MathAction SandBoxDoublePowerSeries

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  - SandBoxDoublePowerSeries <-- You are here.

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a!

$\label{eq1}a !$

(1)

Type: Variable(a!)

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!:=operator '!

$\label{eq2}!$

(2)

Type: BasicOperator?

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a(i,j)==(1+(j-1)*i)/!(2*i+4*j)

Type: Void

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a0:=matrix [[a(i,j)*k^i*l^j for i in 0..5] for j in 1..4]

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Compiling function a with type (NonNegativeInteger, PositiveInteger)
       -> Expression(Integer)

$\label{eq3}\left[ \begin{array}{cccccc} {l \over{! \left({4}\right)}}&{{k \ l}\over{! \left({6}\right)}}&{{{{k}^{2}}\ l}\over{! \left({8}\right)}}&{{{{k}^{3}}\ l}\over{! \left({1 0}\right)}}&{{{{k}^{4}}\ l}\over{! \left({12}\right)}}&{{{{k}^{5}}\ l}\over{! \left({14}\right)}} \ {{{l}^{2}}\over{! \left({8}\right)}}&{{2 \ k \ {{l}^{2}}}\over{! \left({10}\right)}}&{{3 \ {{k}^{2}}\ {{l}^{2}}}\over{! \left({1 2}\right)}}&{{4 \ {{k}^{3}}\ {{l}^{2}}}\over{! \left({14}\right)}}&{{5 \ {{k}^{4}}\ {{l}^{2}}}\over{! \left({16}\right)}}&{{6 \ {{k}^{5}}\ {{l}^{2}}}\over{! \left({18}\right)}} \ {{{l}^{3}}\over{! \left({12}\right)}}&{{3 \ k \ {{l}^{3}}}\over{! \left({14}\right)}}&{{5 \ {{k}^{2}}\ {{l}^{3}}}\over{! \left({1 6}\right)}}&{{7 \ {{k}^{3}}\ {{l}^{3}}}\over{! \left({18}\right)}}&{{9 \ {{k}^{4}}\ {{l}^{3}}}\over{! \left({20}\right)}}&{{{11}\ {{k}^{5}}\ {{l}^{3}}}\over{! \left({22}\right)}} \ {{{l}^{4}}\over{! \left({16}\right)}}&{{4 \ k \ {{l}^{4}}}\over{! \left({18}\right)}}&{{7 \ {{k}^{2}}\ {{l}^{4}}}\over{! \left({2 0}\right)}}&{{{10}\ {{k}^{3}}\ {{l}^{4}}}\over{! \left({22}\right)}}&{{{1 3}\ {{k}^{4}}\ {{l}^{4}}}\over{! \left({24}\right)}}&{{{16}\ {{k}^{5}}\ {{l}^{4}}}\over{! \left({26}\right)}}$

(3)

Type: Matrix(Expression(Integer))

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a(i:INT,j:INT):FRAC INT == (1+(j-1)*i)/Gamma(1+2*i+4*j)

   Function declaration a : (Integer, Integer) -> Fraction(Integer) has
      been added to workspace.
   Compiled code for a has been cleared.
   1 old definition(s) deleted for function or rule a

Type: Void

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aa:DMP([k,l], FRAC INT) := 1+reduce(+,concat [[a(i,j)*k^i*l^j for i in 0..3] for j in 1..3])

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Compiling function a with type (Integer, Integer) -> Fraction(
      Integer)

$\label{eq4}\begin{array}{@{}l} \displaystyle {{1 \over{914624815104000}}\ {{k}^{3}}\ {{l}^{3}}}+{{1 \over{2 1794572800}}\ {{k}^{3}}\ {{l}^{2}}}+ \ \ \displaystyle {{1 \over{3628800}}\ {{k}^{3}}\ l}+{{1 \over{4184557977600}}\ {{k}^{2}}\ {{l}^{3}}}+{{1 \over{159667200}}\ {{k}^{2}}\ {{l}^{2}}}+ \ \ \displaystyle {{1 \over{40320}}\ {{k}^{2}}\ l}+{{1 \over{29059430400}}\ k \ {{l}^{3}}}+{{1 \over{1814400}}\ k \ {{l}^{2}}}+{{1 \over{7 20}}\ k \ l}+ \ \ \displaystyle {{1 \over{479001600}}\ {{l}^{3}}}+{{1 \over{40320}}\ {{l}^{2}}}+{{1 \over{24}}\ l}+ 1$

(4)

Type: DistributedMultivariatePolynomial?([k,l],Fraction(Integer))

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eval(aa,[k=0,l=l])

