*gg(x,y-dy)*a(x-dx,y)*gg(x-dx,y)/(16*dx^2*dy^2)
-f(x,y)*a(x,y+dy)*gg(x,y+dy)*gg(x-dx,y)/(16*dx^2*dy^2)
-a(x,y)*f(x,y)*gg(x,y+dy)*gg(x-dx,y)/(16*dx^2*dy^2)
-f(x,y)*a(x,y-dy)*gg(x,y-dy)*gg(x-dx,y)/(16*dx^2*dy^2)
-a(x,y)*f(x,y)*gg(x,y-dy)*gg(x-dx,y)/(16*dx^2*dy^2)
+f(x,y)*gg(x,y+dy)^2*a(x-dx,y)/(32*dx^2*dy^2)
+f(x,y)*gg(x,y-dy)^2*a(x-dx,y)/(32*dx^2*dy^2)
-f(x,y)*gg(x,y)^2*a(x-dx,y)/(16*dx^2*dy^2)
+f(x,y)*a(x,y+dy)*gg(x,y+dy)^2/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*gg(x,y+dy)^2/(16*dx^2*dy^2)
-f(x,y)*gg(x,y)^2*a(x,y+dy)/(16*dx^2*dy^2)
+f(x,y)*a(x,y-dy)*gg(x,y-dy)^2/(16*dx^2*dy^2)
+a(x,y)*f(x,y)*gg(x,y-dy)^2/(16*dx^2*dy^2)
-f(x,y)*gg(x,y)^2*a(x,y-dy)/(16*dx^2*dy^2)
-a(x,y)*f(x,y)*gg(x,y)^2/(8*dx^2*dy^2)$
comment We define abbreviations for the partial derivatives;
operator ax,ay,fx,fy,gx,gy;
operator axx,axy,ayy,fxx,fxy,fyy,gxx,gxy,gyy;
operator axxx,axxy,axyy,ayyy,fxxx,fxxy,fxyy,fyyy,gxxx,gxxy,gxyy,gyyy;
operator axxxy,axxyy,axyyy,fxxxy,fxxyy,fxyyy,
gxxxx,gxxxy,gxxyy,gxyyy,gyyyy;
operator axxxyy,axxyyy,fxxyyy,fxxxyy,gxxxxy,gxxxyy,gxxyyy,gxyyyy;
operator gxxxxyy,gxxxyyy,gxxyyyy;
operator_diff_rules := {
df(a(~x,~y),~x) => ax(x,y),
df(a(~x,~y),~y) => ay(x,y),
df(f(~x,~y),~x) => fx(x,y),
df(f(~x,~y),~y) => fy(x,y),
df(gg(~x,~y),~x) => gx(x,y),
df(gg(~x,~y),~y) => gy(x,y),
df(ax(~x,~y),~x) => axx(x,y),
df(ax(~x,~y),~y) => axy(x,y),
df(ay(~x,~y),~x) => axy(x,y),
df(ay(~x,~y),~y) => ayy(x,y),
df(fx(~x,~y),~x) => fxx(x,y),
df(fx(~x,~y),~y) => fxy(x,y),
df(fy(~x,~y),~x) => fxy(x,y),
df(fy(~x,~y),~y) => fyy(x,y),
df(gx(~x,~y),~x) => gxx(x,y),
df(gx(~x,~y),~y) => gxy(x,y),
df(gy(~x,~y),~x) => gxy(x,y),
df(gy(~x,~y),~y) => gyy(x,y),
df(axx(~x,~y),~x) => axxx(x,y),
df(axy(~x,~y),~x) => axxy(x,y),
df(ayy(~x,~y),~x) => axyy(x,y),
df(ayy(~x,~y),~y) => ayyy(x,y),
df(fxx(~x,~y),~x) => fxxx(x,y),
df(fxy(~x,~y),~x) => fxxy(x,y),
df(fxy(~x,~y),~y) => fxyy(x,y),
df(fyy(~x,~y),~x) => fxyy(x,y),
df(fyy(~x,~y),~y) => fyyy(x,y),
df(gxx(~x,~y),~x) => gxxx(x,y),
df(gxx(~x,~y),~y) => gxxy(x,y),
df(gxy(~x,~y),~x) => gxxy(x,y),
df(gxy(~x,~y),~y) => gxyy(x,y),
