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last edited 5 years ago by hemmecke |
Edit detail for TaylorSeries revision 2 of 8
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Editor: Ralf Hemmecke
Time: 2014/08/23 23:02:42 GMT+0 |
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changed: -M==>SparseMultivariateTaylorSeries(Q,V,P) M==>TaylorSeries(Q,V,P)
Example for multivariate Taylor series expansion
In order to work with multivariate Taylor series one first has to do a few preparation steps in order to create an appropriate domain.
fricas
(1) -> Z==>Integer
Type: Void
fricas
Q==>Fraction Z
Type: Void
fricas
vl: List Symbol := [x,y]
![\label{eq1}\left[ x , \: y \right]](images/3908585630146559634-16.0px.png "\label{eq1}\left[ x , \: y \right]") | (1) |
Type: List(Symbol)
fricas
V==>OrderedVariableList vl
Type: Void
fricas
E ==> DirectProduct(2,NonNegativeInteger)
Type: Void
fricas
P==>SparseMultivariatePolynomial(Q,V)
Type: Void
fricas
M==>TaylorSeries(Q,V, P)
Type: Void
fricas
X:=monomial(1$M,x, 1)
The constructor TaylorSeries takes 1 argument and you have given 3 .
Naive solution --Bill Page, Fri, 22 Aug 2014 21:19:19 +0000 reply
This is not what I expected:
fricas
x:=taylor 'x
 | (2) |
fricas
y:=taylor 'y
 | (3) |
fricas
sinh(x)*cosh(y)
![\label{eq4}\begin{array}{@{}l}
\displaystyle
x +{{\frac{1}{6}}\ {{x}^{3}}}+{{\frac{1}{120}}\ {{x}^{5}}}+{{\frac{1}{5
040}}\ {{x}^{7}}}+
\
\
\displaystyle
{{\frac{1}{362880}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}+ \
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{2}}\ x}+{{\frac{1}{12}}\ {{x}^{3}}}+{{\frac{1}{24
0}}\ {{x}^{5}}}+
\
\
\displaystyle
{{\frac{1}{10080}}\ {{x}^{7}}}+{{\frac{1}{725760}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}](images/5232264141317905180-16.0px.png "\label{eq4}\begin{array}{@{}l}
\displaystyle
x +{{\frac{1}{6}}\ {{x}^{3}}}+{{\frac{1}{120}}\ {{x}^{5}}}+{{\frac{1}{5
040}}\ {{x}^{7}}}+
\
\
\displaystyle
{{\frac{1}{362880}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}+ \
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{2}}\ x}+{{\frac{1}{12}}\ {{x}^{3}}}+{{\frac{1}{24
0}}\ {{x}^{5}}}+
\
\
\displaystyle
{{\frac{1}{10080}}\ {{x}^{7}}}+{{\frac{1}{725760}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}") | (4) |
Can it be converted somehow to the solution above?
Most simple solution
There is a domain in FriCAS? that is similar to the Polynomial(Q) domain.
Then the input is as simple as above.
fricas
T==>TaylorSeries Fraction Integer
Type: Void
fricas
xt:T := 'x
 | (5) |
Type: TaylorSeries?(Fraction(Integer))
fricas
yt:T := 'y
 | (6) |
Type: TaylorSeries?(Fraction(Integer))
fricas
sinh(xt)*cosh(yt)
![\label{eq7}\begin{array}{@{}l}
\displaystyle
x +{\left({{\frac{1}{2}}\ x \ {{y}^{2}}}+{{\frac{1}{6}}\ {{x}^{3}}}\right)}+
\
\
\displaystyle
{\left({{\frac{1}{24}}\ x \ {{y}^{4}}}+{{\frac{1}{12}}\ {{x}^{3}}\ {{y}^{2}}}+{{\frac{1}{120}}\ {{x}^{5}}}\right)}+
\
\
\displaystyle
{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{720}}\ x \ {{y}^{6}}}+{{\frac{1}{144}}\ {{x}^{3}}\ {{y}^{4}}}+{{\frac{1}{240}}\ {{x}^{5}}\ {{y}^{2}}}+
\
\
\displaystyle
{{\frac{1}{5040}}\ {{x}^{7}}}](images/4651981819716650341-16.0px.png "\label{eq7}\begin{array}{@{}l}
\displaystyle
x +{\left({{\frac{1}{2}}\ x \ {{y}^{2}}}+{{\frac{1}{6}}\ {{x}^{3}}}\right)}+
\
\
\displaystyle
{\left({{\frac{1}{24}}\ x \ {{y}^{4}}}+{{\frac{1}{12}}\ {{x}^{3}}\ {{y}^{2}}}+{{\frac{1}{120}}\ {{x}^{5}}}\right)}+
\
\
\displaystyle
{\left({
\begin{array}{@{}l}
\displaystyle
{{\frac{1}{720}}\ x \ {{y}^{6}}}+{{\frac{1}{144}}\ {{x}^{3}}\ {{y}^{4}}}+{{\frac{1}{240}}\ {{x}^{5}}\ {{y}^{2}}}+
\
\
\displaystyle
{{\frac{1}{5040}}\ {{x}^{7}}}") | (7) |
Type: TaylorSeries?(Fraction(Integer))