series expansion.nb

一个mathematica求级数展开的代码

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Notebook[{
Cell[BoxData[{
    \(\(Clear["\<Global`*\>"];\)\), "\[IndentingNewLine]", 
    \(\(d = Sqrt[4\ Pi/\ Sqrt\ [3]];\)\), "\[IndentingNewLine]", 
    \(\(Array[\ Rx, {3, 6}];\)\), "\[IndentingNewLine]", 
    \(\(Rx = {{d, 0.5\ \ d, \(-0.5\)\ \ d, \(-d\), \(-0.5\)\ \ d, 
            0.5\ \ d}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {1.5\ \ d, 
            0.0, \(-1.5\)\ \ d, \(-1.5\)\ \ d, 0.0, 
            1.5\ d\ }, \[IndentingNewLine]\t\t\t\t\ \ {2.0\ d, 
            d, \(-d\), \(-2.0\)\ d, \(-d\), d}};\)\), "\n", 
    \(\(Array[\ Ry, {3, 6}];\)\), "\[IndentingNewLine]", 
    \(\(Ry = {{0.0, Sqrt[3.0]/2.0\ \ d, Sqrt[3.0]/2.0\ \ d, 
            0.0, \(-Sqrt[3.0]\)/2.0\ \ d, \(-Sqrt[3.0]\)/
                2.0\ \ d}, \[IndentingNewLine]\t\t\t\t\ \ \ {Sqrt[3.0]/
                2.0\ \ d, Sqrt[3.0]\ \ d, 
            Sqrt[3.0]/2.0\ \ d, \(-Sqrt[3.0]\)/
                2.0\ \ d, \(-Sqrt[3.0]\)\ \ d, \(-Sqrt[3.0]\)/
                2.0\ \ d\ }, \[IndentingNewLine]\t\t\t\t\ \ {0.0, 
            Sqrt[3.0]\ \ d, Sqrt[3.0]\ \ d, 
            0.0, \(-Sqrt[3.0]\)\ \ d, \(-Sqrt[
                  3.0]\)\ \ d}};\)\), "\[IndentingNewLine]", 
    \(\(F[kx_, ky_, lx_, ly_, kpx_, kpy_, lpx_, 
          lpy_] := \ \[ExponentialE]\^\(\(-0.5\)\ \((kx - kpx)\)\^2 - 0.5 \
\((ky - kpy)\)\^2\  + \ I\ \((ky - kpy)\) \((kx - lpx)\)\)\ \ \((1.0 + \
\[ExponentialE]\^\(\(-0.5\)\ d\^2\)\ \ \ Sum\ [\[ExponentialE]\^\(Rx[\([1, j]\
\)] \((ky - kpy)\) - Ry[\([1, j]\)] \((kx - kpx)\) + \ I\ \((\ \(-Rx\ [\([1, \
j]\)]\) \((kx - lpx)\) - Ry\ [\([1, j]\)] \((ky - lpy)\)\ )\)\), {j, 1, 
                    6}] + \ \[ExponentialE]\^\(\(-1.5\)\ d\^2\)\ \ Sum\ [\
\[ExponentialE]\^\(Rx[\([2, j]\)] \((ky - kpy)\) - Ry[\([2, j]\)] \((kx - \
kpx)\) + \ I\ \((\ \(-Rx\ [\([2, j]\)]\) \((kx - lpx)\) - Ry\ [\([2, j]\)] \
\((ky - lpy)\)\ )\)\), {j, 1, 
                    6}] + \[ExponentialE]\^\(\(-2.0\)\ d\^2\)\ \ Sum\ [\
\[ExponentialE]\^\(Rx[\([3, j]\)] \((ky - kpy)\) - Ry[\([3, j]\)] \((kx - \
kpx)\) + \ I\ \((\ \(-Rx\ [\([3, j]\)]\) \((kx - lpx)\) - Ry\ [\([3, j]\)] \
\((ky - lpy)\)\ )\)\), {j, 1, 6}])\);\)\), "\[IndentingNewLine]", 
    \(\(gamma\ [\ kx_, 
          ky_]\  := \(\[ExponentialE]\^\(\(-I\)\ kx\ \ ky - 0.5\ kx\^2 - 
                0.5\ ky\^2\)\) \((1.0 + \[ExponentialE]\^\(\(-0.5\)\ d\^2\)\ \
Sum[\[ExponentialE]\^\(\(\ \)\(\((I\ kx + \ ky)\)\ \ \((\ \ Rx[\([1, j]\)] + \
I\ Ry[\([1, j]\)])\)\)\), {j, 1, 
                    6}] + \[ExponentialE]\^\(\(-1.5\)\ d\^2\)\ Sum[\
\[ExponentialE]\^\(\(\ \)\(\((I\ kx + \ ky)\)\ \ \((Rx\ [\([2, j]\)]\  + \ I\ \
Ry\ [\([2, j]\)]\ )\)\)\), {j, 1, 
                    6}] + \[ExponentialE]\^\(\(-2.0\)\ d\^2\)\ Sum[\
