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TowardSpikeInConformalGauge system:sage

Calculo transfor. que me llevan spikes Martin a Conformal gauge

Comprobemos que las soluciones de Martin no están en el gauge conforme 

#auto
sig, sig1, sig2, tau, omg, a, b, tta, r0, r1, x1, x2, u, x0n, x_1n, x1n, x2n  = var('sigma, sigma1, sigma2, tau, omega, a, b, theta, ro,r1, x1, x2, u, x0n, x_1n, x1n, x2n')

#auto
rho = function('rho', sig);
rho=arccosh(sqrt(tan(sig+pi/2)^2*cosh(2*r0)^2/(tan(sig+pi/2)^2-sinh(2*r0)^2)))/2
theta=tau+sig;
rho.show()
<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2
right) }}}
#auto
x0=cosh(rho)*cos(tau); x_1=cosh(rho)*sin(tau);
x1=sinh(rho)*cos(sig+tau); x2=sinh(rho)*sin(sig+tau);
x = vector([x0, x_1, x1, x2]);

for i in range(0,4):
    x[i].show()
<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(\tau\right) \cosh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2
right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\tau\right) \cosh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(\sigma + \tau\right) \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma + \tau\right) \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)
}}}
#auto
dtx=diff(x,tau);
dsx=diff(x,sig);

#auto
dtx2= -dtx[0]*dtx[0]-dtx[1]*dtx[1]+dtx[2]*dtx[2]+dtx[3]*dtx[3]; sdtx2 =dtx2.simplify_full(); sdtx2.show()
\newcommand{\Bold}[1]{\mathbf{#1}}-1
#auto
dsx2= -dsx[0]*dsx[0]-dsx[1]*dsx[1]+dsx[2]*dsx[2]+dsx[3]*dsx[3]; sdsx2 =dsx2.simplify_full(); sdsx2.show()
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(4 \, \cosh\left(\mbox{ro}\right)^{8} - 8 \, \cosh\left(\mbox{ro}\right)^{6} + 5 \, \cosh\left(\mbox{ro}\right)^{4} - \cosh\left(\mbox{ro}\right)^{2}\right)} \sin\left(\sigma\right)^{2} \cosh\left(\frac{1}{2} \, {\rm arccosh}\left(\frac{\sqrt{-\sin\left(\sigma\right) + 1} \sqrt{\sin\left(\sigma\right) + 1} {\left(2 \, \cosh\left(\mbox{ro}\right)^{2} - 1\right)}}{\sqrt{-{\left(2 \, \cosh\left(\mbox{ro}\right)^{2} - 1\right)} \sin\left(\sigma\right) + 1} \sqrt{{\left(2 \, \cosh\left(\mbox{ro}\right)^{2} - 1\right)} \sin\left(\sigma\right) + 1}}\right)\right)^{2} + {\left({\left(16 \, \cosh\left(\mbox{ro}\right)^{8} - 32 \, \cosh\left(\mbox{ro}\right)^{6} + 24 \, \cosh\left(\mbox{ro}\right)^{4} - 8 \, \cosh\left(\mbox{ro}\right)^{2} + 1\right)} \sin\left(\sigma\right)^{4} - {\left(4 \, \cosh\left(\mbox{ro}\right)^{8} - 8 \, \cosh\left(\mbox{ro}\right)^{6} + 13 \, \cosh\left(\mbox{ro}\right)^{4} - 9 \, \cosh\left(\mbox{ro}\right)^{2} + 2\right)} \sin\left(\sigma\right)^{2} + 1\right)} \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\frac{\sqrt{-\sin\left(\sigma\right) + 1} \sqrt{\sin\left(\sigma\right) + 1} {\left(2 \, \cosh\left(\mbox{ro}\right)^{2} - 1\right)}}{\sqrt{-{\left(2 \, \cosh\left(\mbox{ro}\right)^{2} - 1\right)} \sin\left(\sigma\right) + 1} \sqrt{{\left(2 \, \cosh\left(\mbox{ro}\right)^{2} - 1\right)} \sin\left(\sigma\right) + 1}}\right)\right)^{2}}{{\left(16 \, \cosh\left(\mbox{ro}\right)^{8} - 32 \, \cosh\left(\mbox{ro}\right)^{6} + 24 \, \cosh\left(\mbox{ro}\right)^{4} - 8 \, \cosh\left(\mbox{ro}\right)^{2} + 1\right)} \sin\left(\sigma\right)^{4} - 2 \, {\left(4 \, \cosh\left(\mbox{ro}\right)^{4} - 4 \, \cosh\left(\mbox{ro}\right)^{2} + 1\right)} \sin\left(\sigma\right)^{2} + 1}
#auto
dtxdsx= -dtx[0]*dsx[0]-dtx[1]*dsx[1]+dtx[2]*dsx[2]+dtx[3]*dsx[3]; sdtxdsx =dtxdsx.simplify_full(); sdtxdsx.show()
<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\frac{{\left(\sinh\left(\mbox{ro}\right)^{2} + \cosh\left(\mbox{ro}\right)^{2}\right)} {\left| \cos\left(\sigma\right) \right|}}{\sqrt{-4 \, \sinh\left(\mbox{ro}\right)^{2} \cosh\left(\mbox{ro}\right)^{2} + {\left(4 \, \sinh\left(\mbox{ro}\right)^{2} \cosh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma\right)^{2
right)\right)^{2} }}}

As a suppose, este anzats no esta en el gauge conforme

Induced metric on AdS3 (fast rotating string ω=1) 

#auto
t=tau;
sp = vector([t,rho, theta]);
g = matrix(SR,[[-cosh(rho)^2,0,0],[0,1,0],[0,0,sinh(rho)^2]]); pretty_print(g)
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} -\cosh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2
right)\right)^{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)^{2} \end{array}\right) }}}
#auto
dspt = diff(sp,tau);
dsps = diff(sp, sig);

#auto
h11=(dspt*g*dspt).simplify_full(); h12 = dspt*g*dsps; h21 = dsps*g*dspt; h22 =dsps*g*dsps;

#auto
h=matrix(SR,[[h11,h12],[h21,h22]]); pretty_print(h)
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} -1 & \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2
right)\right)^{2} \\ \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)^{2} & \sinh\left(\frac{1}{2} \, {\rm arccosh}\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}}\right)\right)^{2} + \frac{{\left(\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}\right)} {\left(\frac{{\left(\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} + 1\right)} \tan\left(\frac{1}{2} \, \pi + \sigma\right)^{3} \cosh\left(2 \, \mbox{ro}\right)^{2}}{{\left(\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}\right)}^{2}} + \frac{-{\left(\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} + 1\right)} \tan\left(\frac{1}{2} \, \pi + \sigma\right) \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}\right)}^{2}}{4 \, {\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}} - 1\right)} {\left(\sqrt{\frac{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}}{\tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} - \sinh\left(2 \, \mbox{ro}\right)^{2}}} + 1\right)} \tan\left(\frac{1}{2} \, \pi + \sigma\right)^{2} \cosh\left(2 \, \mbox{ro}\right)^{2}} \end{array}\right) }}}

∫ƒ(σ)dσ=σ' 

#Si la worldsheet es lorentziana

#auto
I1= integral(tan(sig)^2*r1^2/(tan(sig)^2-r0^2), sig); I1.simplify_trig().show()
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{r_{1}^{2} \mbox{ro} \log\left(\frac{-\mbox{ro} \cos\left(\sigma\right) - \sin\left(\sigma\right)}{\cos\left(\sigma\right)}\right) - r_{1}^{2} \mbox{ro} \log\left(\frac{\mbox{ro} \cos\left(\sigma\right) + \sin\left(\sigma\right)}{\cos\left(\sigma\right)}\right) + 2 \, r_{1}^{2} \arctan\left(\frac{\sin\left(\sigma\right)}{\cos\left(\sigma\right)}\right)}{2 \, {\left(\mbox{ro}^{2} + 1\right)}}
diff(I1,sig).simplify_full().show()
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{-r_{1}^{2} \sin\left(\sigma\right)^{2}}{\mbox{ro}^{2} \cos\left(\sigma\right)^{2} - \sin\left(\sigma\right)^{2}}
#auto
#I5= integral(sinh(2*r0)/(sinh(2*u)*sqrt(sinh(2*u)^2-sinh(2*r0)^2)), (u, r0, r1) ); I5.show()

∫ƒ(σ)dτ=τ' 

#auto
assume(1-a^2<0)

