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Display information for equation id:math.1775.47 on revision:1775

* Page found: Lippmann- Schwinger- Gleichung (eq math.1775.47)

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Lippmann- Schwinger- Gleichung Eq: math.1777.47

Hash: 1215f54fc2610218579d535a8227e1ab

TeX (original user input):

\begin{align}

& \begin{matrix}

\lim   \\

\rho \to \infty   \\

\end{matrix}\int_{0}^{\pi }{d\Phi i{{e}^{2i\Phi }}}\frac{{{\rho }^{2}}{{e}^{i\rho R\cos \Phi }}{{e}^{-}}^{\rho R\sin \Phi }}{{{{\bar{k}}}^{2}}-{{\rho }^{2}}{{e}^{2i\Phi }}+i\eta }=0 \\

& da \\

& \begin{matrix}

\lim   \\

\rho \to \infty   \\

\end{matrix}{{e}^{-}}^{\rho R\sin \Phi }=0 \\

& \Rightarrow \begin{matrix}

\lim   \\

\rho \to \infty   \\

\end{matrix}\oint\limits_{{}}{dq}q\frac{{{e}^{iqR}}}{{{{\bar{k}}}^{2}}-{{{\bar{q}}}^{2}}+i\eta }=\int_{-\infty }^{\infty }{dq}q\frac{{{e}^{iqR}}}{{{{\bar{k}}}^{2}}-{{{\bar{q}}}^{2}}+i\eta } \\

\end{align}

TeX (checked):

{\begin{aligned}&{\begin{matrix}\lim \\\rho \to \infty \\\end{matrix}}\int _{0}^{\pi }{d\Phi i{{e}^{2i\Phi }}}{\frac {{{\rho }^{2}}{{e}^{i\rho R\cos \Phi }}{{e}^{-}}^{\rho R\sin \Phi }}{{{\bar {k}}^{2}}-{{\rho }^{2}}{{e}^{2i\Phi }}+i\eta }}=0\\&da\\&{\begin{matrix}\lim \\\rho \to \infty \\\end{matrix}}{{e}^{-}}^{\rho R\sin \Phi }=0\\&\Rightarrow {\begin{matrix}\lim \\\rho \to \infty \\\end{matrix}}\oint \limits _{}{dq}q{\frac {{e}^{iqR}}{{{\bar {k}}^{2}}-{{\bar {q}}^{2}}+i\eta }}=\int _{-\infty }^{\infty }{dq}q{\frac {{e}^{iqR}}{{{\bar {k}}^{2}}-{{\bar {q}}^{2}}+i\eta }}\\\end{aligned}}

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MathML (4.882 KB / 644 B) :

\begin{matrix} \begin{matrix} lim \\ ρ \to \infty \end{matrix} \int_{0}^{π} d Φ i e^{2 i Φ} \frac{ρ^{2} e^{i ρ R \cos Φ} {e^{-}}^{ρ R \sin Φ}}{{\bar{k}}^{2} - ρ^{2} e^{2 i Φ} + i η} = 0 \\ d a \\ \begin{matrix} lim \\ ρ \to \infty \end{matrix} {e^{-}}^{ρ R \sin Φ} = 0 \\ \Rightarrow \begin{matrix} lim \\ ρ \to \infty \end{matrix} \oint_{} d q q \frac{e^{i q R}}{{\bar{k}}^{2} - {\bar{q}}^{2} + i η} = \int_{- \infty}^{\infty} d q q \frac{e^{i q R}}{{\bar{k}}^{2} - {\bar{q}}^{2} + i η} \end{matrix}

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Identifiers

$ρ$

$π$

$Φ$

$i$

$e$

$i$

$Φ$

$ρ$

$e$

$i$

$ρ$

$R$

$Φ$

$e$

$ρ$

$R$

$Φ$

$\bar{k}$

$ρ$

$e$

$i$

$Φ$

$i$

$η$

$d$

$a$

$ρ$

$e$

$ρ$

$R$

$Φ$

$ρ$

$d$

$q$

$q$

$e$

$i$

$q$

$R$

$\bar{k}$

$\bar{q}$

$i$

$η$

$q$

$q$

$e$

$i$

$q$

$R$

$\bar{k}$

$\bar{q}$

$i$

$η$

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