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Editor: Bill Page Time: 2014/09/20 05:19:28 GMT+0
Note:

changed:
-1 The Main Result
**Sum and product**

changed:
--- specify n
\end{axiom}

Verification of calculations in the paper

  (version date: September 11, 2014)

1 The Main Result

  Explicit expression for the group-polynomial coefficient functions $g_i(r_1,r_2, ...)$
\begin{axiom}

Exponential of endomorphism with minimal polynomial

First define some specialized operations

Collect terms in x with given factor k.

fricas

)set message type off

collect(x,k)==
  t:=factor(leadingMonomial univariate(numer(x), k::Kernel Expression Integer)) / _
     factor(univariate(denom x, k::Kernel Expression Integer))
  factor multivariate(numer t,k::Kernel Expression Integer)/ _
  factor multivariate(denom t,k::Kernel Expression Integer)

collect(exp(x)/exp(y),exp(x));

   There are 3 exposed and 1 unexposed library operations named 
      multivariate having 2 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                          )display op multivariate
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named 
      multivariate with argument type(s) 
Factored(SparseUnivariatePolynomial(SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))))
                         Kernel(Expression(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.

Choose n items from a list. Returns list of size binomial(#a,n) of lists.

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

Sum and product

fricas

sum(x)==reduce(+,x,0)

product(x)==reduce(*,x,1)

Verification of calculations in the paper

(version date: September 11, 2014)

The Main Result

Explicit expression for the group-polynomial coefficient functions

fricas

f(i,j) == sum [ product x for x in choose([r[q]::Expression Integer for q in 1..n|q~=j],n-i-1)]

groupPolyCoeff(i) == (-1)^(i+n+1)*reduce(+,[exp(r[j])/reduce(*,[r[j]-r[m] for m in 1..n | j~=m])*f(i,j) for j in 1..n])

Polynomial of degree 2

fricas

n:=2

$\label{eq1}2$

(1)

fricas

eq2_1:= m[X]=(x-r[1])*(x-r[2])

$\label{eq2}{m_{X}}={{{x}^{2}}+{{\left(-{r_{2}}-{r_{1}}\right)}\ x}+{{r_{1}}\ {r_{2}}}}$

(2)

fricas

eq2_2:= exp(X)=g[0]+g[1]*X

$\label{eq3}{{e}^{X}}={{{g_{1}}\ X}+{g_{0}}}$

(3)

fricas

eq2_3a:= g[0]=groupPolyCoeff(0)

fricas

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

fricas

Compiling function product with type List(Expression(Integer)) -> 
      Expression(Integer)

fricas

Compiling function sum with type List(Expression(Integer)) -> 
      Expression(Integer)

fricas

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

fricas

Compiling function groupPolyCoeff with type NonNegativeInteger -> 
      Expression(Integer)

$\label{eq4}{g_{0}}={{-{{r_{1}}\ {{e}^{r_{2}}}}+{{r_{2}}\ {{e}^{r_{1}}}}}\over{{r_{2}}-{r_{1}}}}$

(4)

fricas

eq2_3b:= g[1]=groupPolyCoeff(1)

fricas

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

fricas

Compiling function groupPolyCoeff with type PositiveInteger -> 
      Expression(Integer)

$\label{eq5}{g_{1}}={{{{e}^{r_{2}}}-{{e}^{r_{1}}}}\over{{r_{2}}-{r_{1}}}}$

(5)

Example 2.1

fricas

eval(eq2_1,[r[2]=-r[1]])

$\label{eq6}{m_{X}}={{{x}^{2}}-{{r_{1}}^{2}}}$

(6)

fricas

eq2_4:= eval(eval(eq2_2,[eq2_3a,eq2_3b]),r[2]=-r[1])

$\label{eq7}{{e}^{X}}={{{{\left(X +{r_{1}}\right)}\ {{e}^{r_{1}}}}+{{\left(- X +{r_{1}}\right)}\ {{e}^{-{r_{1}}}}}}\over{2 \ {r_{1}}}}$

(7)

fricas

htrigs rhs %

$\label{eq8}{{X \ {\sinh \left({r_{1}}\right)}}+{{r_{1}}\ {\cosh \left({r_{1}}\right)}}}\over{r_{1}}$

(8)

Polynomial of degree 3

fricas

n:=3

$\label{eq9}3$

(9)

fricas

eq3_1:= m[X]=(x-r[1])*(x-r[2])*(x-r[3])

$\label{eq10}\begin{array}{@{}l} \displaystyle {m_{X}}={ \begin{array}{@{}l} \displaystyle {{x}^{3}}+{{\left(-{r_{3}}-{r_{2}}-{r_{1}}\right)}\ {{x}^{2}}}+{{\left({{\left({r_{2}}+{r_{1}}\right)}\ {r_{3}}}+{{r_{1}}\ {r_{2}}}\right)}\ x}- \ \ \displaystyle {{r_{1}}\ {r_{2}}\ {r_{3}}}$

(10)

fricas

eq3_2:= exp(X)=g[0]+g[1]*X+g[2]*X^2

$\label{eq11}{{e}^{X}}={{{g_{2}}\ {{X}^{2}}}+{{g_{1}}\ X}+{g_{0}}}$

(11)

fricas

eq3_3a:= g[0]=groupPolyCoeff(0)

$\label{eq12}\begin{array}{@{}l} \displaystyle {g_{0}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{e}^{r_{3}}}}+ \ \ \displaystyle {{\left(-{{r_{1}}\ {{r_{3}}^{2}}}+{{{r_{1}}^{2}}\ {r_{3}}}\right)}\ {{e}^{r_{2}}}}+ \ \ \displaystyle {{\left({{r_{2}}\ {{r_{3}}^{2}}}-{{{r_{2}}^{2}}\ {r_{3}}}\right)}\ {{e}^{r_{1}}}}$

(12)

fricas

eq3_3b:= g[1]=groupPolyCoeff(1)

$\label{eq13}{g_{1}}={{{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {{e}^{r_{3}}}}+{{\left({{r_{3}}^{2}}-{{r_{1}}^{2}}\right)}\ {{e}^{r_{2}}}}+{{\left(-{{r_{3}}^{2}}+{{r_{2}}^{2}}\right)}\ {{e}^{r_{1}}}}}\over{{{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}}}$

(13)

fricas

eq3_3c:= g[2]=groupPolyCoeff(2)

$\label{eq14}{g_{2}}={{{{\left({r_{2}}-{r_{1}}\right)}\ {{e}^{r_{3}}}}+{{\left(-{r_{3}}+{r_{1}}\right)}\ {{e}^{r_{2}}}}+{{\left({r_{3}}-{r_{2}}\right)}\ {{e}^{r_{1}}}}}\over{{{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+{{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}}}$