df(gyy(~x,~y),~x) => gxyy(x,y),
df(gyy(~x,~y),~y) => gyyy(x,y),
df(axyy(~x,~y),~x) => axxyy(x,y),
df(axxy(~x,~y),~x) => axxxy(x,y),
df(ayyy(~x,~y),~x) => axyyy(x,y),
df(fxxy(~x,~y),~x) => fxxxy(x,y),
df(fxyy(~x,~y),~x) => fxxyy(x,y),
df(fyyy(~x,~y),~x) => fxyyy(x,y),
df(gxxx(~x,~y),~x) => gxxxx(x,y),
df(gxxy(~x,~y),~x) => gxxxy(x,y),
df(gxyy(~x,~y),~x) => gxxyy(x,y),
df(gyyy(~x,~y),~x) => gxyyy(x,y),
df(gyyy(~x,~y),~y) => gyyyy(x,y),
df(axxyy(~x,~y),~x) => axxxyy(x,y),
df(axyyy(~x,~y),~x) => axxyyy(x,y),
df(fxxyy(~x,~y),~x) => fxxxyy(x,y),
df(fxyyy(~x,~y),~x) => fxxyyy(x,y),
df(gxxxy(~x,~y),~x) => gxxxxy(x,y),
df(gxxyy(~x,~y),~x) => gxxxyy(x,y),
df(gxyyy(~x,~y),~x) => gxxyyy(x,y),
df(gyyyy(~x,~y),~x) => gxyyyy(x,y),
df(gxxxyy(~x,~y),~x) => gxxxxyy(x,y),
df(gxxyyy(~x,~y),~x) => gxxxyyy(x,y),
df(gxyyyy(~x,~y),~x) => gxxyyyy(x,y)
};
operator_diff_rules := {df(a(~x,~y),~x) => ax(x,y),
df(a(~x,~y),~y) => ay(x,y),
df(f(~x,~y),~x) => fx(x,y),
df(f(~x,~y),~y) => fy(x,y),
df(gg(~x,~y),~x) => gx(x,y),
df(gg(~x,~y),~y) => gy(x,y),
df(ax(~x,~y),~x) => axx(x,y),
df(ax(~x,~y),~y) => axy(x,y),
df(ay(~x,~y),~x) => axy(x,y),
df(ay(~x,~y),~y) => ayy(x,y),
df(fx(~x,~y),~x) => fxx(x,y),
df(fx(~x,~y),~y) => fxy(x,y),
df(fy(~x,~y),~x) => fxy(x,y),
df(fy(~x,~y),~y) => fyy(x,y),
df(gx(~x,~y),~x) => gxx(x,y),
df(gx(~x,~y),~y) => gxy(x,y),
df(gy(~x,~y),~x) => gxy(x,y),
df(gy(~x,~y),~y) => gyy(x,y),
df(axx(~x,~y),~x) => axxx(x,y),
df(axy(~x,~y),~x) => axxy(x,y),
df(ayy(~x,~y),~x) => axyy(x,y),
df(ayy(~x,~y),~y) => ayyy(x,y),
df(fxx(~x,~y),~x) => fxxx(x,y),
df(fxy(~x,~y),~x) => fxxy(x,y),
df(fxy(~x,~y),~y) => fxyy(x,y),
df(fyy(~x,~y),~x) => fxyy(x,y),
df(fyy(~x,~y),~y) => fyyy(x,y),
df(gxx(~x,~y),~x) => gxxx(x,y),
df(gxx(~x,~y),~y) => gxxy(x,y),
df(gxy(~x,~y),~x) => gxxy(x,y),
df(gxy(~x,~y),~y) => gxyy(x,y),
df(gyy(~x,~y),~x) => gxyy(x,y),
df(gyy(~x,~y),~y) => gyyy(x,y),
df(axyy(~x,~y),~x) => axxyy(x,y),
df(axxy(~x,~y),~x) => axxxy(x,y),
df(ayyy(~x,~y),~x) => axyyy(x,y),
df(fxxy(~x,~y),~x) => fxxxy(x,y),
df(fxyy(~x,~y),~x) => fxxyy(x,y),