\[ExponentialE]\^\(I\ \((kx - I\ ky)\) \((Rx[\([3, j]\)] + I\ Ry[\([3, j]\)])\
\)\), {j, 1, 6}])\);\)\), "\[IndentingNewLine]", 
    \(\(Cgamma[kx_, 
          ky_] := \(\[ExponentialE]\^\(I\ kx\ \ ky - 0.5\ kx\^2 - 
                0.5\ ky\^2\)\) \((1.0 + \[ExponentialE]\^\(\(-0.5\)\ d\^2\)\ \
Sum[\[ExponentialE]\^\(\(\ \)\(\((\(-I\)\ kx + \ ky)\)\ \ \((\ \ Rx[\([1, \
j]\)] - I\ Ry[\([1, j]\)])\)\)\), {j, 1, 
                    6}] + \[ExponentialE]\^\(\(-1.5\)\ d\^2\)\ Sum[\
\[ExponentialE]\^\(\(\ \)\(\((\(-I\)\ kx + \ ky)\)\ \ \((Rx\ [\([2, j]\)]\  - \
\ I\ Ry\ [\([2, j]\)]\ )\)\)\), {j, 1, 
                    6}] + \[ExponentialE]\^\(\(-2.0\)\ d\^2\)\ Sum[\
\[ExponentialE]\^\(\(-I\)\ \((kx + I\ ky)\) \((Rx[\([3, j]\)] - I\ Ry[\([3, \
j]\)])\)\), {j, 1, 6}])\);\)\), "\[IndentingNewLine]", 
    \(\(absgamma[kx_, ky_] := 
        Sqrt[\ gamma[kx, ky]\ Cgamma[kx, ky]\ ];\)\), "\[IndentingNewLine]", 
    \(\(gammap[kx_, ky_]\  := 
        Sqrt[\ gamma[kx, ky]/
            absgamma[kx, ky]\ \ ];\)\), "\[IndentingNewLine]", 
    \(\(phase[kx_, ky_, lx_, ly_, kpx_, kpy_, lpx_, lpy_] := 
        gammap[kx, ky]\ \(gammap[lx, ly]/gammap[kpx, kpy]\)/
            gammap[lpx, lpy];\)\), "\[IndentingNewLine]", 
    \(\(beta[kx_, ky_] := 
        1.0 + \[ExponentialE]\^\(\(-0.5\)\ d\^2\)\ Sum[\[ExponentialE]\^\(I\ \
\((kx\ \ Rx[\([1, j]\)] + ky\ \ Ry[\([1, j]\)])\)\), {j, 1, 
                6}] + \[ExponentialE]\^\(\(-1.5\)\ d\^2\)\ \
Sum[\[ExponentialE]\^\(I\ \((\ kx\ \ Rx[\([2, j]\)] + ky\ \ Ry[\([2, j]\)])\)\
\), {j, 1, 
                6}] + \[ExponentialE]\^\(\(-2.0\)\ d\^2\)\ \
Sum[\[ExponentialE]\^\(I\ \((kx\ \ Rx[\([3, j]\)] + ky\ \ Ry[\([3, \
j]\)])\)\), {j, 1, 6}];\)\), "\[IndentingNewLine]", 
    \(\(au = \(-1.0\);\)\), "\[IndentingNewLine]", 
    \(\(v = \(\(-au\)/2.0\)/beta[0.0, 0.0];\)\), "\[IndentingNewLine]", 
    \(\(g[kx_, ky_] := 
        au + 4.0\ \ v\ \ \ beta[kx, ky];\)\), "\[IndentingNewLine]", 
    \(\(energy[kx_, ky_] := 
        Sqrt[g[kx, ky]^2 - 
            4.0\ v^2\ \ absgamma[kx, ky]^2\ ]\ ;\)\), "\[IndentingNewLine]", 
    \(\(fa[kx_, ky_] := 
        Sqrt[g[kx, ky] + 2.0\ v\ absgamma[kx, ky]] + 
          Sqrt[g[kx, ky] - 
              2.0\ v\ absgamma[kx, ky]];\)\), "\[IndentingNewLine]", 
    \(\(fm[kx_, ky_] := 
        Sqrt[g[kx, ky] + 2.0\ v\ absgamma[kx, ky]] - 
          Sqrt[g[kx, ky] - 
              2.0\ v\ absgamma[kx, 
                  ky]];\)\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\(Fp[kx_, ky_, lx_, ly_, kpx_, kpy_, lpx_, lpy_] := 
        F[kx, ky, lx, ly, kpx, kpy, lpx, lpy]\ \ \ phase[kx, ky, lx, ly, kpx, 
            kpy, lpx, lpy];\)\[IndentingNewLine]\), "\n", 
    \(\(f1[kx_, ky_, lx_, 
          ly_] := \((Fp[\(-kx\) - lx, \(-ky\) - ly, lx, ly, \(-kx\), \(-ky\), 
                0.0, 0.0]\ fa[kx + lx, ky + ly]\ fa[lx, ly]\ fm[kx, 
