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Display information for equation id:math.1439.121 on revision:1439

* Page found: Elektromagnetische Wellen (eq math.1439.121)

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Retardierte Potenziale Eq: math.2129.36

Hash: 6cd471a363b6e5807c9af57d7dd35e48

TeX (original user input):

\begin{align}
& {{d}^{3}}q={{q}^{2}}dq\sin \vartheta d\vartheta d\phi  \\
& \bar{q}\bar{s}=qs\cos \vartheta  \\
& G(\bar{s},\tau )=\frac{c}{{{\left( 2\pi  \right)}^{3}}}\int\limits_{0}^{\infty }{{}}dqq\left( \frac{{{e}^{-icq\tau }}-{{e}^{icq\tau }}}{-2i} \right)\int\limits_{-1}^{1}{{}}d\cos \vartheta {{e}^{iqs\cos \vartheta }}\int\limits_{0}^{2\pi }{{}}d\phi  \\
& \int\limits_{-1}^{1}{{}}d\cos \vartheta {{e}^{iqs\cos \vartheta }}=\frac{{{e}^{iqs}}-{{e}^{-iqs}}}{iqs} \\
& \xi :=cq \\
& \Rightarrow G(\bar{s},\tau )=\frac{c}{2{{\left( 2\pi  \right)}^{2}}s}\int\limits_{0}^{\infty }{{}}d\xi \left\{ {{e}^{i\left( \tau -\frac{s}{c} \right)\xi }}+{{e}^{-i\left( \tau -\frac{s}{c} \right)\xi }}-{{e}^{i\left( \tau +\frac{s}{c} \right)\xi }}-{{e}^{-i\left( \tau +\frac{s}{c} \right)\xi }} \right\} \\
& \Rightarrow G(\bar{s},\tau )=\frac{c}{4\pi s}\int\limits_{0}^{\infty }{{}}d\xi \left\{ \delta \left( \tau -\frac{s}{c} \right)-\delta \left( \tau +\frac{s}{c} \right) \right\} \\
& \delta \left( \tau +\frac{s}{c} \right)=0\quad f\ddot{u}r\ \tau >0 \\
\end{align}

TeX (checked):

{\begin{aligned}&{{d}^{3}}q={{q}^{2}}dq\sin \vartheta d\vartheta d\phi \\&{\bar {q}}{\bar {s}}=qs\cos \vartheta \\&G({\bar {s}},\tau )={\frac {c}{{\left(2\pi \right)}^{3}}}\int \limits _{0}^{\infty }{}dqq\left({\frac {{{e}^{-icq\tau }}-{{e}^{icq\tau }}}{-2i}}\right)\int \limits _{-1}^{1}{}d\cos \vartheta {{e}^{iqs\cos \vartheta }}\int \limits _{0}^{2\pi }{}d\phi \\&\int \limits _{-1}^{1}{}d\cos \vartheta {{e}^{iqs\cos \vartheta }}={\frac {{{e}^{iqs}}-{{e}^{-iqs}}}{iqs}}\\&\xi :=cq\\&\Rightarrow G({\bar {s}},\tau )={\frac {c}{2{{\left(2\pi \right)}^{2}}s}}\int \limits _{0}^{\infty }{}d\xi \left\{{{e}^{i\left(\tau -{\frac {s}{c}}\right)\xi }}+{{e}^{-i\left(\tau -{\frac {s}{c}}\right)\xi }}-{{e}^{i\left(\tau +{\frac {s}{c}}\right)\xi }}-{{e}^{-i\left(\tau +{\frac {s}{c}}\right)\xi }}\right\}\\&\Rightarrow G({\bar {s}},\tau )={\frac {c}{4\pi s}}\int \limits _{0}^{\infty }{}d\xi \left\{\delta \left(\tau -{\frac {s}{c}}\right)-\delta \left(\tau +{\frac {s}{c}}\right)\right\}\\&\delta \left(\tau +{\frac {s}{c}}\right)=0\quad f{\ddot {u}}r\ \tau >0\\\end{aligned}}