$\label{eq5}{{1 \over{479001600}}\ {{l}^{3}}}+{{1 \over{40320}}\ {{l}^{2}}}+{{1 \over{24}}\ l}+ 1$

(5)

Type: DistributedMultivariatePolynomial?([k,l],Polynomial(Fraction(Integer)))

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eval(aa,[k=k,l=0])

$\label{eq6}1$

(6)

Type: DistributedMultivariatePolynomial?([k,l],Polynomial(Fraction(Integer)))

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eval(aa,[k=1.0,l=1.0])

$\label{eq7}1.0431059938 \<u> 809735313$

(7)

Type: Polynomial(Float)

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eval(aa,[k=0,l=0])

$\label{eq8}1$

(8)

Type: DistributedMultivariatePolynomial?([k,l],Polynomial(Fraction(Integer)))

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eval(aa,[k=-1,l=-1])::Float

$\label{eq9}0.9597219508 \<u> 1267275382$

(9)

Type: Float

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x:TaylorSeries FRAC INT

Type: Void

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y:TaylorSeries FRAC INT

Type: Void

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cos(x*y)+sinh(x*y)+cosh(x*y)*sinh(x*y)

$\label{eq10}1 +{2 \ x \ y}-{{1 \over 2}\ {{x}^{2}}\ {{y}^{2}}}+{{5 \over 6}\ {{x}^{3}}\ {{y}^{3}}}+{{1 \over{24}}\ {{x}^{4}}\ {{y}^{4}}}+{{{17}\over{120}}\ {{x}^{5}}\ {{y}^{5}}}+{O \left({11}\right)}$

(10)

Type: TaylorSeries(Fraction(Integer))

Q: Why doesn't this work?

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cosh(sqrt x)

   >> Error detected within library code:
   ^: rational power does not exist

Because sqrt(x) does not have a Taylor series. Even though cosh(sqrt(x)) does have a Taylor series, it can not be constructed this way.

Ans: Use GSERIES

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x:GeneralUnivariatePowerSeries(FRAC INT,'x,0)

Type: Void

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cosh(sqrt x)

$\label{eq11}1 +{{1 \over 2}\ x}+{{1 \over{24}}\ {{x}^{2}}}+{{1 \over{720}}\ {{x}^{3}}}+{{1 \over{40320}}\ {{x}^{4}}}+{{1 \over{3628800}}\ {{x}^{5}}}+{O \left({{x}^{{11}\over 2}}\right)}$

(11)

Type: GeneralUnivariatePowerSeries?(Fraction(Integer),x,0)

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poly(s,n)==reduce(+,[ [(i.c)*variable(s)^(i.k) for i in terms s].j for j in 1..n])

Type: Void

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poly(sinh(sqrt x),5)

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Compiling function poly with type (GeneralUnivariatePowerSeries(
      Fraction(Integer),x,0), PositiveInteger) -> Expression(Integer)

$\label{eq12}{{\left({{x}^{4}}+{{72}\ {{x}^{3}}}+{{3024}\ {{x}^{2}}}+{{604 80}\ x}+{362880}\right)}\ {\sqrt{x}}}\over{362880}$

(12)

Type: Expression(Integer)

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y:GeneralUnivariatePowerSeries(GeneralUnivariatePowerSeries(FRAC INT,'x,0),'y,0)

Type: Void

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poly2(ss,n,m)==reduce(+,[ [poly(i.c,n)*variable(ss)^(i.k) for i in terms ss].j for j in 1..m])

Type: Void

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poly2((2*cosh(sqrt x)-x)*sinh(sqrt y)/sqrt(y),3,4)::DMP(['x,'y],FRAC INT)

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Compiling function poly2 with type (GeneralUnivariatePowerSeries(
      GeneralUnivariatePowerSeries(Fraction(Integer),x,0),y,0), 
      PositiveInteger, PositiveInteger) -> Expression(Integer)

$\label{eq13}\begin{array}{@{}l} \displaystyle {{1 \over{1814400}}\ {{x}^{3}}\ {{y}^{3}}}+{{1 \over{43200}}\ {{x}^{3}}\ {{y}^{2}}}+{{1 \over{2160}}\ {{x}^{3}}\ y}+{{1 \over{3 60}}\ {{x}^{3}}}+ \ \ \displaystyle {{1 \over{60480}}\ {{x}^{2}}\ {{y}^{3}}}+{{1 \over{1440}}\ {{x}^{2}}\ {{y}^{2}}}+{{1 \over{72}}\ {{x}^{2}}\ y}+{{1 \over{12}}\ {{x}^{2}}}+{{1 \over{2520}}\ {{y}^{3}}}+ \ \ \displaystyle {{1 \over{60}}\ {{y}^{2}}}+{{1 \over 3}\ y}+ 2$