                ky]\[IndentingNewLine] - 
            Fp[kx, ky, 0.0, 0.0, kx + lx, ky + ly, \(-lx\), \(-ly\)]\ fm[
                kx + lx, ky + ly]\ fm[lx, ly]\ fa[kx, 
                ky]\ \[IndentingNewLine] + 
            Fp[kx, ky, \(-kx\) - lx, \(-ky\) - ly, \(-lx\), \(-ly\), 0.0, 
                0.0]\ fa[kx + lx, ky + ly]\ fm[lx, ly]\ fa[kx, 
                ky]\ \[IndentingNewLine] - 
            Fp[lx, ly, 0.0, 0.0, \(-kx\), \(-ky\), kx + lx, ky + ly]\ fm[
                kx + lx, ky + ly]\ fa[lx, ly]\ fm[kx, 
                ky]\ \[IndentingNewLine]\ \ )\);\)\), "\[IndentingNewLine]", 
    \(\(f2[kx_, ky_, lx_, ly_] := \((\ 
          Fp[kx + lx, ky + ly, 0.0, 0.0, kx, ky, lx, ly]\ fm[kx + lx, 
                ky + ly] fa[lx, ly]\ fa[kx, ky]\n\t\t\t\t\t\t - 
            Fp[\(-kx\), \(-ky\), \(-lx\), \(-ly\), \(-kx\) - lx, \(-ky\) - 
                  ly, 0.0, 0.0]\ fa[kx + lx, ky + ly]\ fm[lx, ly]\ fm[kx, 
                ky]\n\t\t\t\t\t\t + 
            Fp[\(-lx\), \(-ly\), 0.0, 0.0, kx, 
                ky, \(-kx\) - lx, \(-ky\) - ly]\ fa[kx + lx, ky + ly]\ fm[lx, 
                ly]\ fa[kx, ky]\n\t\t\t\t\t\t - 
            Fp[\(-kx\), \(-ky\), kx + lx, ky + ly, lx, ly, 0.0, 0.0]\ fm[
                kx + lx, ky + ly] 
              fa[lx, ly]\ fm[kx, ky]\[IndentingNewLine]\t\t\t\t\t\ \ \ \ \  + 
            Fp[\(-kx\), \(-ky\), 0.0, 0.0, \(-kx\) - lx, \(-ky\) - ly, lx, 
                ly]\ fa[kx + lx, ky + ly] 
              fa[lx, ly]\ fm[kx, ky]\n\t\t\t\t\t\t - 
            Fp[kx + lx, ky + ly, \(-lx\), \(-ly\), kx, ky, 0.0, 0.0]\ fm[
                kx + lx, ky + ly] 
              fm[lx, ly]\ fa[kx, ky]\n\t\t\t\t\t\t\ \ )\);\)\), "\n", 
    \(\(ff[kx_, ky_, lx_, ly_] := 
        f1[kx, ky, lx, ly]\ f2[kx, ky, lx, 
            ly];\)\[IndentingNewLine]\t\t\t\t\t\t\), "\[IndentingNewLine]", 
    \(\(tot[kx_, ky_, lx_, ly_] := 
        1.0/\((Pi\ Pi\ energy[kx, ky]\ \ energy[lx, ly]\ )\)\ \((beta[
                  kx - lx, ky - ly]\ \ g[kx, ky]\ \ g[lx, ly]/
                  8.0 + \((v\ \ absgamma[kx, ky]^2\ beta[lx, ly]\ g[lx, ly] + 
                      v\ absgamma[lx, ly]^2\ beta[kx, ky] g[kx, ky] - 
                      beta[kx, ky] beta[lx, ly] g[kx, ky] 
                        g[lx, ly]\ \ )\)/\((\ 
                    4.0\ \ beta[0.0, 0.0]\ )\)\[IndentingNewLine]\(-\(v\ \ ff[
                        kx, ky, lx, 
                        ly]\ /\((64.0\ \ energy[kx + lx, 
                            ky + ly] \((energy[kx, ky] + energy[lx, ly] + 
                              energy[kx + lx, 
                                ky + ly])\))\)\)\)\ \ \ \ \ \ \ \ )\);\)\
\[IndentingNewLine]\ \ \ \[IndentingNewLine]\[IndentingNewLine]\
\[IndentingNewLine]\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(\)}], "Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(t1Fp = 
