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Display information for equation id:math.1777.49 on revision:1777

* Page found: Lippmann- Schwinger- Gleichung (eq math.1777.49)

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Lippmann- Schwinger- Gleichung Eq: math.1777.49

Hash: 039a2c0ff1b04908f7eb4a2e495e4ef7

TeX (original user input):

\begin{align}
& RES{{\left. \frac{q{{e}^{iqR}}}{{{{\bar{k}}}^{2}}-{{{\bar{q}}}^{2}}+i\eta } \right|}_{q={{q}_{1}}}}=RES{{\left. \frac{q{{e}^{iqR}}}{\left( \sqrt{{{{\bar{k}}}^{2}}+i\eta }-q \right)\left( \sqrt{{{{\bar{k}}}^{2}}+i\eta }+q \right)} \right|}_{{{q}_{1}}\equiv \sqrt{{{{\bar{k}}}^{2}}+i\eta }}} \\
& \sqrt{{{{\bar{k}}}^{2}}+i\eta }={{q}_{1}}=-{{q}_{2}} \\
& RES{{\left. \frac{q{{e}^{iqR}}}{\left( \sqrt{{{{\bar{k}}}^{2}}+i\eta }-q \right)\left( \sqrt{{{{\bar{k}}}^{2}}+i\eta }+q \right)} \right|}_{{{q}_{1}}\equiv \sqrt{{{{\bar{k}}}^{2}}+i\eta }}}=\begin{matrix}
\lim   \\
q->q1  \\
\end{matrix}\frac{\left( q-{{q}_{1}} \right)q{{e}^{iqR}}}{\left( {{q}_{1}}-q \right)\left( q-{{q}_{2}} \right)}=\frac{{{q}_{1}}{{e}^{i{{q}_{1}}R}}}{\left( {{q}_{1}}-{{q}_{2}} \right)} \\
& =-\frac{{{e}^{i\sqrt{{{{\bar{k}}}^{2}}+i\eta }R}}}{2} \\
& \begin{matrix}
\lim   \\
\eta \to 0  \\
\end{matrix}-\frac{{{e}^{i\sqrt{{{{\bar{k}}}^{2}}+i\eta }R}}}{2}=-\frac{{{e}^{ikR}}}{2} \\
\end{align}

TeX (checked):

{\begin{aligned}&RES{{\left.{\frac {q{{e}^{iqR}}}{{{\bar {k}}^{2}}-{{\bar {q}}^{2}}+i\eta }}\right|}_{q={{q}_{1}}}}=RES{{\left.{\frac {q{{e}^{iqR}}}{\left({\sqrt {{{\bar {k}}^{2}}+i\eta }}-q\right)\left({\sqrt {{{\bar {k}}^{2}}+i\eta }}+q\right)}}\right|}_{{{q}_{1}}\equiv {\sqrt {{{\bar {k}}^{2}}+i\eta }}}}\\&{\sqrt {{{\bar {k}}^{2}}+i\eta }}={{q}_{1}}=-{{q}_{2}}\\&RES{{\left.{\frac {q{{e}^{iqR}}}{\left({\sqrt {{{\bar {k}}^{2}}+i\eta }}-q\right)\left({\sqrt {{{\bar {k}}^{2}}+i\eta }}+q\right)}}\right|}_{{{q}_{1}}\equiv {\sqrt {{{\bar {k}}^{2}}+i\eta }}}}={\begin{matrix}\lim \\q->q1\\\end{matrix}}{\frac {\left(q-{{q}_{1}}\right)q{{e}^{iqR}}}{\left({{q}_{1}}-q\right)\left(q-{{q}_{2}}\right)}}={\frac {{{q}_{1}}{{e}^{i{{q}_{1}}R}}}{\left({{q}_{1}}-{{q}_{2}}\right)}}\\&=-{\frac {{e}^{i{\sqrt {{{\bar {k}}^{2}}+i\eta }}R}}{2}}\\&{\begin{matrix}\lim \\\eta \to 0\\\end{matrix}}-{\frac {{e}^{i{\sqrt {{{\bar {k}}^{2}}+i\eta }}R}}{2}}=-{\frac {{e}^{ikR}}{2}}\\\end{aligned}}

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\begin{aligned} R E S {\frac{q e^{i q R}}{{\bar{k}}^{2} - {\bar{q}}^{2} + i η} |}_{q = q_{1}} = R E S {\frac{q e^{i q R}}{(\sqrt{{\bar{k}}^{2} + i η} - q) (\sqrt{{\bar{k}}^{2} + i η} + q)} |}_{q_{1} \equiv \sqrt{{\bar{k}}^{2} + i η}} \\ \sqrt{{\bar{k}}^{2} + i η} = q_{1} = - q_{2} \\ R E S {\frac{q e^{i q R}}{(\sqrt{{\bar{k}}^{2} + i η} - q) (\sqrt{{\bar{k}}^{2} + i η} + q)} |}_{q_{1} \equiv \sqrt{{\bar{k}}^{2} + i η}} = \begin{matrix} \lim \\ q - > q 1 \end{matrix} \frac{(q - q_{1}) q e^{i q R}}{(q_{1} - q) (q - q_{2})} = \frac{q_{1} e^{i q_{1} R}}{(q_{1} - q_{2})} \\ = - \frac{e^{i \sqrt{{\bar{k}}^{2} + i η} R}}{2} \\ \begin{matrix} \lim \\ η \to 0 \end{matrix} - \frac{e^{i \sqrt{{\bar{k}}^{2} + i η} R}}{2} = - \frac{e^{i k R}}{2} \end{aligned}

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Identifiers

$R$

$E$

$S$

$q$

$e$

$i$

$q$

$R$

$\bar{k}$

$\bar{q}$

$i$

$η$

$q$

$q_{1}$

$R$

$E$

$S$

$q$

$e$

$i$

$q$

$R$

$\bar{k}$

$i$

$η$

$q$

$\bar{k}$

$i$

$η$

$q$

$q_{1}$

$\bar{k}$

$i$

$η$

$\bar{k}$

$i$

$η$

$q_{1}$

$q_{2}$

$R$

$E$

$S$

$q$

$e$

$i$

$q$

$R$

$\bar{k}$

$i$

$η$

$q$

$\bar{k}$

$i$

$η$

$q$

$q_{1}$

$\bar{k}$

$i$

$η$

$q$

$q$

$q$

$q_{1}$

$q$

$e$

$i$

$q$

$R$

$q_{1}$

$q$

$q$

$q_{2}$

$q_{1}$

$e$

$i$

$q_{1}$

$R$

$q_{1}$

$q_{2}$

$e$

$i$

$\bar{k}$

$i$

$η$

$R$

$η$

$e$

$i$

$\bar{k}$

$i$

$η$

$R$

$e$

$i$

$k$

$R$

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