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\begin{matrix} d^{3} q = q^{2} d q \sin ϑ d ϑ d ϕ \\ \bar{q} \bar{s} = q s \cos ϑ \\ G (\bar{s}, τ) = \frac{c}{{(2 π)}^{3}} \int_{0}^{\infty} d q q (\frac{e^{- i c q τ} - e^{i c q τ}}{- 2 i}) \int_{- 1}^{1} d \cos ϑ e^{i q s \cos ϑ} \int_{0}^{2 π} d ϕ \\ \int_{- 1}^{1} d \cos ϑ e^{i q s \cos ϑ} = \frac{e^{i q s} - e^{- i q s}}{i q s} \\ ξ := c q \\ \Rightarrow G (\bar{s}, τ) = \frac{c}{2 {(2 π)}^{2} s} \int_{0}^{\infty} d ξ {e^{i (τ - \frac{s}{c}) ξ} + e^{- i (τ - \frac{s}{c}) ξ} - e^{i (τ + \frac{s}{c}) ξ} - e^{- i (τ + \frac{s}{c}) ξ}} \\ \Rightarrow G (\bar{s}, τ) = \frac{c}{4 π s} \int_{0}^{\infty} d ξ {δ (τ - \frac{s}{c}) - δ (τ + \frac{s}{c})} \\ δ (τ + \frac{s}{c}) = 0 f \ddot{u} r τ > 0 \end{matrix}

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data-mjx-texclass="ORD"><mi>i</mi><mi>q</mi><mi>s</mi></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>ξ</mi><mo stretchy="false">:=</mo><mi>c</mi><mi>q</mi></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mi>G</mi><mo stretchy="false">(</mo><mover><mi>s</mi><mo>¯</mo></mover><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>2</mn><mi>π</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>s</mi></mrow></mrow></mfrac></mrow><munderover><mo form="prefix" stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><mi>d</mi><mi>ξ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>τ</mi><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>ξ</mi></mrow></mrow></msup><mo stretchy="false">+</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>i</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>τ</mi><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><mo 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data-mjx-texclass="CLOSE">}</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mi>G</mi><mo stretchy="false">(</mo><mover><mi>s</mi><mo>¯</mo></mover><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>4</mn><mi>π</mi><mi>s</mi></mrow></mrow></mfrac></mrow><munderover><mo form="prefix" stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><mi>d</mi><mi>ξ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mi>δ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>τ</mi><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">−</mo><mi>δ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>τ</mi><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">}</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>δ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>τ</mi><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><mn>0</mn><mspace width="1em"></mspace><mi>f</mi><mover><mi>u</mi><mo>¨</mo></mover><mi>r</mi><mtext>&#160;</mtext><mi>τ</mi><mo>&gt;</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

$d$

$q$

$q$

$d$

$q$

$ϑ$

$d$

$ϑ$

$d$

$ϕ$

$\bar{q}$

$\bar{s}$

$q$

$s$

$ϑ$

$G$

$\bar{s}$

$τ$

$c$

$π$

$q$

$q$

$e$

$i$

$c$

$q$

$τ$

$e$

$i$

$c$

$q$

$τ$

$i$

$ϑ$

$e$

$i$

$q$

$s$

$ϑ$

$π$

$ϕ$

$ϑ$

$e$

$i$

$q$

$s$

$ϑ$

$e$

$i$

$q$

$s$

$e$

$i$

$q$

$s$

$i$

$q$

$s$

$ξ$

$c$

$q$

$G$

$\bar{s}$

$τ$

$c$

$π$

$s$

$ξ$

$e$

$i$

$τ$

$s$

$c$

$ξ$

$e$

$i$

$τ$

$s$

$c$

$ξ$

$e$

$i$

$τ$

$s$

$c$

$ξ$

$e$

$i$

$τ$

$s$

$c$

$ξ$

$G$

$\bar{s}$

$τ$

$c$

$π$

$s$

$ξ$

$δ$

$τ$

$s$

$c$

$δ$

$τ$

$s$

$c$

$δ$

$τ$

$s$

$c$

$f$

$\ddot{u}$

$r$

$τ$

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