      Series[Fp[\(-k\) - l, \(-k\) - l, l, l, \(-k\), \(-k\), 0.0, 0.0]\ fa[
            k + l, k + l]\ fa[l, l]\ fm[k, k], {k, 0, 3}, {l, 0, 
          3}]\[IndentingNewLine]
    t2Fp = 
      Series[\(-Fp[k, k, 0.0, 0.0, k + l, k + l, \(-l\), \(-l\)]\)\ fm[k + l, 
            k + l]\ fm[l, l]\ fa[k, k]\ , {k, 0, 3}, {l, 0, 
          3}]\[IndentingNewLine]
    t3Fp = 
      Series[Fp[k, k, \(-k\) - l, \(-k\) - l, \(-l\), \(-l\), 0.0, 0.0]\ fa[
            k + l, k + l]\ fm[l, l]\ fa[k, k]\ , {k, 0, 3}, {l, 0, 
          3}]\[IndentingNewLine]
    t4Fp = 
      Series[\(-Fp[l, l, 0.0, 0.0, \(-k\), \(-k\), k + l, k + l]\)\ fm[k + l, 
            k + l]\ fa[l, l]\ fm[k, k]\ , {k, 0, 3}, {l, 0, 
          3}]\[IndentingNewLine]\[IndentingNewLine]
    t5Fp = \ \ Series[
        Fp[k + l, k + l, 0.0, 0.0, k, k, l, l]\ fm[k + l, k + l] 
          fa[l, l]\ fa[k, k], {k, 0, 3}, {l, 0, 3}]\n
    t6Fp = 
      Series[\(-Fp[\(-k\), \(-k\), \(-l\), \(-l\), \(-k\) - l, \(-k\) - l, 
              0.0, 0.0]\)\ fa[k + l, k + l]\ fm[l, l]\ fm[k, k], {k, 0, 
          3}, {l, 0, 3}]\n
    t7Fp = 
      Series[\ Fp[\(-l\), \(-l\), 0.0, 0.0, k, k, \(-k\) - l, \(-k\) - l]\ fa[
            k + l, k + l]\ fm[l, l]\ fa[k, k], {k, 0, 3}, {l, 0, 3}]\ \n
    t8Fp = 
      Series[\(-Fp[\(-k\), \(-k\), k + l, k + l, l, l, 0.0, 0.0]\)\ fm[k + l, 
            k + l] fa[l, l]\ fm[k, k], {k, 0, 3}, {l, 0, 
          3}]\[IndentingNewLine]
    t9Fp = 
      Series[\ Fp[\(-k\), \(-k\), 0.0, 0.0, \(-k\) - l, \(-k\) - l, l, l]\ fa[
            k + l, k + l] fa[l, l]\ fm[k, k], {k, 0, 3}, {l, 0, 3}]\ \n
    t10Fp = 
      Series[\(-Fp[k + l, k + l, \(-l\), \(-l\), k, k, 0.0, 0.0]\)\ fm[k + l, 
            k + l] fm[l, l]\ fa[k, k], {k, 0, 3}, {l, 0, 
          3}]\n\t\t\t\t\t\t\ \ \[IndentingNewLine]
    \)\)\)], "Input"],

Cell[BoxData[{
    \(\(tot[kx_, ky_, lx_, ly_] := 
        1.0/tt \((\ tm + nu\ /de\ )\);\)\ \ \ \), "\[IndentingNewLine]", 
    \(tt = \(\(Pi\)\(\ \)\(Pi\)\(\ \)\(energy[kx, ky]\)\(\ \ \)\(energy[lx, 
          ly]\)\(\ \)\)\), "\[IndentingNewLine]", 
    \(tm = 
      beta[kx - lx, ky - ly]\ \ g[kx, ky]\ \ g[lx, ly]/
            8.0 + \((v\ \ absgamma[kx, ky]^2\ beta[lx, ly]\ g[lx, ly] + 
              v\ absgamma[lx, ly]^2\ beta[kx, ky] g[kx, ky] - 
              beta[kx, ky] beta[lx, ly] g[kx, ky] g[lx, ly]\ \ )\)/\((\ 
            4.0\ \ beta[0.0, 0.0]\ )\)\), "\[IndentingNewLine]", 
    \(nu = \(-v\)\ \ ff[kx, ky, lx, ly]\), "\[IndentingNewLine]", 
    \(\(\(de = 
      64.0\ \ energy[kx + lx, 
          ky + ly] \((energy[kx, ky] + energy[lx, ly] + 
            energy[kx + lx, 
              ky + ly])\)\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)
    \)\), "\[IndentingNewLine]", 
    \(\ \)}], "Input"],