#auto
I2= integral(1/(a*cosh(2*sig)+1), sig); I2.show()
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\arctan\left(\frac{a e^{\left(2 \, \sigma\right)} + 1}{\sqrt{a^{2} - 1}}\right)}{\sqrt{a^{2} - 1}}
#auto
I2.diff(sig).show()
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, a e^{\left(2 \, \sigma\right)}}{{\left(\frac{{\left(a e^{\left(2 \, \sigma\right)} + 1\right)}^{2}}{a^{2} - 1} + 1\right)} {\left(a^{2} - 1\right)}}

Metrica inducida en las nuevas coordenadas (moño) 

#auto
tau1 =(sig1+arctan((cosh(2*r0)-1)*tanh(sig2)/sinh(2*r0))).simplify_full();
rho1 = (arccosh(sqrt((cosh(2*r0)*cosh(2*sig2)+1)/2))).simplify_full();
tta1 =sig1+arctan((cosh(2*r0)-1)*tanh(sig2)/sinh(2*r0))+arctan(-sinh(2*r0)/tanh(2*sig2)); 
tau1.show(); rho1.show(); tta1.show()
\newcommand{\Bold}[1]{\mathbf{#1}}\sigma_{1} + \arctan\left(\frac{\sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)}{\cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm arccosh}\left(\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sigma_{1} + \arctan\left(\frac{{\left(\cosh\left(2 \, \mbox{ro}\right) - 1\right)} \tanh\left(\sigma_{2}\right)}{\sinh\left(2 \, \mbox{ro}\right)}\right) + \arctan\left(\frac{-\sinh\left(2 \, \mbox{ro}\right)}{\tanh\left(2 \, \sigma_{2}\right)}\right)
#auto
spn = vector([tau1,rho1, tta1]); 
gn = matrix(SR,[[-cosh(rho1)^2,0,0],[0,1,0],[0,0,sinh(rho1)^2]]); pretty_print(gn)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(−(2 sinh⁡(ro)2+1)cosh⁡(σ2)2+sinh⁡(ro)20001000((2 sinh⁡(ro)2+1)cosh⁡(σ2)2−sinh⁡(ro)2−1)((2 sinh⁡(ro)2+1)cosh⁡(σ2)2−sinh⁡(ro)2+1)\begin{array}{rrr} -{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} + \sinh\left(\mbox{ro}\right)^{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & {\left(\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} - 1\right)} {\left(\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} + 1\right)} \end{array}\right)
#auto
dspns1 = diff(spn,sig1);
dspns2 = diff(spn,sig2);

#auto
for in range(0,5):
Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_22.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Zm9yIGluIHJhbmdlKDAsNSk6"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpvVazdC/___code___.py", line 3 for in range(_sage_const_0 ,_sage_const_5 ): ^ SyntaxError: invalid syntax
#auto
h11n=(dspns1*gn*dspns1).simplify_full(); h12n = (dspns1*gn*dspns2).simplify_full(); h21n = (dspns2*gn*dspns1).simplify_full(); h22n =(dspns2*gn*dspns2).simplify_full();

#auto
hn=matrix(SR,[[h11n,h12n],[h21n,h22n]]); pretty_print(hn)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(−1001\begin{array}{rr} -1 & 0 \\ 0 & 1 \end{array}\right)

:) 

#auto
signa=arctan(-sinh(2*r0)/tanh(2*sig2));

#auto
xdxp=-diff(tau1, sig2)+diff(signa, sig2)*(cosh(2*r0)*cosh(2*sig2)-1)/2; xdxp.show()
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\frac{\sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)^{2}}{\cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)^{2}} + \frac{-\sinh\left(\mbox{ro}\right)}{\cosh\left(\mbox{ro}\right)}}{\frac{\sinh\left(\mbox{ro}\right)^{2} \sinh\left(\sigma_{2}\right)^{2}}{\cosh\left(\mbox{ro}\right)^{2} \cosh\left(\sigma_{2}\right)^{2}} + 1} + \frac{-{\left(\tanh\left(2 \, \sigma_{2}\right)^{2} - 1\right)} {\left(\cosh\left(2 \, \mbox{ro}\right) \cosh\left(2 \, \sigma_{2}\right) - 1\right)} \sinh\left(2 \, \mbox{ro}\right)}{{\left(\frac{\sinh\left(2 \, \mbox{ro}\right)^{2}}{\tanh\left(2 \, \sigma_{2}\right)^{2}} + 1\right)} \tanh\left(2 \, \sigma_{2}\right)^{2}}
#auto
#xdxp.simplify_full().show()

calculemos las restricciones de virasoro en las nuevas coordenadas 

#auto
x0n=cosh(rho1)*cos(tau1); x_1n=cosh(rho1)*sin(tau1);
x1n=sinh(rho1)*cos(tta1); x2n=sinh(rho1)*sin(tta1);
xn = vector([x0n, x_1n, x1n, x2n]);

for i in range(0,4):
    xn[i].show()
\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} \cos\left(\sigma_{1} + \arctan\left(\frac{\sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)}{\cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)}\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} \sin\left(\sigma_{1} + \arctan\left(\frac{\sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)}{\cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)}\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} - 1} \sqrt{\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} + 1} \cos\left(\sigma_{1} + \arctan\left(\frac{{\left(\cosh\left(2 \, \mbox{ro}\right) - 1\right)} \tanh\left(\sigma_{2}\right)}{\sinh\left(2 \, \mbox{ro}\right)}\right) + \arctan\left(\frac{-\sinh\left(2 \, \mbox{ro}\right)}{\tanh\left(2 \, \sigma_{2}\right)}\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} - 1} \sqrt{\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} + 1} \sin\left(\sigma_{1} + \arctan\left(\frac{{\left(\cosh\left(2 \, \mbox{ro}\right) - 1\right)} \tanh\left(\sigma_{2}\right)}{\sinh\left(2 \, \mbox{ro}\right)}\right) + \arctan\left(\frac{-\sinh\left(2 \, \mbox{ro}\right)}{\tanh\left(2 \, \sigma_{2}\right)}\right)\right)
#auto
cos(tau1).simplify_full().show(); sin(tau1).simplify_full().show();
<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\frac{-\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right) - \cos\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)}{\sqrt{\sinh\left(\mbox{ro}\right)^{2} \sinh\left(\sigma_{2}\right)^{2} + \cosh\left(\mbox{ro}\right)^{2} \cosh\left(\sigma_{2}\right)^{2
/div>
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)}{\sqrt{\sinh\left(\mbox{ro}\right)^{2} \sinh\left(\sigma_{2}\right)^{2} + \cosh\left(\mbox{ro}\right)^{2} \cosh\left(\sigma_{2}\right)^{2}}}
}}}
#auto
assume(sinh(r0)>0);
(cos(tta1)).simplify_full().factor().simplify_full().show()
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)} {\left| \sinh\left(\sigma_{2}\right) \right|}}{\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2} - 1} \sinh\left(\sigma_{2}\right)}
#auto
assume(sinh(r0)>0);
(sin(tta1)).simplify_full().factor().simplify_full().show()
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left(\sin\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right) - \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)\right)} {\left| \sinh\left(\sigma_{2}\right) \right|}}{\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2} - 1} \sinh\left(\sigma_{2}\right)}

coordenadas en el gauge conforme simplificadas 

#auto
x0s=-(sin(sigma1)*sinh(ro)*sinh(sigma2) -cos(sigma1)*cosh(ro)*cosh(sigma2));
x_1s=(sin(sigma1)*cosh(ro)*cosh(sigma2) +cos(sigma1)*sinh(ro)*sinh(sigma2));
x1s=(sin(sigma1)*sinh(ro)*cosh(sigma2) +cos(sigma1)*sinh(sigma2)*cosh(ro));
x2s=(sin(sigma1)*sinh(sigma2)*cosh(ro) -cos(sigma1)*sinh(ro)*cosh(sigma2));
xs = vector([x0s, x_1s, x1s, x2s]);

