︠1343af5b-691b-414b-a15b-9da65d32b0f0︠
var('x,n1,n2,n3,n4,n5')
w=[n1,n2,n3,n4,n5] #[w-1,w-2,w-3,...]
def phi(x,w): #returns iteration of the inverse Gauss map along trajectory w
if (len(w)>1):
return 1/(w[-1]+phi(x,w[0:-1]))
else:
return 1/(w[0]+x)
def dphidx(x,w): #returns returns expansion rate along finite trajectory w
ret=1
for j in range(len(w)):
ret*=-1*(phi(x,w[0:j+1]))^2
return ret
def S(x,w): #returns S_omega as defined in paper for a word of finite length
ret=0
for j in range(len(w)):
ret+=dphidx(x,w[0:j+1])*(-2/phi(x,w[0:j+1]))
return -retprint
def CF(w): #returns continuous fraction expansion of w
if (len(w)>1):
return 1/(w[0]+CF(w[1:]))
else:
return 1/(w[0])
show(S(x,w))
show(-2/(x+1/CF(w)))
g=S(x,w)+2/(x+1/CF(w))
show(g.full_simplify())
show('Thus the two terms above are the same!')
︡b330efff-84ca-48de-810a-14b52db1055e︡{"stdout": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2}{n_{1} + x} + \\frac{2}{{\\left(n_{2} + \\frac{1}{n_{1} + x}\\right)} {\\left(n_{1} + x\\right)}^{2}} - \\frac{2}{{\\left(n_{3} + \\frac{1}{n_{2} + \\frac{1}{n_{1} + x}}\\right)} {\\left(n_{2} + \\frac{1}{n_{1} + x}\\right)}^{2} {\\left(n_{1} + x\\right)}^{2}} + \\frac{2}{{\\left(n_{4} + \\frac{1}{n_{3} + \\frac{1}{n_{2} + \\frac{1}{n_{1} + x"}︡
︠b6e5ae6b-c0bc-41f8-b311-b5421a4ef16di︠
%html
right)} {\left(n_{3} + \frac{1}{n_{2} + \frac{1}{n_{1} + x}}\right)}^{2} {\left(n_{2} + \frac{1}{n_{1} + x}\right)}^{2} {\left(n_{1} + x\right)}^{2}} - \frac{2}{{\left(n_{5} + \frac{1}{n_{4} + \frac{1}{n_{3} + \frac{1}{n_{2} + \frac{1}{n_{1} + x}}}}\right)} {\left(n_{4} + \frac{1}{n_{3} + \frac{1}{n_{2} + \frac{1}{n_{1} + x}}}\right)}^{2} {\left(n_{3} + \frac{1}{n_{2} + \frac{1}{n_{1} + x}}\right)}^{2} {\left(n_{2} + \frac{1}{n_{1} + x}\right)}^{2} {\left(n_{1} + x\right)}^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2}{n_{1} + x + \frac{1}{n_{2} + \frac{1}{n_{3} + \frac{1}{n_{4} + \frac{1}{n_{5}}}}}}
\newcommand{\Bold}[1]{\mathbf{#1}}0
\newcommand{\Bold}[1]{\mathbf{#1}}\verb|Thus|\phantom{x}\verb|the|\phantom{x}\verb|two|\phantom{x}\verb|terms|\phantom{x}\verb|above|\phantom{x}\verb|are|\phantom{x}\verb|the|\phantom{x}\verb|same!|
}}}
︡114d0645-9dae-4bd0-878d-75d623b46c3c︡{"html": "right)} {\\left(n_{3} + \\frac{1}{n_{2} + \\frac{1}{n_{1} + x}}\\right)}^{2} {\\left(n_{2} + \\frac{1}{n_{1} + x}\\right)}^{2} {\\left(n_{1} + x\\right)}^{2}} - \\frac{2}{{\\left(n_{5} + \\frac{1}{n_{4} + \\frac{1}{n_{3} + \\frac{1}{n_{2} + \\frac{1}{n_{1} + x}}}}\\right)} {\\left(n_{4} + \\frac{1}{n_{3} + \\frac{1}{n_{2} + \\frac{1}{n_{1} + x}}}\\right)}^{2} {\\left(n_{3} + \\frac{1}{n_{2} + \\frac{1}{n_{1} + x}}\\right)}^{2} {\\left(n_{2} + \\frac{1}{n_{1} + x}\\right)}^{2} {\\left(n_{1} + x\\right)}^{2}}