(14)

Example 3.1

fricas

eval(eq3_1,[r[2]=-r[1],r[3]=0])

$\label{eq15}{m_{X}}={{{x}^{3}}-{{{r_{1}}^{2}}\ x}}$

(15)

fricas

eq3_4:= eval(eval(eq3_2,[eq3_3a,eq3_3b,eq3_3c]),[r[2]=-r[1],r[3]=0])

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

(16)

fricas

htrigs rhs eq3_4

$\label{eq17}{{{r_{1}}\ X \ {\sinh \left({r_{1}}\right)}}+{{{X}^{2}}\ {\cosh \left({r_{1}}\right)}}-{{X}^{2}}+{{r_{1}}^{2}}}\over{{r_{1}}^{2}}$

(17)

Comment 3.2 (Rescaled enomorphism)

fricas

eq3_6:= X' = sinh(r[1])/r[1]*X

$\label{eq18}X' ={{X \ {\sinh \left({r_{1}}\right)}}\over{r_{1}}}$

(18)

fricas

eq3_7:= γ = cosh(r[1])

$\label{eq19}�� ={\cosh \left({r_{1}}\right)}$

(19)

fricas

eq3_8:= exp(X) = 1+X'+X'^2/(1+γ)

$\label{eq20}{{e}^{X}}={{{{\left(X' + 1 \right)}\ ��}+{{X'}^{2}}+ X' + 1}\over{�� + 1}}$

(20)

fricas

test(normalize(rhs eval(eq3_8,[eq3_6,eq3_7]) - rhs eq3_4)=0)

$\label{eq21} \mbox{\rm true}$

(21)

Exercise 3.3

fricas

eval(eq3_1,[r[3]=r[2]])

$\label{eq22}{m_{X}}={{{x}^{3}}+{{\left(-{2 \ {r_{2}}}-{r_{1}}\right)}\ {{x}^{2}}}+{{\left({{r_{2}}^{2}}+{2 \ {r_{1}}\ {r_{2}}}\right)}\ x}-{{r_{1}}\ {{r_{2}}^{2}}}}$

(22)

fricas

eq3_9a:=lhs eq3_3a = limit(rhs eq3_3a,r[3]=r[2])

$\label{eq23}{g_{0}}={{{{\left({{r_{1}}\ {{r_{2}}^{2}}}+{{\left(-{{r_{1}}^{2}}-{2 \ {r_{1}}}\right)}\ {r_{2}}}+{{r_{1}}^{2}}\right)}\ {{e}^{r_{2}}}}+{{{r_{2}}^{2}}\ {{e}^{r_{1}}}}}\over{{{r_{2}}^{2}}-{2 \ {r_{1}}\ {r_{2}}}+{{r_{1}}^{2}}}}$

(23)

fricas

(numer rhs eq3_9a)/factor(denom rhs eq3_9a)

$\label{eq24}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{1 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{1}}\ {{r_{2}}^{2}}}+ \ \ \displaystyle {{\left(-{{1 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {{r_{1}}^{2}}}-{{2 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{1}}}\right)}\ {r_{2}}}+ \ \ \displaystyle {{1 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {{r_{1}}^{2}}}$

(24)

fricas

eq3_9b:=lhs eq3_3b = limit(rhs eq3_3b,r[3]=r[2])

$\label{eq25}{g_{1}}={{{{\left(-{{r_{2}}^{2}}+{2 \ {r_{2}}}+{{r_{1}}^{2}}\right)}\ {{e}^{r_{2}}}}-{2 \ {r_{2}}\ {{e}^{r_{1}}}}}\over{{{r_{2}}^{2}}-{2 \ {r_{1}}\ {r_{2}}}+{{r_{1}}^{2}}}}$

(25)

fricas

(numer rhs eq3_9b)/factor(denom rhs eq3_9b)

$\label{eq26}\begin{array}{@{}l} \displaystyle {{\left(-{{1 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {{r_{2}}^{2}}}+{{2 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{2}}}+{{1 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {{r_{1}}^{2}}}\right)}\ {{e}^{r_{2}}}}- \ \ \displaystyle {{2 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{2}}\ {{e}^{r_{1}}}}$

(26)

fricas

eq3_9c:=lhs eq3_3c = limit(rhs eq3_3c,r[3]=r[2])

$\label{eq27}{g_{2}}={{{{\left({r_{2}}-{r_{1}}- 1 \right)}\ {{e}^{r_{2}}}}+{{e}^{r_{1}}}}\over{{{r_{2}}^{2}}-{2 \ {r_{1}}\ {r_{2}}}+{{r_{1}}^{2}}}}$

(27)

fricas

(numer rhs eq3_9c)/factor(denom rhs eq3_9c)

$\label{eq28}\begin{array}{@{}l} \displaystyle {{\left({{1 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{2}}}-{{1 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {r_{1}}}-{1 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\right)}\ {{e}^{r_{2}}}}+ \ \ \displaystyle {{1 \over{{\left({r_{2}}-{r_{1}}\right)}^{2}}}\ {{e}^{r_{1}}}}$

(28)

Polynomial of degree 4

fricas

n:=4

$\label{eq29}4$

(29)

fricas

eq4_1:= m[X]=(x-r[1])*(x-r[2])*(x-r[3])*(x-r[4])

$\label{eq30}\begin{array}{@{}l} \displaystyle {m_{X}}={ \begin{array}{@{}l} \displaystyle {{x}^{4}}+{{\left(-{r_{4}}-{r_{3}}-{r_{2}}-{r_{1}}\right)}\ {{x}^{3}}}+ \ \ \displaystyle {{\left({{\left({r_{3}}+{r_{2}}+{r_{1}}\right)}\ {r_{4}}}+{{\left({r_{2}}+{r_{1}}\right)}\ {r_{3}}}+{{r_{1}}\ {r_{2}}}\right)}\ {{x}^{2}}}+ \ \ \displaystyle {{\left({{\left({{\left(-{r_{2}}-{r_{1}}\right)}\ {r_{3}}}-{{r_{1}}\ {r_{2}}}\right)}\ {r_{4}}}-{{r_{1}}\ {r_{2}}\ {r_{3}}}\right)}\ x}+ \ \ \displaystyle {{r_{1}}\ {r_{2}}\ {r_{3}}\ {r_{4}}}$

(30)

fricas

eq4_2:= g[0]=groupPolyCoeff(0)

$\label{eq31}\begin{array}{@{}l} \displaystyle {g_{0}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left(-{{r_{1}}\ {{r_{2}}^{2}}}+{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{3}}^{3}}}+ \ \ \displaystyle {{\left({{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{r_{3}}^{2}}}+ \ \ \displaystyle {{\left(-{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}\right)}\ {r_{3}}}$