for i in range(0,4):
    xs[i].show()
\newcommand{\Bold}[1]{\mathbf{#1}}-\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right) - \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)
#auto
dtxn=diff(xs,sig1); 
dsxn=diff(xs,sig2);
dtxn;
(-sin(sigma1)*cosh(ro)*cosh(sigma2) - cos(sigma1)*sinh(ro)*sinh(sigma2), -sin(sigma1)*sinh(ro)*sinh(sigma2) + cos(sigma1)*cosh(ro)*cosh(sigma2), -sin(sigma1)*sinh(sigma2)*cosh(ro) + cos(sigma1)*sinh(ro)*cosh(sigma2), sin(sigma1)*sinh(ro)*cosh(sigma2) + cos(sigma1)*sinh(sigma2)*cosh(ro))
#auto
dsxn[1].simplify().show();
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)
#auto
dtxndsxn= -dtxn[0]*dsxn[0]-dtxn[1]*dsxn[1]+dtxn[2]*dsxn[2]+dtxn[3]*dsxn[3]; dtxndsxn.show()
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\sin\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) - \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)\right)} {\left(\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)} - {\left(\sin\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)\right)} {\left(\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) - \cos\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)} - {\left(\sin\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right) - \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)\right)} {\left(\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)\right)} + {\left(\sin\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)\right)} {\left(\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right) - \cos\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)\right)}
#auto
sdtxndsxn =dtxndsxn.simplify_full(); sdtxndsxn.show()
\newcommand{\Bold}[1]{\mathbf{#1}}0
#auto
dtxn2= -dtxn[0]*dtxn[0]-dtxn[1]*dtxn[1]+dtxn[2]*dtxn[2]+dtxn[3]*dtxn[3]; sdtxn2 =dtxn2.simplify_full(); sdtxn2.show()
\newcommand{\Bold}[1]{\mathbf{#1}}-1
#auto
dsxn2= -dsxn[0]*dsxn[0]-dsxn[1]*dsxn[1]+dsxn[2]*dsxn[2]+dsxn[3]*dsxn[3]; sdsxn2 =dsxn2.simplify_full(); sdsxn2.show()
\newcommand{\Bold}[1]{\mathbf{#1}}1
sdtxn2+sdsxn2
0
N(ln(4));
N(ln(1/4))
Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_84.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Tihsbig0KSk7Ck4obG4oMS80KSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpaq3CE6/___code___.py", line 3, in <module> N(ln(_sage_const_4 )); File "free_module_element.pyx", line 2555, in sage.modules.free_module_element.FreeModuleElement_generic_dense.__call__ (sage/modules/free_module_element.c:17127) File "expression.pyx", line 3426, in sage.symbolic.expression.Expression.__call__ (sage/symbolic/expression.cpp:15476) File "ring.pyx", line 638, in sage.symbolic.ring.SymbolicRing._call_element_ (sage/symbolic/ring.cpp:6460) ValueError: the number of arguments must be less than or equal to 0
#parametric_plot((cos(u),sin(u)^3),(u,0,2*pi),rgbcolor=hue(0.6))

plot(sinh(1)/tanh(2*u), (u,1,3))
Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_86.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cGxvdChzaW5oKDEpL3RhbmgoMip1KSwgKHUsMSwzKSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmp1WAgB2/___code___.py", line 3, in <module> exec compile(u'plot(sinh(_sage_const_1 )/tanh(_sage_const_2 *u), (u,_sage_const_1 ,_sage_const_3 ))' + '\n', '', 'single') File "", line 1, in <module> NameError: name 'u' is not defined
plot(arctan(sinh(1)/tanh(2*u)), (u,-2,2))
Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_87.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cGxvdChhcmN0YW4oc2luaCgxKS90YW5oKDIqdSkpLCAodSwtMiwyKSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpk6NsNd/___code___.py", line 3, in <module> exec compile(u'plot(arctan(sinh(_sage_const_1 )/tanh(_sage_const_2 *u)), (u,-_sage_const_2 ,_sage_const_2 ))' + '\n', '', 'single') File "", line 1, in <module> NameError: name 'u' is not defined
plot(arctan(u),(-100,100))
Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_88.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cGxvdChhcmN0YW4odSksKC0xMDAsMTAwKSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpMHyhP0/___code___.py", line 3, in <module> exec compile(u'plot(arctan(u),(-_sage_const_100 ,_sage_const_100 ))' + '\n', '', 'single') File "", line 1, in <module> NameError: name 'u' is not defined
<h3 style="color: #800080;">Intentemos respetir las cuentas en el caso coordenadas worldsheet z (metrica inducida euclidea) </h3>
Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_90.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("PGgzIHN0eWxlPSJjb2xvcjogIzgwMDA4MDsiPkludGVudGVtb3MgcmVzcGV0aXIgbGFzIGN1ZW50YXMgZW4gZWwgY2FzbyBjb29yZGVuYWRhcyB3b3JsZHNoZWV0IHogKG1ldHJpY2EgaW5kdWNpZGEgZXVjbGlkZWEpIDwvaDM+"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpcmku9g/___code___.py", line 2 <h3 style="color: #800080;">Intentemos respetir las cuentas en el caso coordenadas worldsheet z (metrica inducida euclidea) </h3> ^ SyntaxError: invalid syntax
#integral tau(moño_sigma)

#auto
Ie1= integral(1/(a*cos(2*sig)+1), sig); Ie1
Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_92.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("SWUxPSBpbnRlZ3JhbCgxLyhhKmNvcygyKnNpZykrMSksIHNpZyk7IEllMQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpQfWS49/___code___.py", line 3, in <module> exec compile(u'Ie1= integral(_sage_const_1 /(a*cos(_sage_const_2 *sig)+_sage_const_1 ), sig); Ie1' + '\n', '', 'single') File "", line 1, in <module> NameError: name 'a' is not defined
(log(((a - 1)*sin(2*sigma)/(cos(2*sigma) + 1) - sqrt(a^2 - 1))/((a -
1)*sin(2*sigma)/(cos(2*sigma) + 1) + sqrt(a^2 - 1)))).simplify_full().show()
Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_93.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("KGxvZygoKGEgLSAxKSpzaW4oMipzaWdtYSkvKGNvcygyKnNpZ21hKSArIDEpIC0gc3FydChhXjIgLSAxKSkvKChhIC0KMSkqc2luKDIqc2lnbWEpLyhjb3MoMipzaWdtYSkgKyAxKSArIHNxcnQoYV4yIC0gMSkpKSkuc2ltcGxpZnlfZnVsbCgpLnNob3coKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpxT2arr/___code___.py", line 3, in <module> (log(((a - _sage_const_1 )*sin(_sage_const_2 *sigma)/(cos(_sage_const_2 *sigma) + _sage_const_1 ) - sqrt(a**_sage_const_2 - _sage_const_1 ))/((a - NameError: name 'a' is not defined
#auto
tae =(sig1-(1/2)*ln((-sinh(r0)*sin(sig2)+cosh(r0)*cos(sig2))/(sinh(r0)*sin(sig2)+cosh(r0)*cos(sig2)))).simplify_full();
rhoe = (arccosh(sqrt((cosh(2*r0)*cos(2*sig2)+1)/2))).simplify_full();
ttae =(sig1-(1/2)*ln((-sinh(r0)*sin(sig2)+cosh(r0)*cos(sig2))/(sinh(r0)*sin(sig2)+cosh(r0)*cos(sig2))))+arctan(-I*sinh(2*r0)/tan(2*sig2)); 
tae.show(); rhoe.show(); ttae.show()
\newcommand{\Bold}[1]{\mathbf{#1}}\sigma_{1} - \frac{1}{2} \, \log\left(-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right) + \frac{1}{2} \, \log\left(\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm arccosh}\left(\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sigma_{1} - \frac{1}{2} \, \log\left(\frac{-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) - \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)}{\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)}\right) + \arctan\left(\frac{-i \, \sinh\left(2 \, \mbox{ro}\right)}{\tan\left(2 \, \sigma_{2}\right)}\right)
#auto
spe = vector([tae,rhoe, ttae]); 
ge = matrix(SR,[[-cosh(rho1)^2,0,0],[0,1,0],[0,0,sinh(rho1)^2]]); pretty_print(ge)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(−(2 sinh⁡(ro)2+1)cosh⁡(σ2)2+sinh⁡(ro)20001000((2 sinh⁡(ro)2+1)cosh⁡(σ2)2−sinh⁡(ro)2−1)((2 sinh⁡(ro)2+1)cosh⁡(σ2)2−sinh⁡(ro)2+1)\begin{array}{rrr} -{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} + \sinh\left(\mbox{ro}\right)^{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & {\left(\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} - 1\right)} {\left(\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cosh\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} + 1\right)} \end{array}\right)
#auto
dspes1 = diff(spe,sig1);
dspes2 = diff(spe,sig2);

#auto
h11e=(dspes1*ge*dspes1).simplify_full(); h12e = (dspes1*ge*dspes2).simplify_full(); 
h21e =(dspes2*ge*dspes1).simplify_full(); h22e =(dspes2*ge*dspes2).simplify_full().factor().simplify_full();