df(fyyy(~x,~y),~x) => fxyyy(x,y),
df(gxxx(~x,~y),~x) => gxxxx(x,y),
df(gxxy(~x,~y),~x) => gxxxy(x,y),
df(gxyy(~x,~y),~x) => gxxyy(x,y),
df(gyyy(~x,~y),~x) => gxyyy(x,y),
df(gyyy(~x,~y),~y) => gyyyy(x,y),
df(axxyy(~x,~y),~x) => axxxyy(x,y),
df(axyyy(~x,~y),~x) => axxyyy(x,y),
df(fxxyy(~x,~y),~x) => fxxxyy(x,y),
df(fxyyy(~x,~y),~x) => fxxyyy(x,y),
df(gxxxy(~x,~y),~x) => gxxxxy(x,y),
df(gxxyy(~x,~y),~x) => gxxxyy(x,y),
df(gxyyy(~x,~y),~x) => gxxyyy(x,y),
df(gyyyy(~x,~y),~x) => gxyyyy(x,y),
df(gxxxyy(~x,~y),~x) => gxxxxyy(x,y),
df(gxxyyy(~x,~y),~x) => gxxxyyy(x,y),
df(gxyyyy(~x,~y),~x) => gxxyyyy(x,y)}
let operator_diff_rules;
texp := taylor (finite_difference_expression, dx, 0, 1, dy, 0, 1);
texp := a(x,y)*fx(x,y)*gx(x,y)*gyy(x,y) + a(x,y)*fx(x,y)*gxy(x,y)*gy(x,y)
+ 2*a(x,y)*fxy(x,y)*gx(x,y)*gy(x,y) + a(x,y)*fy(x,y)*gx(x,y)*gxy(x,y)
+ a(x,y)*fy(x,y)*gxx(x,y)*gy(x,y) + ax(x,y)*fy(x,y)*gx(x,y)*gy(x,y)
2 2
+ ay(x,y)*fx(x,y)*gx(x,y)*gy(x,y) + O(dx ,dy )
comment You may also try to expand further but this needs a lot
of CPU time. Therefore the following line is commented out;
%texp := taylor (finite_difference_expression, dx, 0, 2, dy, 0, 2);
factor dx,dy;
result := taylortostandard texp;
result := a(x,y)*fx(x,y)*gx(x,y)*gyy(x,y) + a(x,y)*fx(x,y)*gxy(x,y)*gy(x,y)
+ 2*a(x,y)*fxy(x,y)*gx(x,y)*gy(x,y) + a(x,y)*fy(x,y)*gx(x,y)*gxy(x,y)
+ a(x,y)*fy(x,y)*gxx(x,y)*gy(x,y) + ax(x,y)*fy(x,y)*gx(x,y)*gy(x,y)
+ ay(x,y)*fx(x,y)*gx(x,y)*gy(x,y)
derivative_expression - result;
0
clear diff(~f,~arg);
clearrules operator_diff_rules;
clear diff,a,f,gg;
clear ax,ay,fx,fy,gx,gy;
clear axx,axy,ayy,fxx,fxy,fyy,gxx,gxy,gyy;
clear axxx,axxy,axyy,ayyy,fxxx,fxxy,fxyy,fyyy,gxxx,gxxy,gxyy,gyyy;
clear axxxy,axxyy,axyyy,fxxxy,fxxyy,fxyyy,gxxxx,gxxxy,gxxyy,gxyyy,gyyyy;
clear axxxyy,axxyyy,fxxyyy,fxxxyy,gxxxxy,gxxxyy,gxxyyy,gxyyyy;
clear gxxxxyy,gxxxyyy,gxxyyyy;
taylorprintterms := 5;
taylorprintterms := 5
off taylorautoexpand,taylorkeeporiginal;
%%% showtime;
comment An example provided by Alan Barnes: elliptic functions;
% Jacobi's elliptic functions
% sn(x,k), cn(x,k), dn(x,k).