(31)

fricas

eq4_3:= g[1]=groupPolyCoeff(1)

$\label{eq32}\begin{array}{@{}l} \displaystyle {g_{1}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({{r_{2}}^{2}}-{{r_{1}}^{2}}\right)}\ {{r_{3}}^{3}}}+ \ \ \displaystyle {{\left(-{{r_{2}}^{3}}+{{r_{1}}^{3}}\right)}\ {{r_{3}}^{2}}}+{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}- \ \ \displaystyle {{{r_{1}}^{3}}\ {{r_{2}}^{2}}}$

(32)

fricas

eq4_4:= g[2]=groupPolyCoeff(2)

$\label{eq33}\begin{array}{@{}l} \displaystyle {g_{2}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left(-{r_{2}}+{r_{1}}\right)}\ {{r_{3}}^{3}}}+{{\left({{r_{2}}^{3}}-{{r_{1}}^{3}}\right)}\ {r_{3}}}- \ \ \displaystyle {{r_{1}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {r_{2}}}$

(33)

fricas

eq4_5:= g[3]=groupPolyCoeff(3)

$\label{eq34}\begin{array}{@{}l} \displaystyle {g_{3}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+{{\left(-{{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {r_{3}}}+ \ \ \displaystyle {{r_{1}}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}$

(34)

$m_X(x) \equiv (x^2-r_1^2)\ (x^2-r_2^2)$

fricas

eq5_1:=eval(eq4_1,[r[3]=-r[1],r[4]=-r[2]])

$\label{eq35}{m_{X}}={{{x}^{4}}+{{\left(-{{r_{2}}^{2}}-{{r_{1}}^{2}}\right)}\ {{x}^{2}}}+{{{r_{1}}^{2}}\ {{r_{2}}^{2}}}}$

(35)

Corollary 5.1

fricas

eq5_2:= exp(X)=g[0]+g[1]*X+g[2]*X^2+g[3]*X^3

$\label{eq36}{{e}^{X}}={{{g_{3}}\ {{X}^{3}}}+{{g_{2}}\ {{X}^{2}}}+{{g_{1}}\ X}+{g_{0}}}$

(36)

fricas

eq5_3a:= eval(eq4_2,[r[3]=-r[1],r[4]=-r[2]])

$\label{eq37}{g_{0}}={{-{{{r_{1}}^{2}}\ {{e}^{r_{2}}}}+{{{r_{2}}^{2}}\ {{e}^{r_{1}}}}+{{{r_{2}}^{2}}\ {{e}^{-{r_{1}}}}}-{{{r_{1}}^{2}}\ {{e}^{-{r_{2}}}}}}\over{{2 \ {{r_{2}}^{2}}}-{2 \ {{r_{1}}^{2}}}}}$

(37)

fricas

htrigs rhs %

$\label{eq38}{-{{{r_{1}}^{2}}\ {\cosh \left({r_{2}}\right)}}+{{{r_{2}}^{2}}\ {\cosh \left({r_{1}}\right)}}}\over{{{r_{2}}^{2}}-{{r_{1}}^{2}}}$

(38)

fricas

eq5_3b:= eval(eq4_3,[r[3]=-r[1],r[4]=-r[2]])

$\label{eq39}{g_{1}}={{-{{{r_{1}}^{3}}\ {{e}^{r_{2}}}}+{{{r_{2}}^{3}}\ {{e}^{r_{1}}}}-{{{r_{2}}^{3}}\ {{e}^{-{r_{1}}}}}+{{{r_{1}}^{3}}\ {{e}^{-{r_{2}}}}}}\over{{2 \ {r_{1}}\ {{r_{2}}^{3}}}-{2 \ {{r_{1}}^{3}}\ {r_{2}}}}}$

(39)

fricas

htrigs rhs %

$\label{eq40}{-{{{r_{1}}^{3}}\ {\sinh \left({r_{2}}\right)}}+{{{r_{2}}^{3}}\ {\sinh \left({r_{1}}\right)}}}\over{{{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}}$

(40)

fricas

eq5_3c:= eval(eq4_4,[r[3]=-r[1],r[4]=-r[2]])

$\label{eq41}{g_{2}}={{{{e}^{r_{2}}}-{{e}^{r_{1}}}-{{e}^{-{r_{1}}}}+{{e}^{-{r_{2}}}}}\over{{2 \ {{r_{2}}^{2}}}-{2 \ {{r_{1}}^{2}}}}}$

(41)

fricas

htrigs rhs %

$\label{eq42}{{\cosh \left({r_{2}}\right)}-{\cosh \left({r_{1}}\right)}}\over{{{r_{2}}^{2}}-{{r_{1}}^{2}}}$

(42)

fricas

eq5_3d:= eval(eq4_5,[r[3]=-r[1],r[4]=-r[2]])

$\label{eq43}{g_{3}}={{{{r_{1}}\ {{e}^{r_{2}}}}-{{r_{2}}\ {{e}^{r_{1}}}}+{{r_{2}}\ {{e}^{-{r_{1}}}}}-{{r_{1}}\ {{e}^{-{r_{2}}}}}}\over{{2 \ {r_{1}}\ {{r_{2}}^{3}}}-{2 \ {{r_{1}}^{3}}\ {r_{2}}}}}$

(43)

fricas

htrigs rhs %

$\label{eq44}{{{r_{1}}\ {\sinh \left({r_{2}}\right)}}-{{r_{2}}\ {\sinh \left({r_{1}}\right)}}}\over{{{r_{1}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {r_{2}}}}$

(44)

fricas

eq5_4:= eval(eval(eq5_2,[eq4_2,eq4_3,eq4_4,eq4_5]),[r[3]=-r[1],r[4]=-r[2]])

$\label{eq45}\begin{array}{@{}l} \displaystyle {{e}^{X}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({{r_{1}}\ {{X}^{3}}}+{{r_{1}}\ {r_{2}}\ {{X}^{2}}}-{{{r_{1}}^{3}}\ X}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{e}^{r_{2}}}}+ \ \ \displaystyle {{\left(-{{r_{2}}\ {{X}^{3}}}-{{r_{1}}\ {r_{2}}\ {{X}^{2}}}+{{{r_{2}}^{3}}\ X}+{{r_{1}}\ {{r_{2}}^{3}}}\right)}\ {{e}^{r_{1}}}}+ \ \ \displaystyle {{\left({{r_{2}}\ {{X}^{3}}}-{{r_{1}}\ {r_{2}}\ {{X}^{2}}}-{{{r_{2}}^{3}}\ X}+{{r_{1}}\ {{r_{2}}^{3}}}\right)}\ {{e}^{-{r_{1}}}}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{r_{1}}\ {{X}^{3}}}+{{r_{1}}\ {r_{2}}\ {{X}^{2}}}+{{{r_{1}}^{3}}\ X}- \ \ \displaystyle {{{r_{1}}^{3}}\ {r_{2}}}$

