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* Page found: Elektromagnetische Wellen (eq math.1438.222)

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TeX (original user input): 

\begin{align}
& \Phi \left( \bar{r},t \right)=\int_{{}}^{{}}{{{d}^{3}}r\acute{\ }\int_{-\infty }^{t}{dt\acute{\ }}}\frac{\rho \left( \bar{r}\acute{\ },t\acute{\ } \right)}{{{\varepsilon }_{0}}}G\left( \bar{r}-\bar{r}\acute{\ },t-t\acute{\ } \right)=\int_{{}}^{{}}{{{d}^{3}}r\acute{\ }\int_{-\infty }^{t}{dt\acute{\ }}}\frac{\rho \left( \bar{r}\acute{\ } \right)}{{{\varepsilon }_{0}}}{{e}^{-i\omega t\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ },t-t\acute{\ } \right) \\
& t-t\acute{\ }:=\tau  \\
& \Rightarrow \int_{-\infty }^{t}{dt\acute{\ }}{{e}^{-i\omega t\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ },t-t\acute{\ } \right)=\int_{-\infty }^{t}{dt\acute{\ }}{{e}^{-i\omega t\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ },\tau  \right) \\
& =\left[ \int_{0}^{\infty }{d\tau }{{e}^{i\omega \tau }}G\left( \bar{r}-\bar{r}\acute{\ },\tau  \right) \right]{{e}^{-i\omega t}}:=\tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right){{e}^{-i\omega t}} \\
& \int_{0}^{\infty }{d\tau }{{e}^{i\omega \tau }}G\left( \bar{r}-\bar{r}\acute{\ },\tau  \right):=\tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right) \\
\end{align}

TeX (checked): 

{\begin{aligned}&\Phi \left({\bar {r}},t\right)=\int _{}^{}{{{d}^{3}}r{\acute {\ }}\int _{-\infty }^{t}{dt{\acute {\ }}}}{\frac {\rho \left({\bar {r}}{\acute {\ }},t{\acute {\ }}\right)}{{\varepsilon }_{0}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},t-t{\acute {\ }}\right)=\int _{}^{}{{{d}^{3}}r{\acute {\ }}\int _{-\infty }^{t}{dt{\acute {\ }}}}{\frac {\rho \left({\bar {r}}{\acute {\ }}\right)}{{\varepsilon }_{0}}}{{e}^{-i\omega t{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},t-t{\acute {\ }}\right)\\&t-t{\acute {\ }}:=\tau \\&\Rightarrow \int _{-\infty }^{t}{dt{\acute {\ }}}{{e}^{-i\omega t{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},t-t{\acute {\ }}\right)=\int _{-\infty }^{t}{dt{\acute {\ }}}{{e}^{-i\omega t{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},\tau \right)\\&=\left[\int _{0}^{\infty }{d\tau }{{e}^{i\omega \tau }}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},\tau \right)\right]{{e}^{-i\omega t}}:={\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right){{e}^{-i\omega t}}\\&\int _{0}^{\infty }{d\tau }{{e}^{i\omega \tau }}G\left({\bar {r}}-{\bar {r}}{\acute {\ }},\tau \right):={\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\\\end{aligned}}

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Φ(r¯,t)=∫d3r ´∫−∞tdt ´ρ(r¯ ´,t ´)ε0G(r¯−r¯ ´,t−t ´)=∫d3r ´∫−∞tdt ´ρ(r¯ ´)ε0e−iωt ´G(r¯−r¯ ´,t−t ´)t−t ´:=τ⇒∫−∞tdt ´e−iωt ´G(r¯−r¯ ´,t−t ´)=∫−∞tdt ´e−iωt ´G(r¯−r¯ ´,τ)=[∫0∞dτeiωτG(r¯−r¯ ´,τ)]e−iωt:=G~(r¯−r¯ ´)e−iωt∫0∞dτeiωτG(r¯−r¯ ´,τ):=G~(r¯−r¯ ´)
<math xmlns="http://www.w3.org/1998/Math/MathML" class="mwe-math-element mwe-math-element-inline"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mtable displaystyle="true"><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>Φ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo>,</mo><mi>t</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msubsup><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo>,</mo><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><msub><mi>ε</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></mfrac></mrow><mi>G</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">−</mo><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msubsup><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><msub><mi>ε</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></mfrac></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>i</mi><mi>ω</mi><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></msup><mi>G</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">−</mo><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>t</mi><mo stretchy="false">−</mo><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo stretchy="false">:=</mo><mi>τ</mi></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msubsup><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>i</mi><mi>ω</mi><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></msup><mi>G</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">−</mo><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>t</mi></mrow></msubsup><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>i</mi><mi>ω</mi><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></msup><mi>G</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo>,</mo><mi>τ</mi><mo 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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>τ</mi></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>ω</mi><mi>τ</mi></mrow></mrow></msup><mi>G</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo>,</mo><mi>τ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>i</mi><mi>ω</mi><mi>t</mi></mrow></mrow></msup><mo stretchy="false">:=</mo><mover><mi>G</mi><mo>~</mo></mover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mi>i</mi><mi>ω</mi><mi>t</mi></mrow></mrow></msup></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>τ</mi></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>ω</mi><mi>τ</mi></mrow></mrow></msup><mi>G</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo>,</mo><mi>τ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">:=</mo><mover><mi>G</mi><mo>~</mo></mover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Calculated based on the variables occurring on the entire Elektromagnetische Wellen page

Identifiers

  • Φ
  • r¯
  • t
  • r
  •  ´
  • t
  • t
  •  ´
  • ρ
  • r¯
  •  ´
  • t
  •  ´
  • ε0
  • G
  • r¯
  • r¯
  •  ´
  • t
  • t
  •  ´
  • r
  •  ´
  • t
  • t
  •  ´
  • ρ
  • r¯
  •  ´
  • ε0
  • e
  • i
  • ω
  • t
  •  ´
  • G
  • r¯
  • r¯
  •  ´
  • t
  • t
  •  ´
  • t
  • t
  •  ´
  • τ
  • t
  • t
  •  ´
  • e
  • i
  • ω
  • t
  •  ´
  • G
  • r¯
  • r¯
  •  ´
  • t
  • t
  •  ´
  • t
  • t
  •  ´
  • e
  • i
  • ω
  • t
  •  ´
  • G
  • r¯
  • r¯
  •  ´
  • τ
  • τ
  • e
  • i
  • ω
  • τ
  • G
  • r¯
  • r¯
  •  ´
  • τ
  • e
  • i
  • ω
  • t
  • G~
  • r¯
  • r¯
  •  ´
  • e
  • i
  • ω
  • t
  • τ
  • e
  • i
  • ω
  • τ
  • G
  • r¯
  • r¯
  •  ´
  • τ
  • G~
  • r¯
  • r¯
  •  ´

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