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last edited 10 years ago by Bill page |
Edit detail for SandBoxDoublePowerSeries revision 19 of 19
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Editor: Bill page
Time: 2014/08/30 21:32:47 GMT+0 |
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| Note: | ||
changed: -ans1:=eval(a0,sol.1) ans0:=eval(a0,sol.1) a1:=(-k[2]^2*sinh(k[1])/k[1]+k[1]^2*sinh(k[2])/k[2])/(k[1]^2-k[2]^2) ans1:=eval(a1,sol.1)
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a!
 | (1) |
Type: Variable(a!)
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!:=operator '!
 | (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)}}](images/8467506247718380201-16.0px.png "\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](images/8557366300551141511-16.0px.png "\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) |
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eval(aa,[k=0, l=l])
 | (5) |
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eval(aa,[k=k, l=0])
 | (6) |
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eval(aa,[k=1.0, l=1.0])
 | (7) |
Type: Polynomial(Float)
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eval(aa,[k=0, l=0])
 | (8) |
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eval(aa,[k=-1, l=-1])::Float
 | (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)
 | (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)
 | (11) |
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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), | (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),![\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](images/5261789321992057421-16.0px.png "\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) |
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eq1:=(l=-4*(k[1]*k[2])^2)::EQ EXPR INT
 | (14) |
Type: Equation(Expression(Integer))
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eq2:=(k=k[1]^2+k[2]^2)::EQ EXPR INT
 | (15) |
Type: Equation(Expression(Integer))
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sol:=solve([eq1,eq2], [k[1], k[2]]); # sol
 | (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]](images/3228127607051823786-16.0px.png "\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)
 | (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)}}](images/5142701042514555408-16.0px.png "\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)
 | (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)}}](images/3092010834072430137-16.0px.png "\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"](images/6212032351234159128-16.0px.png "[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\"")