Taylor expansion about exp The 2019 Stack Overflow Developer Survey Results Are InFinding...
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Taylor expansion about exp
The 2019 Stack Overflow Developer Survey Results Are InFinding power series representation of $ int_0^{frac{pi }{2}} frac{1}{sqrt {1 - k^2sin^2{x}}};{dx}$Infinite Series $sumlimits_{n=1}^inftyfrac{x^{3n}}{(3n-1)!}$Iterated Sine… Further expansionAsymptotic expansion of $(1+epsilon)^{s/epsilon}$Convergence of a Taylor series expansion involving $tan^{-1}$Taylor series expansion and evaluating an integralSecond Order and Beyond for Multivariable Taylor SeriesSimplify $intlimits_{-infty}^Xexp[-(Ae^x+Bx+Cx^2)]mathrm dx$ and $sumlimits_{n=0}^inftyfrac{(-A)^n}{n!}e^{Dn^2-Kn}$Want to show that asymptotic expansion of $T[phi](x)=int_0^{2pi}sin(alpha |x-y|)phi(y)dx$ converges in the operator norm?How do you write the sines of a binary expansion as an infinite series?
$begingroup$
Recently I am trying to compute a Taylor expansion. I had known the result is
$$begin{aligned}
&expBig(-i(frac{h+delta h}{2}sigma_x+frac{J+delta J}{2}sigma_z)frac{phi}{sqrt{h^2+J^2}}Big)
\
&= expBig(-i(frac{h}{2}sigma_x+frac{J}{2}sigma_z)frac{phi}{sqrt{h^2+J^2}}Big)times cdots
\
×Bigg(I+ifrac{delta h}{2(h^2+J^2)^{3/2}}left((-h^2phi-J^2 sinphi)sigma_x+2 Jsqrt{h^2+J^2} sin^2frac{phi}{2}sigma_y+hJ(sinphi-phi)sigma_z right)
\
&+ifrac{delta J}{2(h^2+J^2)^{3/2}}left(hJ(sinphi-phi)sigma_z+2hsqrt{h^2+J^2}sin^2frac{phi}{2}sigma_y+(-J^2phi-h^2 sinphi)sigma_zright)
\
&+O(delta h^2) +O(delta J^2)Bigg)
end{aligned}$$
In above expression, $sigma$ indicates Pauli Matrices.
I cannot finish this and don't how to do this,some could help me?
sequences-and-series
asked Mar 21 at 4:56
YuXuanLiYuXuanLi
11
$endgroup$
add a comment |
$begingroup$
Recently I am trying to compute a Taylor expansion. I had known the result is
$$begin{aligned}
&expBig(-i(frac{h+delta h}{2}sigma_x+frac{J+delta J}{2}sigma_z)frac{phi}{sqrt{h^2+J^2}}Big)
\
&= expBig(-i(frac{h}{2}sigma_x+frac{J}{2}sigma_z)frac{phi}{sqrt{h^2+J^2}}Big)times cdots
\
×Bigg(I+ifrac{delta h}{2(h^2+J^2)^{3/2}}left((-h^2phi-J^2 sinphi)sigma_x+2 Jsqrt{h^2+J^2} sin^2frac{phi}{2}sigma_y+hJ(sinphi-phi)sigma_z right)
\
&+ifrac{delta J}{2(h^2+J^2)^{3/2}}left(hJ(sinphi-phi)sigma_z+2hsqrt{h^2+J^2}sin^2frac{phi}{2}sigma_y+(-J^2phi-h^2 sinphi)sigma_zright)
\
&+O(delta h^2) +O(delta J^2)Bigg)
end{aligned}$$
In above expression, $sigma$ indicates Pauli Matrices.
I cannot finish this and don't how to do this,some could help me?
sequences-and-series
asked Mar 21 at 4:56
YuXuanLiYuXuanLi
11
$endgroup$
add a comment |
$begingroup$
Recently I am trying to compute a Taylor expansion. I had known the result is
$$begin{aligned}
&expBig(-i(frac{h+delta h}{2}sigma_x+frac{J+delta J}{2}sigma_z)frac{phi}{sqrt{h^2+J^2}}Big)
\
&= expBig(-i(frac{h}{2}sigma_x+frac{J}{2}sigma_z)frac{phi}{sqrt{h^2+J^2}}Big)times cdots
\
×Bigg(I+ifrac{delta h}{2(h^2+J^2)^{3/2}}left((-h^2phi-J^2 sinphi)sigma_x+2 Jsqrt{h^2+J^2} sin^2frac{phi}{2}sigma_y+hJ(sinphi-phi)sigma_z right)
\
&+ifrac{delta J}{2(h^2+J^2)^{3/2}}left(hJ(sinphi-phi)sigma_z+2hsqrt{h^2+J^2}sin^2frac{phi}{2}sigma_y+(-J^2phi-h^2 sinphi)sigma_zright)
\
&+O(delta h^2) +O(delta J^2)Bigg)
end{aligned}$$
In above expression, $sigma$ indicates Pauli Matrices.
I cannot finish this and don't how to do this,some could help me?
sequences-and-series
asked Mar 21 at 4:56
YuXuanLiYuXuanLi
11
$endgroup$
Recently I am trying to compute a Taylor expansion. I had known the result is
$$begin{aligned}
&expBig(-i(frac{h+delta h}{2}sigma_x+frac{J+delta J}{2}sigma_z)frac{phi}{sqrt{h^2+J^2}}Big)
\
&= expBig(-i(frac{h}{2}sigma_x+frac{J}{2}sigma_z)frac{phi}{sqrt{h^2+J^2}}Big)times cdots
\
×Bigg(I+ifrac{delta h}{2(h^2+J^2)^{3/2}}left((-h^2phi-J^2 sinphi)sigma_x+2 Jsqrt{h^2+J^2} sin^2frac{phi}{2}sigma_y+hJ(sinphi-phi)sigma_z right)
\
&+ifrac{delta J}{2(h^2+J^2)^{3/2}}left(hJ(sinphi-phi)sigma_z+2hsqrt{h^2+J^2}sin^2frac{phi}{2}sigma_y+(-J^2phi-h^2 sinphi)sigma_zright)
\
&+O(delta h^2) +O(delta J^2)Bigg)
end{aligned}$$
In above expression, $sigma$ indicates Pauli Matrices.
I cannot finish this and don't how to do this,some could help me?
sequences-and-series
sequences-and-series
asked Mar 21 at 4:56
YuXuanLiYuXuanLi
11
asked Mar 21 at 4:56
YuXuanLiYuXuanLi
11
asked Mar 21 at 4:56
YuXuanLiYuXuanLi
11
asked Mar 21 at 4:56
YuXuanLiYuXuanLi
11
asked Mar 21 at 4:56
YuXuanLiYuXuanLi
11
11
add a comment |
add a comment |
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