(45)

fricas

htrigs rhs %

$\label{eq46}{\left( \begin{array}{@{}l} \displaystyle {{\left({{r_{1}}\ {{X}^{3}}}-{{{r_{1}}^{3}}\ X}\right)}\ {\sinh \left({r_{2}}\right)}}+ \ \ \displaystyle {{\left(-{{r_{2}}\ {{X}^{3}}}+{{{r_{2}}^{3}}\ X}\right)}\ {\sinh \left({r_{1}}\right)}}+ \ \ \displaystyle {{\left({{r_{1}}\ {r_{2}}\ {{X}^{2}}}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {\cosh \left({r_{2}}\right)}}+ \ \ \displaystyle {{\left(-{{r_{1}}\ {r_{2}}\ {{X}^{2}}}+{{r_{1}}\ {{r_{2}}^{3}}}\right)}\ {\cosh \left({r_{1}}\right)}}$

(46)

Definition 5.2

fricas

eq5_5a:= Y[1] = 1/(2*r[1]^2-r[1]^2-r[2]^2)*(X^3-(r[1]^2+r[2]^2-r[1]^2)*X)

$\label{eq47}{Y_{1}}={{-{{X}^{3}}+{{{r_{2}}^{2}}\ X}}\over{{{r_{2}}^{2}}-{{r_{1}}^{2}}}}$

(47)

fricas

eq5_5b:= Y[2] = 1/(2*r[2]^2-r[1]^2-r[2]^2)*(X^3-(r[1]^2+r[2]^2-r[2]^2)*X)

$\label{eq48}{Y_{2}}={{{{X}^{3}}-{{{r_{1}}^{2}}\ X}}\over{{{r_{2}}^{2}}-{{r_{1}}^{2}}}}$

(48)

Exercise 5.3

fricas

eq5_6a:= X = Y[1]+Y[2]

$\label{eq49}X ={{Y_{2}}+{Y_{1}}}$

(49)

fricas

test(eval(eq5_6a,[eq5_5a,eq5_5b]))

$\label{eq50} \mbox{\rm true}$

(50)

fricas

eq5_6b:= eval(Y[1]*Y[2]=0,[eq5_5a,eq5_5b])

$\label{eq51}{{-{{X}^{6}}+{{\left({{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {{X}^{4}}}-{{{r_{1}}^{2}}\ {{r_{2}}^{2}}\ {{X}^{2}}}}\over{{{r_{2}}^{4}}-{2 \ {{r_{1}}^{2}}\ {{r_{2}}^{2}}}+{{r_{1}}^{4}}}}= 0$

(51)

fricas

eq5_6c:= X^4 = X^4-eval(rhs eq5_1,x=X)

$\label{eq52}{{X}^{4}}={{{\left({{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {{X}^{2}}}-{{{r_{1}}^{2}}\ {{r_{2}}^{2}}}}$

(52)

fricas

eq5_6d:= X^2*(lhs %)=X^2*(rhs %)

$\label{eq53}{{X}^{6}}={{{\left({{r_{2}}^{2}}+{{r_{1}}^{2}}\right)}\ {{X}^{4}}}-{{{r_{1}}^{2}}\ {{r_{2}}^{2}}\ {{X}^{2}}}}$

(53)

fricas

test(_rule(lhs eq5_6d,rhs eq5_6d)(lhs eq5_6b)=rhs eq5_6b)

$\label{eq54} \mbox{\rm true}$

(54)

Comment 5.4 (Rescaling)

fricas

eq5_7:= sinh(r[1])/r[1]*Y[1]+sinh(r[2])/r[2]*Y[2]

$\label{eq55}{{{Y_{2}}\ {r_{1}}\ {\sinh \left({r_{2}}\right)}}+{{Y_{1}}\ {r_{2}}\ {\sinh \left({r_{1}}\right)}}}\over{{r_{1}}\ {r_{2}}}$

(55)

fricas

eq5_8a:= [ Y'[1]=sinh(r[1])/r[1]*Y[1], Y'[2]=sinh(r[2])/r[2]*Y[2] ]

$\label{eq56}\left[{{Y'_{1}}={{{Y_{1}}\ {\sinh \left({r_{1}}\right)}}\over{r_{1}}}}, \:{{Y'_{2}}={{{Y_{2}}\ {\sinh \left({r_{2}}\right)}}\over{r_{2}}}}\right]$

(56)

fricas

eq5_8b:= [ γ[1] = cosh(r[1]), γ[2] = cosh(r[2]) ]

$\label{eq57}\left[{{��_{1}}={\cosh \left({r_{1}}\right)}}, \:{{��_{2}}={\cosh \left({r_{2}}\right)}}\right]$

(57)

fricas

eq5_9a:= exp(X) = exp(Y[1])*exp(Y[2])

$\label{eq58}{{e}^{X}}={{{e}^{Y_{1}}}\ {{e}^{Y_{2}}}}$

(58)

fricas

eval(eq5_9a,[eq5_5a,eq5_5b])

$\label{eq59}{{e}^{X}}={{{e}^{{-{{X}^{3}}+{{{r_{2}}^{2}}\ X}}\over{{{r_{2}}^{2}}-{{r_{1}}^{2}}}}}\ {{e}^{{{{X}^{3}}-{{{r_{1}}^{2}}\ X}}\over{{{r_{2}}^{2}}-{{r_{1}}^{2}}}}}}$

(59)

fricas

test(lhs % = simplify expand rhs %)

$\label{eq60} \mbox{\rm true}$

(60)

fricas

eq5_9b:= exp(X) = 1 + Y'[1] +Y'[2] + Y'[1]^2/(1+γ[1]) + Y'[2]^2/(1+γ[2])

$\label{eq61}\begin{array}{@{}l} \displaystyle {{e}^{X}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({Y'_{2}}+{Y'_{1}}+ 1 \right)}\ {��_{1}}}+{Y'_{2}}+{{Y'_{1}}^{2}}+ \ \ \displaystyle {Y'_{1}}+ 1$

(61)

fricas

normalize eval(eval(rhs eq5_9b,concat [eq5_8a,eq5_8b]),[eq5_5a,eq5_5b])