% The modulus and complementary modulus are denoted by K and K!'
% usually written mathematically as k and k' respectively
%
% epsilon(x,k) is the incomplete elliptic integral of the second kind
% usually written mathematically as E(x,k)
%
% KK(k) is the complete elliptic integral of the first kind
% usually written mathematically as K(k)
% K(k) = arcsn(1,k)
% KK!'(k) is the complementary complete integral
% usually written mathematically as K'(k)
% NB. K'(k) = K(k')
% EE(k) is the complete elliptic integral of the second kind
% usually written mathematically as E(k)
% EE!'(k) is the complementary complete integral
% usually written mathematically as E'(k)
% NB. E'(k) = E(k')
operator sn, cn, dn, epsilon;
elliptic_rules := {
% Differentiation rules for basic functions
df(sn(~x,~k),~x) => cn(x,k)*dn(x,k),
df(cn(~x,~k),~x) => -sn(x,k)*dn(x,k),
df(dn(~x,~k),~x) => -k^2*sn(x,k)*cn(x,k),
df(epsilon(~x,~k),~x)=> dn(x,k)^2,
% k-derivatives
% DF Lawden Elliptic Functions & Applications Springer (1989)
df(sn(~x,~k),~k) => (k*sn(x,k)*cn(x,k)^2
-epsilon(x,k)*cn(x,k)*dn(x,k)/k)/(1-k^2)
+ x*cn(x,k)*dn(x,k)/k,
df(cn(~x,~k),~k) => (-k*sn(x,k)^2*cn(x,k)
+epsilon(x,k)*sn(x,k)*dn(x,k)/k)/(1-k^2)
- x*sn(x,k)*dn(x,k)/k,
df(dn(~x,~k),~k) => k*(-sn(x,k)^2*dn(x,k)
+epsilon(x,k)*sn(x,k)*cn(x,k))/(1-k^2)
- k*x*sn(x,k)*cn(x,k),
df(epsilon(~x,~k),~k) => k*(sn(x,k)*cn(x,k)*dn(x,k)
-epsilon(x,k)*cn(x,k)^2)/(1-k^2)
-k*x*sn(x,k)^2,
% parity properties
sn(-~x,~k) => -sn(x,k),
cn(-~x,~k) => cn(x,k),
dn(-~x,~k) => dn(x,k),
epsilon(-~x,~k) => -epsilon(x,k),
sn(~x,-~k) => sn(x,k),
cn(~x,-~k) => cn(x,k),
dn(~x,-~k) => dn(x,k),
epsilon(~x,-~k) => epsilon(x,k),
% behaviour at zero
sn(0,~k) => 0,
cn(0,~k) => 1,
dn(0,~k) => 1,
epsilon(0,~k) => 0,
% degenerate cases of modulus
sn(~x,0) => sin(x),
cn(~x,0) => cos(x),
dn(~x,0) => 1,
epsilon(~x,0) => x,
sn(~x,1) => tanh(x),
cn(~x,1) => 1/cosh(x),
dn(~x,1) => 1/cosh(x),
epsilon(~x,1) => tanh(x)
};
elliptic_rules := {df(sn(~x,~k),~x) => cn(x,k)*dn(x,k),
df(cn(~x,~k),~x) => - sn(x,k)*dn(x,k),
2