#auto
he=matrix(SR,[[h11e,h12e],[h21e,h22e]]); pretty_print(he)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(−1(i+1) sinh⁡(ro)5cosh⁡(ro)+(4 sinh⁡(ro)5cosh⁡(ro)+4 sinh⁡(ro)3cosh⁡(ro)+sinh⁡(ro)cosh⁡(ro))sin⁡(σ2)4+(i+1) sinh⁡(ro)3cosh⁡(ro)−((2i+4) sinh⁡(ro)5cosh⁡(ro)+(i+4) sinh⁡(ro)3cosh⁡(ro)+sinh⁡(ro)cosh⁡(ro))sin⁡(σ2)2−((−2i) sinh⁡(ro)5cosh⁡(ro)+(−3i) sinh⁡(ro)3cosh⁡(ro)+((4i) sinh⁡(ro)5cosh⁡(ro)+(4i) sinh⁡(ro)3cosh⁡(ro)+i sinh⁡(ro)cosh⁡(ro))sin⁡(σ2)2−i sinh⁡(ro)cosh⁡(ro))sinh⁡(σ2)2sin⁡(σ2)6sinh⁡(ro)4cosh⁡(ro)2−cos⁡(σ2)6sinh⁡(ro)2cosh⁡(ro)4−(2 sinh⁡(ro)4cosh⁡(ro)2+(cosh⁡(ro)4+1)sinh⁡(ro)2)sin⁡(σ2)4cos⁡(σ2)2+(sinh⁡(ro)4cosh⁡(ro)2+2 sinh⁡(ro)2cosh⁡(ro)4+cosh⁡(ro)2)sin⁡(σ2)2cos⁡(σ2)4(i+1) sinh⁡(ro)5cosh⁡(ro)+(4 sinh⁡(ro)5cosh⁡(ro)+4 sinh⁡(ro)3cosh⁡(ro)+sinh⁡(ro)cosh⁡(ro))sin⁡(σ2)4+(i+1) sinh⁡(ro)3cosh⁡(ro)−((2i+4) sinh⁡(ro)5cosh⁡(ro)+(i+4) sinh⁡(ro)3cosh⁡(ro)+sinh⁡(ro)cosh⁡(ro))sin⁡(σ2)2−((−2i) sinh⁡(ro)5cosh⁡(ro)+(−3i) sinh⁡(ro)3cosh⁡(ro)+((4i) sinh⁡(ro)5cosh⁡(ro)+(4i) sinh⁡(ro)3cosh⁡(ro)+i sinh⁡(ro)cosh⁡(ro))sin⁡(σ2)2−i sinh⁡(ro)cosh⁡(ro))sinh⁡(σ2)2sin⁡(σ2)6sinh⁡(ro)4cosh⁡(ro)2−cos⁡(σ2)6sinh⁡(ro)2cosh⁡(ro)4−(2 sinh⁡(ro)4cosh⁡(ro)2+(cosh⁡(ro)4+1)sinh⁡(ro)2)sin⁡(σ2)4cos⁡(σ2)2+(sinh⁡(ro)4cosh⁡(ro)2+2 sinh⁡(ro)2cosh⁡(ro)4+cosh⁡(ro)2)sin⁡(σ2)2cos⁡(σ2)4−(256 cosh⁡(ro)16−1024 cosh⁡(ro)14+1792 cosh⁡(ro)12−1792 cosh⁡(ro)10+1120 cosh⁡(ro)8−448 cosh⁡(ro)6+112 cosh⁡(ro)4−16 cosh⁡(ro)2+1)cos⁡(σ2)16+(2i+1) cosh⁡(ro)16−(1024 cosh⁡(ro)16−4224 cosh⁡(ro)14+7616 cosh⁡(ro)12−7840 cosh⁡(ro)10+5040 cosh⁡(ro)8−2072 cosh⁡(ro)6+532 cosh⁡(ro)4−78 cosh⁡(ro)2+5)cos⁡(σ2)14+(−8i−5) cosh⁡(ro)14+(1792 cosh⁡(ro)16−7616 cosh⁡(ro)14+14080 cosh⁡(ro)12−14800 cosh⁡(ro)10+9680 cosh⁡(ro)8−4036 cosh⁡(ro)6+1048 cosh⁡(ro)4−155 cosh⁡(ro)2+10)cos⁡(σ2)12+(12i+10) cosh⁡(ro)12+((−64i−1792) cosh⁡(ro)16+(224i+7808) cosh⁡(ro)14+(−320i−14688) cosh⁡(ro)12+(240i+15600) cosh⁡(ro)10+(−100i−10240) cosh⁡(ro)8+(22i+4256) cosh⁡(ro)6+(−2i−1094) cosh⁡(ro)4+159 cosh⁡(ro)2−10)cos⁡(σ2)10+(−8i−10) cosh⁡(ro)10+((160i+1120) cosh⁡(ro)16+(−576i−4976) cosh⁡(ro)14+(848i+9440) cosh⁡(ro)12+(−656i−9992) cosh⁡(ro)10+(282i+6454) cosh⁡(ro)8+(−64i−2603) cosh⁡(ro)6+(6i+639) cosh⁡(ro)4−87 cosh⁡(ro)2+5)cos⁡(σ2)8+(2i+5) cosh⁡(ro)8+((−160i−448) cosh⁡(ro)16+(592i+2024) cosh⁡(ro)14+(−888i−3844) cosh⁡(ro)12+(692i+3994) cosh⁡(ro)10+(−296i−2471) cosh⁡(ro)8+(66i+925) cosh⁡(ro)6+(−6i−202) cosh⁡(ro)4+23 cosh⁡(ro)2−1)cos⁡(σ2)6−cosh⁡(ro)6+((80i+112) cosh⁡(ro)16+(−304i−516) cosh⁡(ro)14+(460i+980) cosh⁡(ro)12+(−352i−989) cosh⁡(ro)10+(142i+569) cosh⁡(ro)8+(−28i−185) cosh⁡(ro)6+(2i+31) cosh⁡(ro)4−2 cosh⁡(ro)2)cos⁡(σ2)4+((−20i−16) cosh⁡(ro)16+(78i+76) cosh⁡(ro)14+(−118i−146) cosh⁡(ro)12+(86i+144) cosh⁡(ro)10+(−30i−76) cosh⁡(ro)8+(4i+20) cosh⁡(ro)6−2 cosh⁡(ro)4)cos⁡(σ2)2+((−4i) cosh⁡(ro)16+(18i+2) cosh⁡(ro)14+(−32i−9) cosh⁡(ro)12+((128i) cosh⁡(ro)16+(−512i) cosh⁡(ro)14+(864i) cosh⁡(ro)12+(−800i) cosh⁡(ro)10+(440i) cosh⁡(ro)8+(−144i) cosh⁡(ro)6+(26i) cosh⁡(ro)4+(−2i) cosh⁡(ro)2)cos⁡(σ2)10+(28i+16) cosh⁡(ro)10+((−320i) cosh⁡(ro)16+(1312i+32) cosh⁡(ro)14+(−2272i−112) cosh⁡(ro)12+(2160i+160) cosh⁡(ro)10+(−1220i−120) cosh⁡(ro)8+(410i+50) cosh⁡(ro)6+(−76i−11) cosh⁡(ro)4+(6i+1) cosh⁡(ro)2)cos⁡(σ2)8+(−12i−14) cosh⁡(ro)8+((320i) cosh⁡(ro)16+(−1344i−64) cosh⁡(ro)14+(2368i+240) cosh⁡(ro)12+(−2272i−368) cosh⁡(ro)10+(1284i+296) cosh⁡(ro)8+(−428i−132) cosh⁡(ro)6+(78i+31) cosh⁡(ro)4+(−6i−3) cosh⁡(ro)2)cos⁡(σ2)6+(2i+6) cosh⁡(ro)6+((−160i) cosh⁡(ro)16+(688i+48) cosh⁡(ro)14+(−1224i−192) cosh⁡(ro)12+(1164i+312) cosh⁡(ro)10+(−636i−264) cosh⁡(ro)8+(198i+123) cosh⁡(ro)6+(−32i−30) cosh⁡(ro)4+(2i+3) cosh⁡(ro)2)cos⁡(σ2)4−cosh⁡(ro)4+((40i) cosh⁡(ro)16+(−176i−16) cosh⁡(ro)14+(314i+68) cosh⁡(ro)12+(−290i−116) cosh⁡(ro)10+(146i+101) cosh⁡(ro)8+(−38i−47) cosh⁡(ro)6+(4i+11) cosh⁡(ro)4−cosh⁡(ro)2)cos⁡(σ2)2)cosh⁡(σ2)2(256 sinh⁡(ro)16+1024 sinh⁡(ro)14+1792 sinh⁡(ro)12+1792 sinh⁡(ro)10+1120 sinh⁡(ro)8+448 sinh⁡(ro)6+112 sinh⁡(ro)4+16 sinh⁡(ro)2+1)cos⁡(σ2)16+sinh⁡(ro)16−(1024 sinh⁡(ro)16+3968 sinh⁡(ro)14+6720 sinh⁡(ro)12+6496 sinh⁡(ro)10+3920 sinh⁡(ro)8+1512 sinh⁡(ro)6+364 sinh⁡(ro)4+50 sinh⁡(ro)2+3)cos⁡(σ2)14+3 sinh⁡(ro)14+(1792 sinh⁡(ro)16+6720 sinh⁡(ro)14+10944 sinh⁡(ro)12+10096 sinh⁡(ro)10+5760 sinh⁡(ro)8+2076 sinh⁡(ro)6+460 sinh⁡(ro)4+57 sinh⁡(ro)2+3)cos⁡(σ2)12+3 sinh⁡(ro)12−(1792 sinh⁡(ro)16+6496 sinh⁡(ro)14+10096 sinh⁡(ro)12+8752 sinh⁡(ro)10+4600 sinh⁡(ro)8+1486 sinh⁡(ro)6+283 sinh⁡(ro)4+28 sinh⁡(ro)2+1)cos⁡(σ2)10+sinh⁡(ro)10+5 (224 sinh⁡(ro)16+784 sinh⁡(ro)14+1152 sinh⁡(ro)12+920 sinh⁡(ro)10+430 sinh⁡(ro)8+117 sinh⁡(ro)6+17 sinh⁡(ro)4+sinh⁡(ro)2)cos⁡(σ2)8−(448 sinh⁡(ro)16+1512 sinh⁡(ro)14+2076 sinh⁡(ro)12+1486 sinh⁡(ro)10+585 sinh⁡(ro)8+120 sinh⁡(ro)6+10 sinh⁡(ro)4)cos⁡(σ2)6+(112 sinh⁡(ro)16+364 sinh⁡(ro)14+460 sinh⁡(ro)12+283 sinh⁡(ro)10+85 sinh⁡(ro)8+10 sinh⁡(ro)6)cos⁡(σ2)4−(16 sinh⁡(ro)16+50 sinh⁡(ro)14+57 sinh⁡(ro)12+28 sinh⁡(ro)10+5 sinh⁡(ro)8)cos⁡(σ2)2\begin{array}{rr} -1 & \frac{\left(i + 1\right) \, \sinh\left(\mbox{ro}\right)^{5} \cosh\left(\mbox{ro}\right) + {\left(4 \, \sinh\left(\mbox{ro}\right)^{5} \cosh\left(\mbox{ro}\right) + 4 \, \sinh\left(\mbox{ro}\right)^{3} \cosh\left(\mbox{ro}\right) + \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)\right)} \sin\left(\sigma_{2}\right)^{4} + \left(i + 1\right) \, \sinh\left(\mbox{ro}\right)^{3} \cosh\left(\mbox{ro}\right) - {\left(\left(2 i + 4\right) \, \sinh\left(\mbox{ro}\right)^{5} \cosh\left(\mbox{ro}\right) + \left(i + 4\right) \, \sinh\left(\mbox{ro}\right)^{3} \cosh\left(\mbox{ro}\right) + \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)\right)} \sin\left(\sigma_{2}\right)^{2} - {\left(\left(-2 i\right) \, \sinh\left(\mbox{ro}\right)^{5} \cosh\left(\mbox{ro}\right) + \left(-3 i\right) \, \sinh\left(\mbox{ro}\right)^{3} \cosh\left(\mbox{ro}\right) + {\left(\left(4 i\right) \, \sinh\left(\mbox{ro}\right)^{5} \cosh\left(\mbox{ro}\right) + \left(4 i\right) \, \sinh\left(\mbox{ro}\right)^{3} \cosh\left(\mbox{ro}\right) + i \, \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)\right)} \sin\left(\sigma_{2}\right)^{2} - i \, \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)\right)} \sinh\left(\sigma_{2}\right)^{2}}{\sin\left(\sigma_{2}\right)^{6} \sinh\left(\mbox{ro}\right)^{4} \cosh\left(\mbox{ro}\right)^{2} - \cos\left(\sigma_{2}\right)^{6} \sinh\left(\mbox{ro}\right)^{2} \cosh\left(\mbox{ro}\right)^{4} - {\left(2 \, \sinh\left(\mbox{ro}\right)^{4} \cosh\left(\mbox{ro}\right)^{2} + {\left(\cosh\left(\mbox{ro}\right)^{4} + 1\right)} \sinh\left(\mbox{ro}\right)^{2}\right)} \sin\left(\sigma_{2}\right)^{4} \cos\left(\sigma_{2}\right)^{2} + {\left(\sinh\left(\mbox{ro}\right)^{4} \cosh\left(\mbox{ro}\right)^{2} + 2 \, \sinh\left(\mbox{ro}\right)^{2} \cosh\left(\mbox{ro}\right)^{4} + \cosh\left(\mbox{ro}\right)^{2}\right)} \sin\left(\sigma_{2}\right)^{2} \cos\left(\sigma_{2}\right)^{4}} \\ \frac{\left(i + 1\right) \, \sinh\left(\mbox{ro}\right)^{5} \cosh\left(\mbox{ro}\right) + {\left(4 \, \sinh\left(\mbox{ro}\right)^{5} \cosh\left(\mbox{ro}\right) + 4 \, \sinh\left(\mbox{ro}\right)^{3} \cosh\left(\mbox{ro}\right) + \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)\right)} \sin\left(\sigma_{2}\right)^{4} + \left(i + 1\right) \, \sinh\left(\mbox{ro}\right)^{3} \cosh\left(\mbox{ro}\right) - {\left(\left(2 i + 4\right) \, \sinh\left(\mbox{ro}\right)^{5} \cosh\left(\mbox{ro}\right) + \left(i + 4\right) \, \sinh\left(\mbox{ro}\right)^{3} \cosh\left(\mbox{ro}\right) + \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)\right)} \sin\left(\sigma_{2}\right)^{2} - {\left(\left(-2 i\right) \, \sinh\left(\mbox{ro}\right)^{5} \cosh\left(\mbox{ro}\right) + \left(-3 i\right) \, \sinh\left(\mbox{ro}\right)^{3} \cosh\left(\mbox{ro}\right) + {\left(\left(4 i\right) \, \sinh\left(\mbox{ro}\right)^{5} \cosh\left(\mbox{ro}\right) + \left(4 i\right) \, \sinh\left(\mbox{ro}\right)^{3} \cosh\left(\mbox{ro}\right) + i \, \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)\right)} \sin\left(\sigma_{2}\right)^{2} - i \, \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)\right)} \sinh\left(\sigma_{2}\right)^{2}}{\sin\left(\sigma_{2}\right)^{6} \sinh\left(\mbox{ro}\right)^{4} \cosh\left(\mbox{ro}\right)^{2} - \cos\left(\sigma_{2}\right)^{6} \sinh\left(\mbox{ro}\right)^{2} \cosh\left(\mbox{ro}\right)^{4} - {\left(2 \, \sinh\left(\mbox{ro}\right)^{4} \cosh\left(\mbox{ro}\right)^{2} + {\left(\cosh\left(\mbox{ro}\right)^{4} + 1\right)} \sinh\left(\mbox{ro}\right)^{2}\right)} \sin\left(\sigma_{2}\right)^{4} \cos\left(\sigma_{2}\right)^{2} + {\left(\sinh\left(\mbox{ro}\right)^{4} \cosh\left(\mbox{ro}\right)^{2} + 2 \, \sinh\left(\mbox{ro}\right)^{2} \cosh\left(\mbox{ro}\right)^{4} + \cosh\left(\mbox{ro}\right)^{2}\right)} \sin\left(\sigma_{2}\right)^{2} \cos\left(\sigma_{2}\right)^{4}} & \frac{-{\left(256 \, \cosh\left(\mbox{ro}\right)^{16} - 1024 \, \cosh\left(\mbox{ro}\right)^{14} + 1792 \, \cosh\left(\mbox{ro}\right)^{12} - 1792 \, \cosh\left(\mbox{ro}\right)^{10} + 1120 \, \cosh\left(\mbox{ro}\right)^{8} - 448 \, \cosh\left(\mbox{ro}\right)^{6} + 112 \, \cosh\left(\mbox{ro}\right)^{4} - 16 \, \cosh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma_{2}\right)^{16} + \left(2 i + 1\right) \, \cosh\left(\mbox{ro}\right)^{16} - {\left(1024 \, \cosh\left(\mbox{ro}\right)^{16} - 4224 \, \cosh\left(\mbox{ro}\right)^{14} + 7616 \, \cosh\left(\mbox{ro}\right)^{12} - 7840 \, \cosh\left(\mbox{ro}\right)^{10} + 5040 \, \cosh\left(\mbox{ro}\right)^{8} - 2072 \, \cosh\left(\mbox{ro}\right)^{6} + 532 \, \cosh\left(\mbox{ro}\right)^{4} - 78 \, \cosh\left(\mbox{ro}\right)^{2} + 5\right)} \cos\left(\sigma_{2}\right)^{14} + \left(-8 i - 5\right) \, \cosh\left(\mbox{ro}\right)^{14} + {\left(1792 \, \cosh\left(\mbox{ro}\right)^{16} - 7616 \, \cosh\left(\mbox{ro}\right)^{14} + 14080 \, \cosh\left(\mbox{ro}\right)^{12} - 14800 \, \cosh\left(\mbox{ro}\right)^{10} + 9680 \, \cosh\left(\mbox{ro}\right)^{8} - 4036 \, \cosh\left(\mbox{ro}\right)^{6} + 1048 \, \cosh\left(\mbox{ro}\right)^{4} - 155 \, \cosh\left(\mbox{ro}\right)^{2} + 10\right)} \cos\left(\sigma_{2}\right)^{12} + \left(12 i + 10\right) \, \cosh\left(\mbox{ro}\right)^{12} + {\left(\left(-64 i - 1792\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(224 i + 7808\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(-320 i - 14688\right) \, \cosh\left(\mbox{ro}\right)^{12} + \left(240 i + 15600\right) \, \cosh\left(\mbox{ro}\right)^{10} + \left(-100 i - 10240\right) \, \cosh\left(\mbox{ro}\right)^{8} + \left(22 i + 4256\right) \, \cosh\left(\mbox{ro}\right)^{6} + \left(-2 i - 1094\right) \, \cosh\left(\mbox{ro}\right)^{4} + 159 \, \cosh\left(\mbox{ro}\right)^{2} - 10\right)} \cos\left(\sigma_{2}\right)^{10} + \left(-8 i - 10\right) \, \cosh\left(\mbox{ro}\right)^{10} + {\left(\left(160 i + 1120\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(-576 i - 4976\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(848 i + 9440\right) \, \cosh\left(\mbox{ro}\right)^{12} + \left(-656 i - 9992\right) \, \cosh\left(\mbox{ro}\right)^{10} + \left(282 i + 6454\right) \, \cosh\left(\mbox{ro}\right)^{8} + \left(-64 i - 2603\right) \, \cosh\left(\mbox{ro}\right)^{6} + \left(6 i + 639\right) \, \cosh\left(\mbox{ro}\right)^{4} - 87 \, \cosh\left(\mbox{ro}\right)^{2} + 5\right)} \cos\left(\sigma_{2}\right)^{8} + \left(2 i + 5\right) \, \cosh\left(\mbox{ro}\right)^{8} + {\left(\left(-160 i - 448\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(592 i + 2024\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(-888 i - 3844\right) \, \cosh\left(\mbox{ro}\right)^{12} + \left(692 i + 3994\right) \, \cosh\left(\mbox{ro}\right)^{10} + \left(-296 i - 2471\right) \, \cosh\left(\mbox{ro}\right)^{8} + \left(66 i + 925\right) \, \cosh\left(\mbox{ro}\right)^{6} + \left(-6 i - 202\right) \, \cosh\left(\mbox{ro}\right)^{4} + 23 \, \cosh\left(\mbox{ro}\right)^{2} - 1\right)} \cos\left(\sigma_{2}\right)^{6} - \cosh\left(\mbox{ro}\right)^{6} + {\left(\left(80 i + 112\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(-304 i - 516\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(460 i + 980\right) \, \cosh\left(\mbox{ro}\right)^{12} + \left(-352 i - 989\right) \, \cosh\left(\mbox{ro}\right)^{10} + \left(142 i + 569\right) \, \cosh\left(\mbox{ro}\right)^{8} + \left(-28 i - 185\right) \, \cosh\left(\mbox{ro}\right)^{6} + \left(2 i + 31\right) \, \cosh\left(\mbox{ro}\right)^{4} - 2 \, \cosh\left(\mbox{ro}\right)^{2}\right)} \cos\left(\sigma_{2}\right)^{4} + {\left(\left(-20 i - 16\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(78 i + 76\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(-118 i - 146\right) \, \cosh\left(\mbox{ro}\right)^{12} + \left(86 i + 144\right) \, \cosh\left(\mbox{ro}\right)^{10} + \left(-30 