$\label{eq62}{\left( \begin{array}{@{}l} \displaystyle { \begin{array}{@{}l} \displaystyle {\left({ \begin{array}{@{}l} \displaystyle {{{r_{1}}^{2}}\ {{X}^{6}}}-{2 \ {{r_{1}}^{4}}\ {{X}^{4}}}+{{\left({{{r_{1}}^{2}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{4}}\ {r_{2}}}\right)}\ {{X}^{3}}}+ \ \ \displaystyle {{{r_{1}}^{6}}\ {{X}^{2}}}+{{\left(-{{{r_{1}}^{4}}\ {{r_{2}}^{3}}}+{{{r_{1}}^{6}}\ {r_{2}}}\right)}\ X}$

(62)

fricas

_rule(lhs eq5_6d,rhs eq5_6d)(%)

$\label{eq63}{\left( \begin{array}{@{}l} \displaystyle {{\left({{{r_{1}}^{2}}\ {{X}^{4}}}+{{{r_{1}}^{2}}\ {r_{2}}\ {{X}^{3}}}-{{{r_{1}}^{4}}\ {{X}^{2}}}-{{{r_{1}}^{4}}\ {r_{2}}\ X}\right)}\ {{e}^{r_{1}}}\ {{{e}^{r_{2}}}^{2}}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{{r_{2}}^{2}}\ {{X}^{4}}}-{{r_{1}}\ {{r_{2}}^{2}}\ {{X}^{3}}}+ \ \ \displaystyle {{{r_{2}}^{4}}\ {{X}^{2}}}+{{r_{1}}\ {{r_{2}}^{4}}\ X}$

(63)

fricas

_rule(lhs eq5_6c,rhs eq5_6c)(%)

$\label{eq64}{\left( \begin{array}{@{}l} \displaystyle {{\left({{r_{1}}\ {{X}^{3}}}+{{r_{1}}\ {r_{2}}\ {{X}^{2}}}-{{{r_{1}}^{3}}\ X}-{{{r_{1}}^{3}}\ {r_{2}}}\right)}\ {{e}^{r_{1}}}\ {{{e}^{r_{2}}}^{2}}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{r_{2}}\ {{X}^{3}}}-{{r_{1}}\ {r_{2}}\ {{X}^{2}}}+ \ \ \displaystyle {{{r_{2}}^{3}}\ X}+{{r_{1}}\ {{r_{2}}^{3}}}$

(64)

fricas

test(normalize(% - rhs eq5_4) = 0)

$\label{eq65} \mbox{\rm true}$

(65)

Multiple roots for polynomial of degree four

Exercise 6.1

fricas

eval(eq4_1,r[4]=r[3])

$\label{eq66}\begin{array}{@{}l} \displaystyle {m_{X}}={ \begin{array}{@{}l} \displaystyle {{x}^{4}}+{{\left(-{2 \ {r_{3}}}-{r_{2}}-{r_{1}}\right)}\ {{x}^{3}}}+ \ \ \displaystyle {{\left({{r_{3}}^{2}}+{{\left({2 \ {r_{2}}}+{2 \ {r_{1}}}\right)}\ {r_{3}}}+{{r_{1}}\ {r_{2}}}\right)}\ {{x}^{2}}}+ \ \ \displaystyle {{\left({{\left(-{r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}-{2 \ {r_{1}}\ {r_{2}}\ {r_{3}}}\right)}\ x}+{{r_{1}}\ {r_{2}}\ {{r_{3}}^{2}}}$

(66)

fricas

eq6_1a:=lhs eq4_2 = limit(rhs eq4_2,r[4]=r[3])

$\label{eq67}\begin{array}{@{}l} \displaystyle {g_{0}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left(-{{r_{1}}\ {{r_{2}}^{2}}}+{{{r_{1}}^{2}}\ {r_{2}}}\right)}\ {{r_{3}}^{3}}}+ \ \ \displaystyle {{\left({{{r_{1}}\ {{r_{2}}^{3}}}+{3 \ {r_{1}}\ {{r_{2}}^{2}}}+{{\left(-{{r_{1}}^{3}}-{3 \ {{r_{1}}^{2}}}\right)}\ {r_{2}}}}\right)}\ {{r_{3}}^{2}}}+ \ \ \displaystyle {{\left({{{\left(-{{r_{1}}^{2}}-{2 \ {r_{1}}}\right)}\ {{r_{2}}^{3}}}+{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}+{2 \ {{r_{1}}^{3}}\ {r_{2}}}}\right)}\ {r_{3}}}+ \ \ \displaystyle {{{r_{1}}^{2}}\ {{r_{2}}^{3}}}-{{{r_{1}}^{3}}\ {{r_{2}}^{2}}}$

(67)

fricas

collect(rhs(eq6_1a), exp(r[1]))

$\label{eq68}{{r_{2}}\ {{r_{3}}^{2}}\ {{e}^{r_{1}}}}\over{{\left({r_{2}}-{r_{1}}\right)}\ {{\left({r_{3}}-{r_{1}}\right)}^{2}}}$

(68)

fricas

collect(rhs(eq6_1a), exp(r[2]))

$\label{eq69}-{{{r_{1}}\ {{r_{3}}^{2}}\ {{e}^{r_{2}}}}\over{{\left({r_{2}}-{r_{1}}\right)}\ {{\left({r_{3}}-{r_{2}}\right)}^{2}}}}$

(69)

fricas

collect(rhs(eq6_1a), exp(r[3]))

$\label{eq70}\begin{array}{@{}l} \displaystyle -{{\left( \begin{array}{@{}l} \displaystyle {r_{1}}\ {r_{2}}\ {\left({ \begin{array}{@{}l} \displaystyle {{r_{3}}^{3}}+{{\left(-{r_{2}}-{r_{1}}- 3 \right)}\ {{r_{3}}^{2}}}+ \ \ \displaystyle {{\left({{\left({r_{1}}+ 2 \right)}\ {r_{2}}}+{2 \ {r_{1}}}\right)}\ {r_{3}}}-{{r_{1}}\ {r_{2}}}$

(70)

fricas

eq6_1b:=lhs eq4_3 = limit(rhs eq4_3,r[4]=r[3])

$\label{eq71}\begin{array}{@{}l} \displaystyle {g_{1}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({{r_{2}}^{2}}-{{r_{1}}^{2}}\right)}\ {{r_{3}}^{3}}}+ \ \ \displaystyle {{\left(-{{r_{2}}^{3}}-{3 \ {{r_{2}}^{2}}}+{{r_{1}}^{3}}+{3 \ {{r_{1}}^{2}}}\right)}\ {{r_{3}}^{2}}}+ \ \ \displaystyle {{\left({2 \ {{r_{2}}^{3}}}-{2 \ {{r_{1}}^{3}}}\right)}\ {r_{3}}}+{{{r_{1}}^{2}}\ {{r_{2}}^{3}}}- \ \ \displaystyle {{{r_{1}}^{3}}\ {{r_{2}}^{2}}}$