df(dn(~x,~k),~x) => - k *sn(x,k)*cn(x,k),
2
df(epsilon(~x,~k),~x) => dn(x,k) ,
2 dn(x,k)
k*sn(x,k)*cn(x,k) - epsilon(x,k)*cn(x,k)*---------
k
df(sn(~x,~k),~k) => -----------------------------------------------------
2
1 - k
dn(x,k)
+ x*cn(x,k)*---------,
k
2 dn(x,k)
- k*sn(x,k) *cn(x,k) + epsilon(x,k)*sn(x,k)*---------
k
df(cn(~x,~k),~k) => --------------------------------------------------------
2
1 - k
dn(x,k)
- x*sn(x,k)*---------,
k
2
- sn(x,k) *dn(x,k) + epsilon(x,k)*sn(x,k)*cn(x,k)
df(dn(~x,~k),~k) => k*----------------------------------------------------
2
1 - k
- k*x*sn(x,k)*cn(x,k),
df(epsilon(~x,~k),~k)
2
sn(x,k)*cn(x,k)*dn(x,k) - epsilon(x,k)*cn(x,k) 2
=> k*------------------------------------------------- - k*x*sn(x,k) ,
2
1 - k
sn( - ~x,~k) => - sn(x,k),
cn( - ~x,~k) => cn(x,k),
dn( - ~x,~k) => dn(x,k),
epsilon( - ~x,~k) => - epsilon(x,k),
sn(~x, - ~k) => sn(x,k),
cn(~x, - ~k) => cn(x,k),
dn(~x, - ~k) => dn(x,k),
epsilon(~x, - ~k) => epsilon(x,k),
sn(0,~k) => 0,
cn(0,~k) => 1,
dn(0,~k) => 1,
epsilon(0,~k) => 0,
sn(~x,0) => sin(x),
cn(~x,0) => cos(x),
dn(~x,0) => 1,
epsilon(~x,0) => x,
sn(~x,1) => tanh(x),
1
cn(~x,1) => ---------,
cosh(x)
1
dn(~x,1) => ---------,
cosh(x)
epsilon(~x,1) => tanh(x)}
let elliptic_rules;
hugo := taylor(sn(x,k),k,0,6);
2 2
cos(x)*(cos(x) *x + cos(x)*sin(x) + sin(x) *x - 2*x) 2
hugo := sin(x) + ------------------------------------------------------*k + (
4
5 4 2 4
cos(x) *x - 2*cos(x) *sin(x)*x + 5*cos(x) *sin(x)
3 2 3 2 3 2
- 10*cos(x) *sin(x) *x + 6*cos(x) *x - 4*cos(x) *sin(x) *x
2 3 2 2 2
+ cos(x) *sin(x) + 8*cos(x) *sin(x)*x + 4*cos(x) *sin(x)
4 2
- 11*cos(x)*sin(x) *x + 30*cos(x)*sin(x) *x - 16*cos(x)*x
5 2 3 2 2 4
- 2*sin(x) *x + 8*sin(x) *x - 8*sin(x)*x )/64*k + (
7 3 7 6 2
- 6*cos(x) *x + 17*cos(x) *x - 99*cos(x) *sin(x)*x
6 5 2 3 5 2
+ 21*cos(x) *sin(x) - 18*cos(x) *sin(x) *x - 71*cos(x) *sin(x) *x
5 3 5 4 3 2
+ 36*cos(x) *x - 18*cos(x) *x - 135*cos(x) *sin(x) *x
4 3 4 2 4
- 133*cos(x) *sin(x) + 324*cos(x) *sin(x)*x + 172*cos(x) *sin(x)
3 4 3 3 4
- 18*cos(x) *sin(x) *x - 13*cos(x) *sin(x) *x