i - 76\right) \, \cosh\left(\mbox{ro}\right)^{8} + \left(4 i + 20\right) \, \cosh\left(\mbox{ro}\right)^{6} - 2 \, \cosh\left(\mbox{ro}\right)^{4}\right)} \cos\left(\sigma_{2}\right)^{2} + {\left(\left(-4 i\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(18 i + 2\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(-32 i - 9\right) \, \cosh\left(\mbox{ro}\right)^{12} + {\left(\left(128 i\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(-512 i\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(864 i\right) \, \cosh\left(\mbox{ro}\right)^{12} + \left(-800 i\right) \, \cosh\left(\mbox{ro}\right)^{10} + \left(440 i\right) \, \cosh\left(\mbox{ro}\right)^{8} + \left(-144 i\right) \, \cosh\left(\mbox{ro}\right)^{6} + \left(26 i\right) \, \cosh\left(\mbox{ro}\right)^{4} + \left(-2 i\right) \, \cosh\left(\mbox{ro}\right)^{2}\right)} \cos\left(\sigma_{2}\right)^{10} + \left(28 i + 16\right) \, \cosh\left(\mbox{ro}\right)^{10} + {\left(\left(-320 i\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(1312 i + 32\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(-2272 i - 112\right) \, \cosh\left(\mbox{ro}\right)^{12} + \left(2160 i + 160\right) \, \cosh\left(\mbox{ro}\right)^{10} + \left(-1220 i - 120\right) \, \cosh\left(\mbox{ro}\right)^{8} + \left(410 i + 50\right) \, \cosh\left(\mbox{ro}\right)^{6} + \left(-76 i - 11\right) \, \cosh\left(\mbox{ro}\right)^{4} + \left(6 i + 1\right) \, \cosh\left(\mbox{ro}\right)^{2}\right)} \cos\left(\sigma_{2}\right)^{8} + \left(-12 i - 14\right) \, \cosh\left(\mbox{ro}\right)^{8} + {\left(\left(320 i\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(-1344 i - 64\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(2368 i + 240\right) \, \cosh\left(\mbox{ro}\right)^{12} + \left(-2272 i - 368\right) \, \cosh\left(\mbox{ro}\right)^{10} + \left(1284 i + 296\right) \, \cosh\left(\mbox{ro}\right)^{8} + \left(-428 i - 132\right) \, \cosh\left(\mbox{ro}\right)^{6} + \left(78 i + 31\right) \, \cosh\left(\mbox{ro}\right)^{4} + \left(-6 i - 3\right) \, \cosh\left(\mbox{ro}\right)^{2}\right)} \cos\left(\sigma_{2}\right)^{6} + \left(2 i + 6\right) \, \cosh\left(\mbox{ro}\right)^{6} + {\left(\left(-160 i\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(688 i + 48\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(-1224 i - 192\right) \, \cosh\left(\mbox{ro}\right)^{12} + \left(1164 i + 312\right) \, \cosh\left(\mbox{ro}\right)^{10} + \left(-636 i - 264\right) \, \cosh\left(\mbox{ro}\right)^{8} + \left(198 i + 123\right) \, \cosh\left(\mbox{ro}\right)^{6} + \left(-32 i - 30\right) \, \cosh\left(\mbox{ro}\right)^{4} + \left(2 i + 3\right) \, \cosh\left(\mbox{ro}\right)^{2}\right)} \cos\left(\sigma_{2}\right)^{4} - \cosh\left(\mbox{ro}\right)^{4} + {\left(\left(40 i\right) \, \cosh\left(\mbox{ro}\right)^{16} + \left(-176 i - 16\right) \, \cosh\left(\mbox{ro}\right)^{14} + \left(314 i + 68\right) \, \cosh\left(\mbox{ro}\right)^{12} + \left(-290 i - 116\right) \, \cosh\left(\mbox{ro}\right)^{10} + \left(146 i + 101\right) \, \cosh\left(\mbox{ro}\right)^{8} + \left(-38 i - 47\right) \, \cosh\left(\mbox{ro}\right)^{6} + \left(4 i + 11\right) \, \cosh\left(\mbox{ro}\right)^{4} - \cosh\left(\mbox{ro}\right)^{2}\right)} \cos\left(\sigma_{2}\right)^{2}\right)} \cosh\left(\sigma_{2}\right)^{2}}{{\left(256 \, \sinh\left(\mbox{ro}\right)^{16} + 1024 \, \sinh\left(\mbox{ro}\right)^{14} + 1792 \, \sinh\left(\mbox{ro}\right)^{12} + 1792 \, \sinh\left(\mbox{ro}\right)^{10} + 1120 \, \sinh\left(\mbox{ro}\right)^{8} + 448 \, \sinh\left(\mbox{ro}\right)^{6} + 112 \, \sinh\left(\mbox{ro}\right)^{4} + 16 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma_{2}\right)^{16} + \sinh\left(\mbox{ro}\right)^{16} - {\left(1024 \, \sinh\left(\mbox{ro}\right)^{16} + 3968 \, \sinh\left(\mbox{ro}\right)^{14} + 6720 \, \sinh\left(\mbox{ro}\right)^{12} + 6496 \, \sinh\left(\mbox{ro}\right)^{10} + 3920 \, \sinh\left(\mbox{ro}\right)^{8} + 1512 \, \sinh\left(\mbox{ro}\right)^{6} + 364 \, \sinh\left(\mbox{ro}\right)^{4} + 50 \, \sinh\left(\mbox{ro}\right)^{2} + 3\right)} \cos\left(\sigma_{2}\right)^{14} + 3 \, \sinh\left(\mbox{ro}\right)^{14} + {\left(1792 \, \sinh\left(\mbox{ro}\right)^{16} + 6720 \, \sinh\left(\mbox{ro}\right)^{14} + 10944 \, \sinh\left(\mbox{ro}\right)^{12} + 10096 \, \sinh\left(\mbox{ro}\right)^{10} + 5760 \, \sinh\left(\mbox{ro}\right)^{8} + 2076 \, \sinh\left(\mbox{ro}\right)^{6} + 460 \, \sinh\left(\mbox{ro}\right)^{4} + 57 \, \sinh\left(\mbox{ro}\right)^{2} + 3\right)} \cos\left(\sigma_{2}\right)^{12} + 3 \, \sinh\left(\mbox{ro}\right)^{12} - {\left(1792 \, \sinh\left(\mbox{ro}\right)^{16} + 6496 \, \sinh\left(\mbox{ro}\right)^{14} + 10096 \, \sinh\left(\mbox{ro}\right)^{12} + 8752 \, \sinh\left(\mbox{ro}\right)^{10} + 4600 \, \sinh\left(\mbox{ro}\right)^{8} + 1486 \, \sinh\left(\mbox{ro}\right)^{6} + 283 \, \sinh\left(\mbox{ro}\right)^{4} + 28 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma_{2}\right)^{10} + \sinh\left(\mbox{ro}\right)^{10} + 5 \, {\left(224 \, \sinh\left(\mbox{ro}\right)^{16} + 784 \, \sinh\left(\mbox{ro}\right)^{14} + 1152 \, \sinh\left(\mbox{ro}\right)^{12} + 920 \, \sinh\left(\mbox{ro}\right)^{10} + 430 \, \sinh\left(\mbox{ro}\right)^{8} + 117 \, \sinh\left(\mbox{ro}\right)^{6} + 17 \, \sinh\left(\mbox{ro}\right)^{4} + \sinh\left(\mbox{ro}\right)^{2}\right)} \cos\left(\sigma_{2}\right)^{8} - {\left(448 \, \sinh\left(\mbox{ro}\right)^{16} + 1512 \, \sinh\left(\mbox{ro}\right)^{14} + 2076 \, \sinh\left(\mbox{ro}\right)^{12} + 1486 \, \sinh\left(\mbox{ro}\right)^{10} + 585 \, \sinh\left(\mbox{ro}\right)^{8} + 120 \, \sinh\left(\mbox{ro}\right)^{6} + 10 \, \sinh\left(\mbox{ro}\right)^{4}\right)} \cos\left(\sigma_{2}\right)^{6} + {\left(112 \, \sinh\left(\mbox{ro}\right)^{16} + 364 \, \sinh\left(\mbox{ro}\right)^{14} + 460 \, \sinh\left(\mbox{ro}\right)^{12} + 283 \, \sinh\left(\mbox{ro}\right)^{10} + 85 \, \sinh\left(\mbox{ro}\right)^{8} + 10 \, \sinh\left(\mbox{ro}\right)^{6}\right)} \cos\left(\sigma_{2}\right)^{4} - {\left(16 \, \sinh\left(\mbox{ro}\right)^{16} + 50 \, \sinh\left(\mbox{ro}\right)^{14} + 57 \, \sinh\left(\mbox{ro}\right)^{12} + 28 \, \sinh\left(\mbox{ro}\right)^{10} + 5 \, \sinh\left(\mbox{ro}\right)^{8}\right)} \cos\left(\sigma_{2}\right)^{2}} \end{array}\right)
#calculemos rescricciones, en las nuevas coordenadas