(71)

fricas

collect(rhs(eq6_1b), exp(r[1]))

$\label{eq72}-{{{r_{3}}\ {\left({r_{3}}+{2 \ {r_{2}}}\right)}\ {{e}^{r_{1}}}}\over{{\left({r_{2}}-{r_{1}}\right)}\ {{\left({r_{3}}-{r_{1}}\right)}^{2}}}}$

(72)

fricas

collect(rhs(eq6_1b), exp(r[2]))

$\label{eq73}{{r_{3}}\ {\left({r_{3}}+{2 \ {r_{1}}}\right)}\ {{e}^{r_{2}}}}\over{{\left({r_{2}}-{r_{1}}\right)}\ {{\left({r_{3}}-{r_{2}}\right)}^{2}}}$

(73)

fricas

collect(rhs(eq6_1b), exp(r[3]))

$\label{eq74}{\left({\left({ \begin{array}{@{}l} \displaystyle {{\left({r_{2}}+{r_{1}}\right)}\ {{r_{3}}^{3}}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{r_{2}}^{2}}+{{\left(-{r_{1}}- 3 \right)}\ {r_{2}}}-{{r_{1}}^{2}}- \ \ \displaystyle {3 \ {r_{1}}}$

(74)

fricas

eq6_1c:=lhs eq4_4 = limit(rhs eq4_4,r[4]=r[3])

$\label{eq75}\begin{array}{@{}l} \displaystyle {g_{2}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left(-{r_{2}}+{r_{1}}\right)}\ {{r_{3}}^{3}}}+{{\left({3 \ {r_{2}}}-{3 \ {r_{1}}}\right)}\ {{r_{3}}^{2}}}+ \ \ \displaystyle {{\left({{r_{2}}^{3}}-{{r_{1}}^{3}}\right)}\ {r_{3}}}+{{\left(-{r_{1}}- 1 \right)}\ {{r_{2}}^{3}}}+ \ \ \displaystyle {{{r_{1}}^{3}}\ {r_{2}}}+{{r_{1}}^{3}}$

(75)

fricas

collect(rhs(eq6_1c), exp(r[1]))

$\label{eq76}{{\left({2 \ {r_{3}}}+{r_{2}}\right)}\ {{e}^{r_{1}}}}\over{{\left({r_{2}}-{r_{1}}\right)}\ {{\left({r_{3}}-{r_{1}}\right)}^{2}}}$

(76)

fricas

collect(rhs(eq6_1c), exp(r[2]))

$\label{eq77}-{{{\left({2 \ {r_{3}}}+{r_{1}}\right)}\ {{e}^{r_{2}}}}\over{{\left({r_{2}}-{r_{1}}\right)}\ {{\left({r_{3}}-{r_{2}}\right)}^{2}}}}$

(77)

fricas

collect(rhs(eq6_1c), exp(r[3]))

$\label{eq78}\begin{array}{@{}l} \displaystyle -{{\left({\left({ \begin{array}{@{}l} \displaystyle {{r_{3}}^{3}}-{3 \ {{r_{3}}^{2}}}+ \ \ \displaystyle {{\left(-{{r_{2}}^{2}}-{{r_{1}}\ {r_{2}}}-{{r_{1}}^{2}}\right)}\ {r_{3}}}+ \ \ \displaystyle {{\left({r_{1}}+ 1 \right)}\ {{r_{2}}^{2}}}+{{\left({{r_{1}}^{2}}+{r_{1}}\right)}\ {r_{2}}}+{{r_{1}}^{2}}$

(78)

fricas

eq6_1d:=lhs eq4_5 = limit(rhs eq4_5,r[4]=r[3])

$\label{eq79}\begin{array}{@{}l} \displaystyle {g_{3}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({r_{2}}-{r_{1}}\right)}\ {{r_{3}}^{2}}}+ \ \ \displaystyle {{\left(-{{r_{2}}^{2}}-{2 \ {r_{2}}}+{{r_{1}}^{2}}+{2 \ {r_{1}}}\right)}\ {r_{3}}}+ \ \ \displaystyle {{\left({r_{1}}+ 1 \right)}\ {{r_{2}}^{2}}}-{{{r_{1}}^{2}}\ {r_{2}}}-{{r_{1}}^{2}}$

(79)

fricas

collect(rhs(eq6_1d), exp(r[1]))

$\label{eq80}-{{{e}^{r_{1}}}\over{{\left({r_{2}}-{r_{1}}\right)}\ {{\left({r_{3}}-{r_{1}}\right)}^{2}}}}$

(80)

fricas

collect(rhs(eq6_1d), exp(r[2]))

$\label{eq81}{{e}^{r_{2}}}\over{{\left({r_{2}}-{r_{1}}\right)}\ {{\left({r_{3}}-{r_{2}}\right)}^{2}}}$

(81)

fricas

collect(rhs(eq6_1d), exp(r[3]))

$\label{eq82}{{\left({{r_{3}}^{2}}+{{\left(-{r_{2}}-{r_{1}}- 2 \right)}\ {r_{3}}}+{{\left({r_{1}}+ 1 \right)}\ {r_{2}}}+{r_{1}}\right)}\ {{e}^{r_{3}}}}\over{{{\left({r_{3}}-{r_{2}}\right)}^{2}}\ {{\left({r_{3}}-{r_{1}}\right)}^{2}}}$

(82)

Exercise 6.3 (Double root)

fricas

eval(eq4_1,[r[3]=r[1],r[4]=r[2]])

$\label{eq83}\begin{array}{@{}l} \displaystyle {m_{X}}={ \begin{array}{@{}l} \displaystyle {{x}^{4}}+{{\left(-{2 \ {r_{2}}}-{2 \ {r_{1}}}\right)}\ {{x}^{3}}}+{{\left({{r_{2}}^{2}}+{4 \ {r_{1}}\ {r_{2}}}+{{r_{1}}^{2}}\right)}\ {{x}^{2}}}+ \ \ \displaystyle {{\left(-{2 \ {r_{1}}\ {{r_{2}}^{2}}}-{2 \ {{r_{1}}^{2}}\ {r_{2}}}\right)}\ x}+{{{r_{1}}^{2}}\ {{r_{2}}^{2}}}$

(83)

fricas

eq6_3a:=lhs eq4_2 = limit(limit(rhs eq4_2,r[3]=r[1]),r[4]=r[2])