(13)

Type: DistributedMultivariatePolynomial?([x,y],Fraction(Integer))

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eq1:=(l=-4*(k[1]*k[2])^2)::EQ EXPR INT

$\label{eq14}l = -{4 \ {{k_{1}}^{2}}\ {{k_{2}}^{2}}}$

(14)

Type: Equation(Expression(Integer))

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eq2:=(k=k[1]^2+k[2]^2)::EQ EXPR INT

$\label{eq15}k ={{{k_{2}}^{2}}+{{k_{1}}^{2}}}$

(15)

Type: Equation(Expression(Integer))

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sol:=solve([eq1,eq2],[k[1],k[2]]); # sol

$\label{eq16}8$

(16)

Type: PositiveInteger?

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sol.1

$\label{eq17}\left[{{k_{1}}= -{{\sqrt{{\sqrt{l +{{k}^{2}}}}+ k}}\over{\sqrt{2}}}}, \:{{k_{2}}= -{{\sqrt{-{\sqrt{l +{{k}^{2}}}}+ k}}\over{\sqrt{2}}}}\right]$

(17)

Type: List(Equation(Expression(Integer)))

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a0:=(-k[2]^2*cosh(k[1])+k[1]^2*cosh(k[2]))/(k[1]^2-k[2]^2)

$\label{eq18}{-{{{k_{1}}^{2}}\ {\cosh \left({k_{2}}\right)}}+{{{k_{2}}^{2}}\ {\cosh \left({k_{1}}\right)}}}\over{{{k_{2}}^{2}}-{{k_{1}}^{2}}}$

(18)

Type: Expression(Integer)

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ans0:=eval(a0,sol.1)

$\label{eq19}{\left( \begin{array}{@{}l} \displaystyle {{\left({\sqrt{l +{{k}^{2}}}}- k \right)}\ {\cosh \left({{\sqrt{{\sqrt{l +{{k}^{2}}}}+ k}}\over{\sqrt{2}}}\right)}}+ \ \ \displaystyle {{\left({\sqrt{l +{{k}^{2}}}}+ k \right)}\ {\cosh \left({{\sqrt{-{\sqrt{l +{{k}^{2}}}}+ k}}\over{\sqrt{2}}}\right)}}$

(19)

Type: Expression(Integer)

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a1:=(-k[2]^2*sinh(k[1])/k[1]+k[1]^2*sinh(k[2])/k[2])/(k[1]^2-k[2]^2)

$\label{eq20}{-{{{k_{1}}^{3}}\ {\sinh \left({k_{2}}\right)}}+{{{k_{2}}^{3}}\ {\sinh \left({k_{1}}\right)}}}\over{{{k_{1}}\ {{k_{2}}^{3}}}-{{{k_{1}}^{3}}\ {k_{2}}}}$

(20)

Type: Expression(Integer)

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ans1:=eval(a1,sol.1)

$\label{eq21}{\left( \begin{array}{@{}l} \displaystyle {{\left({\sqrt{l +{{k}^{2}}}}- k \right)}\ {\sqrt{-{\sqrt{l +{{k}^{2}}}}+ k}}\ {\sinh \left({{\sqrt{{\sqrt{l +{{k}^{2}}}}+ k}}\over{\sqrt{2}}}\right)}}+ \ \ \displaystyle {{\left({\sqrt{l +{{k}^{2}}}}+ k \right)}\ {\sqrt{{\sqrt{l +{{k}^{2}}}}+ k}}\ {\sinh \left({{\sqrt{-{\sqrt{l +{{k}^{2}}}}+ k}}\over{\sqrt{2}}}\right)}}$

(21)

Type: Expression(Integer)

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interpret(eval(ans1,[k='x,l='y])::InputForm)

   >> Error detected within library code:
   ^ rational power of coefficient undefined

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)lib GDRAW

   OldGnuDraw is now explicitly exposed in frame initial 
   OldGnuDraw will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/GDRAW.NRLIB/GDRAW
gnuDraw(ans1,k=-30..60,l=-30..30,"SandBoxDoublePowerSeries1.dat",title=="Generating Function")

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Compiling function %C with type (DoubleFloat, DoubleFloat) -> 
      DoubleFloat

Type: Void

[terminal=pslatex,terminaloptions=color,scale=1.3]
set view 60, 30, 0.85, 1.1
set samples 20, 20
set isosamples 21, 21
set contour base
set cntrparam levels auto 20
set xlabel "k axis"
set ylabel "l axis"
load "SandBoxDoublePowerSeries1.dat"

Subject: Be Bold !!

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