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SandBoxExpOfEnd

Edit detail for SandBoxExpOfEnd revision 2 of 18

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Editor: Bill Page Time: 2014/09/19 06:38:43 GMT+0
Note:

added:
\begin{axiom}
choose(a,n) ==
  j:=[i for i in 1..n]
  r:=[[a(j(i)) for i in 1..n]]
  k:=n
  while k>0 and j(k)+n-k<#a repeat
    j(k):=j(k)+1
    for i in k..n-1 repeat j(i+1):=j(i)+1
    r:=concat(r,[a(j(i)) for i in 1..n])
    k:=n; while j(k)+n-k>=#a and k>1 repeat k:=k-1
  if binomial(#a,n)~=#r then error "error in choose"
  return r
\end{axiom}

\begin{axiom}
f(i,j)==reduce(+,map((x:List Expression Integer):Expression Integer +-> reduce(*,x,1),choose([r[q] for q in 1..n|q~=j],n-i-1)))
n:=2
f(0,1)
f(0,2)
f(1,1)
f(1,2)
n:=3
f(0,1)
f(0,2)
f(0,3)
f(1,1)
f(1,2)
f(1,3)
f(2,1)
f(2,2)
f(2,3)
reduce(+,[exp(r[j])/reduce(*,[r[j]-r[m] for m in 1..3 | j~=m]) for j in 1..3])
\end{axiom}

removed:
-1

fricas

choose(a,n) ==
  j:=[i for i in 1..n]
  r:=[[a(j(i)) for i in 1..n]]
  k:=n
  while k>0 and j(k)+n-k<#a repeat
    j(k):=j(k)+1
    for i in k..n-1 repeat j(i+1):=j(i)+1
    r:=concat(r,[a(j(i)) for i in 1..n])
    k:=n; while j(k)+n-k>=#a and k>1 repeat k:=k-1
  if binomial(#a,n)~=#r then error "error in choose"
  return r

Type: Void

fricas

f(i,j)==reduce(+,map((x:List Expression Integer):Expression Integer +-> reduce(*,x,1),choose([r[q] for q in 1..n|q~=j],n-i-1)))

Type: Void

fricas

n:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

fricas

f(0,1)

fricas

Compiling function choose with type (List(Symbol),Integer) -> List(
      List(Symbol))

fricas

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

$\label{eq2}r_{2}$

(2)

Type: Expression(Integer)

fricas

f(0,2)

$\label{eq3}r_{1}$

(3)

Type: Expression(Integer)

fricas

f(1,1)

fricas

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

$\label{eq4}1$

(4)

Type: Expression(Integer)

fricas

f(1,2)

$\label{eq5}1$

(5)

Type: Expression(Integer)

fricas

n:=3

$\label{eq6}3$

(6)

Type: PositiveInteger?

fricas

f(0,1)

$\label{eq7}{r_{2}}\ {r_{3}}$

(7)

Type: Expression(Integer)

fricas

f(0,2)

$\label{eq8}{r_{1}}\ {r_{3}}$

(8)

Type: Expression(Integer)

fricas

f(0,3)

$\label{eq9}{r_{1}}\ {r_{2}}$

(9)

Type: Expression(Integer)

fricas

f(1,1)

$\label{eq10}{r_{3}}+{r_{2}}$

(10)

Type: Expression(Integer)

fricas

f(1,2)

$\label{eq11}{r_{3}}+{r_{1}}$

(11)

Type: Expression(Integer)

fricas

f(1,3)

$\label{eq12}{r_{2}}+{r_{1}}$

(12)

Type: Expression(Integer)

fricas

f(2,1)

$\label{eq13}1$

(13)

Type: Expression(Integer)

fricas

f(2,2)

$\label{eq14}1$

(14)

Type: Expression(Integer)

fricas

f(2,3)

$\label{eq15}1$

(15)

Type: Expression(Integer)

fricas

reduce(+,[exp(r[j])/reduce(*,[r[j]-r[m] for m in 1..3 | j~=m]) for j in 1..3])

$\label{eq16}{{{\left({r_{2}}-{r_{1}}\right)}\ {{e}^{r_{3}}}}+{{\left(-{r_{3}}+{r_{1}}\right)}\ {{e}^{r_{2}}}}+{{\left({r_{3}}-{r_{2}}\right)}\ {{e}^{r_{1}}}}}\over{{{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}}$

(16)

Type: Expression(Integer)

one

fricas

eq42 := _
  -r2*r3*r4*exp(r1)/((r1-r2)*(r1-r3)*(r1-r4)) + _
  -r1*r3*r4*exp(r2)/((r2-r1)*(r2-r3)*(r2-r4)) + _
  -r1*r2*r4*exp(r3)/((r3-r1)*(r3-r2)*(r3-r4)) + _
  -r1*r2*r3*exp(r4)/((r4-r1)*(r4-r2)*(r4-r3))

$\label{eq17}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left(-{r 1 \ {{r 2}^{2}}}+{{{r 1}^{2}}\ r 2}\right)}\ {{r 3}^{3}}}+ \ \ \displaystyle {{\left({r 1 \ {{r 2}^{3}}}-{{{r 1}^{3}}\ r 2}\right)}\ {{r 3}^{2}}}+ \ \ \displaystyle {{\left(-{{{r 1}^{2}}\ {{r 2}^{3}}}+{{{r 1}^{3}}\ {{r 2}^{2}}}\right)}\ r 3}$

(17)

Type: Expression(Integer)

three

fricas

htrigs eval(eq42, [r3=-r1,r4=-r2])

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

(18)

Type: Expression(Integer)

fricas

htrigs limit(%,r2=r1)

$\label{eq19}{-{r 1 \ {\sinh \left({r 1}\right)}}+{2 \ {\cosh \left({r 1}\right)}}}\over 2$

(19)

Type: Expression(Integer)

fricas

eq43 :=
  (r2*r3 + r3*r4 + r4*r2)*exp(r1)/((r1-r2)*(r1-r3)*(r1-r4)) + _
  (r1*r3 + r3*r4 + r4*r1)*exp(r2)/((r2-r1)*(r2-r3)*(r2-r4)) + _
  (r1*r2 + r2*r4 + r4*r1)*exp(r3)/((r3-r1)*(r3-r2)*(r3-r4)) + _
  (r1*r2 + r2*r3 + r3*r1)*exp(r4)/((r4-r1)*(r4-r2)*(r4-r3))

$\label{eq20}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({{r 2}^{2}}-{{r 1}^{2}}\right)}\ {{r 3}^{3}}}+{{\left(-{{r 2}^{3}}+{{r 1}^{3}}\right)}\ {{r 3}^{2}}}+ \ \ \displaystyle {{{r 1}^{2}}\ {{r 2}^{3}}}-{{{r 1}^{3}}\ {{r 2}^{2}}}$

(20)

Type: Expression(Integer)

fricas

htrigs eval(eq43, [r3=-r1,r4=-r2])

$\label{eq21}{-{{{r 1}^{3}}\ {\sinh \left({r 2}\right)}}+{{{r 2}^{3}}\ {\sinh \left({r 1}\right)}}}\over{{r 1 \ {{r 2}^{3}}}-{{{r 1}^{3}}\ r 2}}$

(21)

Type: Expression(Integer)

fricas

htrigs limit(%,r2=r1)

$\label{eq22}{{3 \ {\sinh \left({r 1}\right)}}-{r 1 \ {\cosh \left({r 1}\right)}}}\over{2 \ r 1}$

(22)

Type: Expression(Integer)