$\label{eq84}\begin{array}{@{}l} \displaystyle {g_{0}}={{\left( \begin{array}{@{}l} \displaystyle {{\left(-{{{r_{1}}^{2}}\ {{r_{2}}^{2}}}+{{\left({{r_{1}}^{3}}+{3 \ {{r_{1}}^{2}}}\right)}\ {r_{2}}}-{{r_{1}}^{3}}\right)}\ {{e}^{r_{2}}}}+ \ \ \displaystyle {{\left({{\left(-{r_{1}}+ 1 \right)}\ {{r_{2}}^{3}}}+{{\left({{r_{1}}^{2}}-{3 \ {r_{1}}}\right)}\ {{r_{2}}^{2}}}\right)}\ {{e}^{r_{1}}}}$

(84)

fricas

collect(rhs(eq6_3a), exp(r[1]))

$\label{eq85}-{{{{r_{2}}^{2}}\ {\left({{\left({r_{1}}- 1 \right)}\ {r_{2}}}-{{r_{1}}^{2}}+{3 \ {r_{1}}}\right)}\ {{e}^{r_{1}}}}\over{{\left({r_{2}}-{r_{1}}\right)}^{3}}}$

(85)

fricas

collect(rhs(eq6_3a), exp(r[2]))

$\label{eq86}-{{{{r_{1}}^{2}}\ {\left({{r_{2}}^{2}}+{{\left(-{r_{1}}- 3 \right)}\ {r_{2}}}+{r_{1}}\right)}\ {{e}^{r_{2}}}}\over{{\left({r_{2}}-{r_{1}}\right)}^{3}}}$

(86)

fricas

eq6_3b:=lhs eq4_3 = limit(limit(rhs eq4_3,r[3]=r[1]),r[4]=r[2])

$\label{eq87}\begin{array}{@{}l} \displaystyle {g_{1}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({2 \ {r_{1}}\ {{r_{2}}^{2}}}+{{\left(-{{r_{1}}^{2}}-{6 \ {r_{1}}}\right)}\ {r_{2}}}-{{r_{1}}^{3}}\right)}\ {{e}^{r_{2}}}}+ \ \ \displaystyle {{\left({{r_{2}}^{3}}+{{r_{1}}\ {{r_{2}}^{2}}}+{{\left(-{2 \ {{r_{1}}^{2}}}+{6 \ {r_{1}}}\right)}\ {r_{2}}}\right)}\ {{e}^{r_{1}}}}$

(87)

fricas

collect(rhs(eq6_3b), exp(r[1]))

$\label{eq88}{{r_{2}}\ {\left({{r_{2}}^{2}}+{{r_{1}}\ {r_{2}}}-{2 \ {{r_{1}}^{2}}}+{6 \ {r_{1}}}\right)}\ {{e}^{r_{1}}}}\over{{\left({r_{2}}-{r_{1}}\right)}^{3}}$

(88)

fricas

collect(rhs(eq6_3b), exp(r[2]))

$\label{eq89}{{r_{1}}\ {\left({2 \ {{r_{2}}^{2}}}+{{\left(-{r_{1}}- 6 \right)}\ {r_{2}}}-{{r_{1}}^{2}}\right)}\ {{e}^{r_{2}}}}\over{{\left({r_{2}}-{r_{1}}\right)}^{3}}$

(89)

fricas

eq6_3c:=lhs eq4_4 = limit(limit(rhs eq4_4,r[3]=r[1]),r[4]=r[2])

$\label{eq90}\begin{array}{@{}l} \displaystyle {g_{2}}={{\left( \begin{array}{@{}l} \displaystyle {{\left(-{{r_{2}}^{2}}+{{\left(-{r_{1}}+ 3 \right)}\ {r_{2}}}+{2 \ {{r_{1}}^{2}}}+{3 \ {r_{1}}}\right)}\ {{e}^{r_{2}}}}+ \ \ \displaystyle {{\left(-{2 \ {{r_{2}}^{2}}}+{{\left({r_{1}}- 3 \right)}\ {r_{2}}}+{{r_{1}}^{2}}-{3 \ {r_{1}}}\right)}\ {{e}^{r_{1}}}}$

(90)

fricas

collect(rhs(eq6_3c), exp(r[1]))

$\label{eq91}-{{{\left({2 \ {{r_{2}}^{2}}}+{{\left(-{r_{1}}+ 3 \right)}\ {r_{2}}}-{{r_{1}}^{2}}+{3 \ {r_{1}}}\right)}\ {{e}^{r_{1}}}}\over{{\left({r_{2}}-{r_{1}}\right)}^{3}}}$

(91)

fricas

collect(rhs(eq6_3c), exp(r[2]))

$\label{eq92}-{{{\left({{r_{2}}^{2}}+{{\left({r_{1}}- 3 \right)}\ {r_{2}}}-{2 \ {{r_{1}}^{2}}}-{3 \ {r_{1}}}\right)}\ {{e}^{r_{2}}}}\over{{\left({r_{2}}-{r_{1}}\right)}^{3}}}$

(92)

fricas

eq6_3d:=lhs eq4_5 = limit(limit(rhs eq4_5,r[3]=r[1]),r[4]=r[2])

$\label{eq93}{g_{3}}={{{{\left({r_{2}}-{r_{1}}- 2 \right)}\ {{e}^{r_{2}}}}+{{\left({r_{2}}-{r_{1}}+ 2 \right)}\ {{e}^{r_{1}}}}}\over{{{r_{2}}^{3}}-{3 \ {r_{1}}\ {{r_{2}}^{2}}}+{3 \ {{r_{1}}^{2}}\ {r_{2}}}-{{r_{1}}^{3}}}}$

(93)

fricas

collect(rhs(eq6_3d), exp(r[1]))

$\label{eq94}{{\left({r_{2}}-{r_{1}}+ 2 \right)}\ {{e}^{r_{1}}}}\over{{\left({r_{2}}-{r_{1}}\right)}^{3}}$

(94)

fricas

collect(rhs(eq6_3d), exp(r[2]))

$\label{eq95}{{\left({r_{2}}-{r_{1}}- 2 \right)}\ {{e}^{r_{2}}}}\over{{\left({r_{2}}-{r_{1}}\right)}^{3}}$

(95)

Exercise 6.4 (Triple root)

fricas

eval(eq4_1,[r[3]=r[2],r[4]=r[2]])