#auto
x0e=cosh(rhoe)*cos(tae); x_1e=cosh(rhoe)*sin(tae);
x1e=sinh(rhoe)*cos(ttae); x2e=sinh(rhoe)*sin(ttae);
xe = vector([x0e, x_1e, x1e, x2e]);

for i in range(0,4):
    xe[i].show()
\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} \cos\left(\sigma_{1} - \frac{1}{2} \, \log\left(-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right) + \frac{1}{2} \, \log\left(\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} \sin\left(\sigma_{1} - \frac{1}{2} \, \log\left(-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right) + \frac{1}{2} \, \log\left(\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} - 1} \sqrt{\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} + 1} \cos\left(\sigma_{1} - \frac{1}{2} \, \log\left(\frac{-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) - \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)}{\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)}\right) + \arctan\left(\frac{-i \, \sinh\left(2 \, \mbox{ro}\right)}{\tan\left(2 \, \sigma_{2}\right)}\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} - 1} \sqrt{\sqrt{{\left(2 \, \sinh\left(\mbox{ro}\right)^{2} + 1\right)} \cos\left(\sigma_{2}\right)^{2} - \sinh\left(\mbox{ro}\right)^{2}} + 1} \sin\left(\sigma_{1} - \frac{1}{2} \, \log\left(\frac{-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) - \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)}{\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)}\right) + \arctan\left(\frac{-i \, \sinh\left(2 \, \mbox{ro}\right)}{\tan\left(2 \, \sigma_{2}\right)}\right)\right)
#auto
cos(tae).show(); sin(tae).show();
\newcommand{\Bold}[1]{\mathbf{#1}}\cos\left(\sigma_{1} - \frac{1}{2} \, \log\left(-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right) + \frac{1}{2} \, \log\left(\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1} - \frac{1}{2} \, \log\left(-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right) + \frac{1}{2} \, \log\left(\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)\right)
#auto
#assume(sinh(r0)>0);
(cos(ttae)).simplify_full().factor().simplify_full().show()
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{{\left({\left(\left(2 i\right) \, \cos\left(\sigma_{1}\right) \cos\left(\sigma_{2}\right)^{2} \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right) - \sin\left(\sigma_{1}\right) \sin\left(\sigma_{2}\right) \cos\left(\sigma_{2}\right) - i \, \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)\right)} \cos\left(\frac{1}{2} \, \log\left(-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)\right) + {\left(\left(2 i\right) \, \sin\left(\sigma_{1}\right) \cos\left(\sigma_{2}\right)^{2} \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right) - i \, \sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right) + \sin\left(\sigma_{2}\right) \cos\left(\sigma_{1}\right) \cos\left(\sigma_{2}\right)\right)} \sin\left(\frac{1}{2} \, \log\left(-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)\right)\right)} \sin\left(\frac{1}{2} \, \log\left(\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)\right) + {\left({\left(\left(-2 i\right) \, \cos\left(\sigma_{1}\right) \cos\left(\sigma_{2}\right)^{2} \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right) + \sin\left(\sigma_{1}\right) \sin\left(\sigma_{2}\right) \cos\left(\sigma_{2}\right) + i \, \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)\right)} \sin\left(\frac{1}{2} \, \log\left(-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)\right) + {\left(\left(2 i\right) \, \sin\left(\sigma_{1}\right) \cos\left(\sigma_{2}\right)^{2} \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right) - i \, \sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right) + \sin\left(\sigma_{2}\right) \cos\left(\sigma_{1}\right) \cos\left(\sigma_{2}\right)\right)} \cos\left(\frac{1}{2} \, \log\left(-\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)\right)\right)} \cos\left(\frac{1}{2} \, \log\left(\sin\left(\sigma_{2}\right) \sinh\left(\mbox{ro}\right) + \cos\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)\right)\right)}{\sqrt{-2 \, \cos\left(\sigma_{2}\right)^{2} \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right) + \sin\left(\sigma_{2}\right) \cos\left(\sigma_{2}\right) + \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)} \sqrt{2 \, \cos\left(\sigma_{2}\right)^{2} \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right) + \sin\left(\sigma_{2}\right) \cos\left(\sigma_{2}\right) - \sinh\left(\mbox{ro}\right) \cosh\left(\mbox{ro}\right)}}
#auto
dtxe=diff(xe,sig1); 
dsxe=diff(xe,sig2);
dtxe;
(-sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2)*sin(sigma1 - 1/2*log(-sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro)) + 1/2*log(sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro))), sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2)*cos(sigma1 - 1/2*log(-sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro)) + 1/2*log(sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro))), -sqrt(sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2) - 1)*sqrt(sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2) + 1)*sin(sigma1 - 1/2*log(-(sin(sigma2)*sinh(ro) - cos(sigma2)*cosh(ro))/(sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro))) + arctan(-I*sinh(2*ro)/tan(2*sigma2))), sqrt(sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2) - 1)*sqrt(sqrt((2*sinh(ro)^2 + 1)*cos(sigma2)^2 - sinh(ro)^2) + 1)*cos(sigma1 - 1/2*log(-(sin(sigma2)*sinh(ro) - cos(sigma2)*cosh(ro))/(sin(sigma2)*sinh(ro) + cos(sigma2)*cosh(ro))) + arctan(-I*sinh(2*ro)/tan(2*sigma2))))
#auto
sum(dtxe[i]*dsxe[i]*adsv[i] for i in range(0,4)).simplify_full()
(((4*I + 4)*cosh(ro)^5 + (-4*I - 4)*cosh(ro)^3 + (I + 1)*cosh(ro))*cos(sigma2)^4*sinh(ro) + ((-4*I - 4)*cosh(ro)^5 + (4*I + 4)*cosh(ro)^3 + (-I - 1)*cosh(ro))*cos(sigma2)^2*sinh(ro) + ((I + 1)*cosh(ro)^5 + (-I - 1)*cosh(ro)^3)*sinh(ro))/(sin(sigma2)^6*sinh(ro)^4*cosh(ro)^2 - cos(sigma2)^6*sinh(ro)^2*cosh(ro)^4 - (2*sinh(ro)^4*cosh(ro)^2 + (cosh(ro)^4 + 1)*sinh(ro)^2)*sin(sigma2)^4*cos(sigma2)^2 + (sinh(ro)^4*cosh(ro)^2 + 2*sinh(ro)^2*cosh(ro)^4 + cosh(ro)^2)*sin(sigma2)^2*cos(sigma2)^4)
((((4*I + 4)*cosh(ro)^5 - (4*I - 4)*cosh(ro)^3 + (I + 1)*cosh(ro))*cos(sigma2)^4*sinh(ro) + (-(4*I - 4)*cosh(ro)^5 + (4*I + 4)*cosh(ro)^3 - (I - 1)*cosh(ro))*cos(sigma2)^2*sinh(ro) + ((I + 1)*cosh(ro)^5 - (I - 1)*cosh(ro)^3)*sinh(ro))).factor().simplify_full().show()
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\left(4 i + 4\right) \, \cosh\left(\mbox{ro}\right)^{5} + \left(-4 i + 4\right) \, \cosh\left(\mbox{ro}\right)^{3} + \left(i + 1\right) \, \cosh\left(\mbox{ro}\right)\right)} \cos\left(\sigma_{2}\right)^{4} \sinh\left(\mbox{ro}\right) + {\left(\left(-4 i + 4\right) \, \cosh\left(\mbox{ro}\right)^{5} + \left(4 i + 4\right) \, \cosh\left(\mbox{ro}\right)^{3} + \left(-i + 1\right) \, \cosh\left(\mbox{ro}\right)\right)} \cos\left(\sigma_{2}\right)^{2} \sinh\left(\mbox{ro}\right) + {\left(\left(i + 1\right) \, \cosh\left(\mbox{ro}\right)^{5} + \left(-i + 1\right) \, \cosh\left(\mbox{ro}\right)^{3}\right)} \sinh\left(\mbox{ro}\right)