3 2 3 3 2 3 3
+ 72*cos(x) *sin(x) *x - 156*cos(x) *sin(x) *x - 72*cos(x) *x
3 2 5 2 2 5
+ 160*cos(x) *x + 27*cos(x) *sin(x) *x - 118*cos(x) *sin(x)
2 3 2 2 2
+ 176*cos(x) *sin(x) - 108*cos(x) *sin(x)*x + 32*cos(x) *sin(x)
6 3 6 4 3
- 6*cos(x)*sin(x) *x + 75*cos(x)*sin(x) *x + 36*cos(x)*sin(x) *x
4 2 3 2
- 498*cos(x)*sin(x) *x - 72*cos(x)*sin(x) *x + 888*cos(x)*sin(x) *x
3 7 2 5 2
+ 48*cos(x)*x - 384*cos(x)*x + 63*sin(x) *x - 324*sin(x) *x
3 2 2 6 7
+ 540*sin(x) *x - 288*sin(x)*x )/2304*k + O(k )
otto := taylor(cn(x,k),k,0,6);
2 2
sin(x)*( - cos(x) *x - cos(x)*sin(x) - sin(x) *x + 2*x) 2
otto := cos(x) + ---------------------------------------------------------*k +
4
5 2 4 3 2 2
( - 2*cos(x) *x - 5*cos(x) *sin(x)*x - 4*cos(x) *sin(x) *x
3 2 3 2 2 3
- 7*cos(x) *sin(x) + 8*cos(x) *x + 2*cos(x) *sin(x) *x
2 4 2 4
+ 2*cos(x) *sin(x)*x - 2*cos(x)*sin(x) *x - 3*cos(x)*sin(x)
2 2 2 2 5
+ 8*cos(x)*sin(x) *x - 4*cos(x)*sin(x) - 8*cos(x)*x + 7*sin(x) *x
3 4 7 2
- 22*sin(x) *x + 16*sin(x)*x)/64*k + ( - 9*cos(x) *x
6 3 6 5 2 2
+ 6*cos(x) *sin(x)*x - 71*cos(x) *sin(x)*x + 135*cos(x) *sin(x) *x
5 2 5 2 4 3 3
- 66*cos(x) *sin(x) - 36*cos(x) *x + 18*cos(x) *sin(x) *x
4 3 4 3 4
- cos(x) *sin(x) *x - 36*cos(x) *sin(x)*x + 18*cos(x) *sin(x)*x
3 4 2 3 4
+ 297*cos(x) *sin(x) *x + 61*cos(x) *sin(x)
3 2 2 3 2 3 2
- 720*cos(x) *sin(x) *x - 208*cos(x) *sin(x) + 252*cos(x) *x
2 5 3 2 5
+ 18*cos(x) *sin(x) *x + 31*cos(x) *sin(x) *x
2 3 3 2 3
- 72*cos(x) *sin(x) *x - 24*cos(x) *sin(x) *x
2 3 2 6 2
+ 72*cos(x) *sin(x)*x + 56*cos(x) *sin(x)*x + 153*cos(x)*sin(x) *x
6 4 2 4
+ 91*cos(x)*sin(x) - 684*cos(x)*sin(x) *x - 212*cos(x)*sin(x)
2 2 2 2
+ 900*cos(x)*sin(x) *x - 32*cos(x)*sin(x) - 288*cos(x)*x
7 3 7 5 3 5
+ 6*sin(x) *x - 39*sin(x) *x - 36*sin(x) *x + 318*sin(x) *x
3 3 3 3
+ 72*sin(x) *x - 672*sin(x) *x - 48*sin(x)*x + 384*sin(x)*x)/2304
6 7
*k + O(k )
taylorcombine(hugo^2 + otto^2);
2 2 7
cos(x) + sin(x) + O(k )
clearrules elliptic_rules;
clear sn, cn, dn, epsilon;
%%% showtime;
comment That's all, folks;
end;
(TIME: taylor 14199 15009)