$\label{eq96}\begin{array}{@{}l} \displaystyle {m_{X}}={ \begin{array}{@{}l} \displaystyle {{x}^{4}}+{{\left(-{3 \ {r_{2}}}-{r_{1}}\right)}\ {{x}^{3}}}+{{\left({3 \ {{r_{2}}^{2}}}+{3 \ {r_{1}}\ {r_{2}}}\right)}\ {{x}^{2}}}+ \ \ \displaystyle {{\left(-{{r_{2}}^{3}}-{3 \ {r_{1}}\ {{r_{2}}^{2}}}\right)}\ x}+{{r_{1}}\ {{r_{2}}^{3}}}$

(96)

fricas

eq6_4a:=lhs eq4_2 = limit(limit(rhs eq4_2,r[3]=r[2]),r[4]=r[2])

$\label{eq97}\begin{array}{@{}l} \displaystyle {g_{0}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{r_{1}}\ {{r_{2}}^{4}}}+{{\left({2 \ {{r_{1}}^{2}}}+{4 \ {r_{1}}}\right)}\ {{r_{2}}^{3}}}+ \ \ \displaystyle {{\left(-{{r_{1}}^{3}}-{6 \ {{r_{1}}^{2}}}-{6 \ {r_{1}}}\right)}\ {{r_{2}}^{2}}}+ \ \ \displaystyle {{\left({2 \ {{r_{1}}^{3}}}+{6 \ {{r_{1}}^{2}}}\right)}\ {r_{2}}}-{2 \ {{r_{1}}^{3}}}$

(97)

fricas

collect(rhs(eq6_4a), exp(r[1]))

$\label{eq98}{{{r_{2}}^{3}}\ {{e}^{r_{1}}}}\over{{\left({r_{2}}-{r_{1}}\right)}^{3}}$

(98)

fricas

collect(rhs(eq6_4a), exp(r[2]))

$\label{eq99}\begin{array}{@{}l} \displaystyle -{{\left( \begin{array}{@{}l} \displaystyle {r_{1}}\ {\left({ \begin{array}{@{}l} \displaystyle {{r_{2}}^{4}}+{{\left(-{2 \ {r_{1}}}- 4 \right)}\ {{r_{2}}^{3}}}+ \ \ \displaystyle {{\left({{r_{1}}^{2}}+{6 \ {r_{1}}}+ 6 \right)}\ {{r_{2}}^{2}}}+ \ \ \displaystyle {{\left(-{2 \ {{r_{1}}^{2}}}-{6 \ {r_{1}}}\right)}\ {r_{2}}}+{2 \ {{r_{1}}^{2}}}$

(99)

fricas

eq6_4b:=lhs eq4_3 = limit(limit(rhs eq4_3,r[3]=r[2]),r[4]=r[2])

$\label{eq100}\begin{array}{@{}l} \displaystyle {g_{1}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{r_{2}}^{4}}-{4 \ {{r_{2}}^{3}}}+{{\left(-{3 \ {{r_{1}}^{2}}}+ 6 \right)}\ {{r_{2}}^{2}}}+ \ \ \displaystyle {{\left({2 \ {{r_{1}}^{3}}}+{6 \ {{r_{1}}^{2}}}\right)}\ {r_{2}}}-{2 \ {{r_{1}}^{3}}}$

(100)

fricas

collect(rhs(eq6_4b), exp(r[1]))

$\label{eq101}-{{6 \ {{r_{2}}^{2}}\ {{e}^{r_{1}}}}\over{2 \ {{\left({r_{2}}-{r_{1}}\right)}^{3}}}}$

(101)

fricas

collect(rhs(eq6_4b), exp(r[2]))

$\label{eq102}{\left({\left({ \begin{array}{@{}l} \displaystyle {{r_{2}}^{4}}-{4 \ {{r_{2}}^{3}}}+{{\left(-{3 \ {{r_{1}}^{2}}}+ 6 \right)}\ {{r_{2}}^{2}}}+ \ \ \displaystyle {{\left({2 \ {{r_{1}}^{3}}}+{6 \ {{r_{1}}^{2}}}\right)}\ {r_{2}}}-{2 \ {{r_{1}}^{3}}}$

(102)

fricas

eq6_4c:=lhs eq4_4 = limit(limit(rhs eq4_4,r[3]=r[2]),r[4]=r[2])

$\label{eq103}\begin{array}{@{}l} \displaystyle {g_{2}}={{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ {{r_{2}}^{3}}}+{{\left({3 \ {r_{1}}}+ 6 \right)}\ {{r_{2}}^{2}}}+ \ \ \displaystyle {{\left(-{6 \ {r_{1}}}- 6 \right)}\ {r_{2}}}-{{r_{1}}^{3}}$

(103)

fricas

collect(rhs(eq6_4c), exp(r[1]))

$\label{eq104}{6 \ {r_{2}}\ {{e}^{r_{1}}}}\over{2 \ {{\left({r_{2}}-{r_{1}}\right)}^{3}}}$

(104)

fricas

collect(rhs(eq6_4c), exp(r[2]))

$\label{eq105}-{{{\left({2 \ {{r_{2}}^{3}}}+{{\left(-{3 \ {r_{1}}}- 6 \right)}\ {{r_{2}}^{2}}}+{{\left({6 \ {r_{1}}}+ 6 \right)}\ {r_{2}}}+{{r_{1}}^{3}}\right)}\ {{e}^{r_{2}}}}\over{2 \ {{\left({r_{2}}-{r_{1}}\right)}^{3}}}}$

(105)

fricas

eq6_4d:=lhs eq4_5 = limit(limit(rhs eq4_5,r[3]=r[2]),r[4]=r[2])

$\label{eq106}{g_{3}}={{{{\left({{r_{2}}^{2}}+{{\left(-{2 \ {r_{1}}}- 2 \right)}\ {r_{2}}}+{{r_{1}}^{2}}+{2 \ {r_{1}}}+ 2 \right)}\ {{e}^{r_{2}}}}-{2 \ {{e}^{r_{1}}}}}\over{{2 \ {{r_{2}}^{3}}}-{6 \ {r_{1}}\ {{r_{2}}^{2}}}+{6 \ {{r_{1}}^{2}}\ {r_{2}}}-{2 \ {{r_{1}}^{3}}}}}$

(106)

fricas

collect(rhs(eq6_4d), exp(r[1]))

$\label{eq107}-{{{e}^{r_{1}}}\over{{\left({r_{2}}-{r_{1}}\right)}^{3}}}$

(107)

fricas

collect(rhs(eq6_4d), exp(r[2]))

$\label{eq108}{{\left({{r_{2}}^{2}}+{{\left(-{2 \ {r_{1}}}- 2 \right)}\ {r_{2}}}+{{r_{1}}^{2}}+{2 \ {r_{1}}}+ 2 \right)}\ {{e}^{r_{2}}}}\over{2 \ {{\left({r_{2}}-{r_{1}}\right)}^{3}}}$

(108)