calculemos α y P(σ\pm) correspondiente al caso lorentziano 

#auto
i, j, k, l, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4 = var('i, j, k, l, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4')

#auto
eps=((j-i)*(k-i)*(l-i)*(k-j)*(l-j)*(l-k))/12

eps(2,4,1,3)
__main__:3: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...) -1
#auto
dsp = dsxn+dtxn;
dsm = dsxn-dtxn; dsp[1].show()
\newcommand{\Bold}[1]{\mathbf{#1}}-\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right) + \sin\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)
adsv=vector([-1,-1,1,1]);

#auto
N = vector(SR,[1,2,3,4]);
for i in range(0,4):
    N[i]=((1/2)*sum(sum(sum(eps(i,j,k,l)*xs[j]*adsv[j]*dsp[k]*adsv[k]*dsm[l]*adsv[l] for j in range(0,4)) for k in range(0,4)) for l in range(0,4))).simplify_full()

#auto
for i in range(0,4):
    N[i].show()
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) - \cos\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) - \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}-\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right) - \cos\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)
#auto
for i in range(0,4):
    xs[i].show()
\newcommand{\Bold}[1]{\mathbf{#1}}-\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1}\right) \cosh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \sinh\left(\sigma_{2}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \cos\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\sin\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right) - \cos\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right)
#auto
prue=vector(SR,[a1, a2, a3, a4]);
prue1=vector(SR,[b1, b2, b3, b4]);
prue2=vector(SR,[c1, c2, c3, c4]);
adsv=vector([-1,-1,1,1]);

#auto
sum(dtxn[i]*dsxn[i]*adsv[i] for i in range(0,4)).simplify_full()
0
#auto
sum(dsp[i]*dsp[i]*adsv[i] for i in range(0,4)).simplify_full()
0
#auto
sum(dsm[i]*dsm[i]*adsv[i] for i in range(0,4)).simplify_full()
0
#auto
sum(sum(sum(eps(0,j,k,l)*prue[j]*prue1[k]*prue2[l] for j in range(0,4)) for k in range(0,4)) for l in range(0,4))
a2*b3*c4 - a2*b4*c3 - a3*b2*c4 + a3*b4*c2 + a4*b2*c3 - a4*b3*c2

Observe que Y es ortogonal a ∂Y \bar∂Y 

#auto
sum(dsp[i]*xs[i]*adsv[i] for i in range(0,4)).simplify_full()
0
#auto
sum(dsm[i]*xs[i]*adsv[i] for i in range(0,4)).simplify_full()
0

Observe que N es ortogonal a ∂Y \bar∂Y 

#auto
sum(N[i]*xs[i]*adsv[i] for i in range(0,4)).simplify_full()
0
#auto
sum(N[i]*dsm[i]*adsv[i] for i in range(0,4)).simplify_full()
0
#auto
sum(N[i]*dsp[i]*adsv[i] for i in range(0,4)).simplify_full()
0
#auto
sum(N[i]*N[i]*adsv[i] for i in range(0,4)).simplify_full()
1

Ahora calculamos p(σ\pm) 

#auto
ddsp =(1/4)*(diff(dsp,sig1)+diff(dsp,sig2));
ddsm =(1/4)*(diff(dsm,sig2)-diff(dsm,sig1)); ddsp[1].show(); ddsm[1].show();
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, \sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) + \frac{1}{2} \, \cos\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \sin\left(\sigma_{1}\right) \sinh\left(\mbox{ro}\right) \cosh\left(\sigma_{2}\right) - \frac{1}{2} \, \cos\left(\sigma_{1}\right) \sinh\left(\sigma_{2}\right) \cosh\left(\mbox{ro}\right)
#auto
p=sum((1/2)*ddsp[i]*N[i]*adsv[i] for i in range(0,4)).simplify_full(); p.show()
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{4}
#auto
pb=sum((1/2)*ddsm[i]*N[i]*adsv[i] for i in range(0,4)).simplify_full(); pb.show()
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{4}