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TaylorSeries

Edit detail for TaylorSeries revision 3 of 8

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Editor: hemmecke Time: 2014/08/23 23:11:12 GMT+0
Note:

added:

  If you are not interested in a specific domain of multivariate
  power series you can find a simpler solution below using a domain of
  power series in infinitely many variables.

changed:
-  There is a domain in FriCAS that is similar to the 'Polynomial(Q)' domain.
-  Then the input is as simple as above.
  There is a domain in FriCAS that is similar to the 'Polynomial(Q)' domain,
  i.e. 'TaylorSeries(Q)' is the domain of power series over 'Q' in infinitely many variables.

  With that domain the input is as simple as above.

changed:
-xt:T := 'x
-yt:T := 'y
-sinh(xt)*cosh(yt)
xx:T := 'x
yy:T := 'y
sinh(xx)*cosh(yy)

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.

If you are not interested in a specific domain of multivariate power series you can find a simpler solution below using a domain of power series in infinitely many variables.

fricas

Z==>Integer

Type: Void

fricas

Q==>Fraction Z

Type: Void

fricas

vl: List Symbol := [x,y]

$\label{eq1}\left[ x , \: y \right]$

(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 
      .
Y:=monomial(1$M,y,1)

   The constructor TaylorSeries takes 1 argument and you have given  3 
      .
sinh(X)*cosh(Y)

$\label{eq2}{\cosh \left({Y}\right)}\ {\sinh \left({X}\right)}$

(2)

Type: Expression(Integer)

Naive solution --Bill Page, Fri, 22 Aug 2014 21:19:19 +0000 reply

This is not what I expected:

fricas

x:=taylor 'x

$\label{eq3}x$

(3)

Type: UnivariateTaylorSeries?(Expression(Integer),x,0)

fricas

y:=taylor 'y

$\label{eq4}y$

(4)

Type: UnivariateTaylorSeries?(Expression(Integer),y,0)

fricas

sinh(x)*cosh(y)

$\label{eq5}\begin{array}{@{}l} \displaystyle x +{{1 \over 6}\ {{x}^{3}}}+{{1 \over{120}}\ {{x}^{5}}}+{{1 \over{5 040}}\ {{x}^{7}}}+{{1 \over{362880}}\ {{x}^{9}}}+{O \left({{x}^{11}}\right)}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{1 \over 2}\ x}+{{1 \over{12}}\ {{x}^{3}}}+{{1 \over{240}}\ {{x}^{5}}}+{{1 \over{10080}}\ {{x}^{7}}}+{{1 \over{725760}}\ {{x}^{9}}}+ \ \ \displaystyle {O \left({{x}^{11}}\right)}$

(5)

Type: UnivariateTaylorSeries?(UnivariateTaylorSeries?(Expression(Integer),x,0),y,0)

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, i.e. TaylorSeries(Q) is the domain of power series over Q in infinitely many variables.

With that domain the input is as simple as above.

fricas

T==>TaylorSeries Fraction Integer

Type: Void

fricas

xx:T := 'x

$\label{eq6}x$

(6)

Type: TaylorSeries?(Fraction(Integer))

fricas

yy:T := 'y

$\label{eq7}y$

(7)

Type: TaylorSeries?(Fraction(Integer))

fricas

sinh(xx)*cosh(yy)

$\label{eq8}\begin{array}{@{}l} \displaystyle x +{\left({{1 \over 2}\ x \ {{y}^{2}}}+{{1 \over 6}\ {{x}^{3}}}\right)}+{\left({{1 \over{24}}\ x \ {{y}^{4}}}+{{1 \over{12}}\ {{x}^{3}}\ {{y}^{2}}}+{{1 \over{120}}\ {{x}^{5}}}\right)}+ \ \ \displaystyle {\left({{1 \over{720}}\ x \ {{y}^{6}}}+{{1 \over{144}}\ {{x}^{3}}\ {{y}^{4}}}+{{1 \over{240}}\ {{x}^{5}}\ {{y}^{2}}}+{{1 \over{5 040}}\ {{x}^{7}}}\right)}+ \ \ \displaystyle {\left({ \begin{array}{@{}l} \displaystyle {{1 \over{40320}}\ x \ {{y}^{8}}}+{{1 \over{4320}}\ {{x}^{3}}\ {{y}^{6}}}+{{1 \over{2880}}\ {{x}^{5}}\ {{y}^{4}}}+ \ \ \displaystyle {{1 \over{10080}}\ {{x}^{7}}\ {{y}^{2}}}+{{1 \over{362880}}\ {{x}^{9}}}$

(8)

Type: TaylorSeries?(Fraction(Integer))