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Display information for equation id:math.1199.674 on revision:1199
* Page found: Elektrodynamik Schöll (eq math.1199.674)
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Hash: ac39c3dd37158d419750fc6d0d031699
TeX (original user input):
\Gamma (\bar{q},\tau ):=\int_{-\infty }^{\infty }{d\omega }\frac{{{e}^{-i\omega \tau }}}{\left( {{q}^{2}}-\frac{{{\omega }^{2}}}{{{c}^{2}}} \right)}=\oint\limits_{C}{d\omega }\frac{{{e}^{-i\omega \tau }}}{\left( {{q}^{2}}-\frac{{{\omega }^{2}}}{{{c}^{2}}} \right)}=2\pi i\sum\limits_{Pole}^{{}}{{}}\operatorname{Re}s\frac{{{e}^{-i\omega \tau }}}{\left( {{q}^{2}}-\frac{{{\omega }^{2}}}{{{c}^{2}}} \right)}
TeX (checked):
\Gamma ({\bar {q}},\tau ):=\int _{-\infty }^{\infty }{d\omega }{\frac {{e}^{-i\omega \tau }}{\left({{q}^{2}}-{\frac {{\omega }^{2}}{{c}^{2}}}\right)}}=\oint \limits _{C}{d\omega }{\frac {{e}^{-i\omega \tau }}{\left({{q}^{2}}-{\frac {{\omega }^{2}}{{c}^{2}}}\right)}}=2\pi i\sum \limits _{Pole}^{}{}\operatorname {Re} s{\frac {{e}^{-i\omega \tau }}{\left({{q}^{2}}-{\frac {{\omega }^{2}}{{c}^{2}}}\right)}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>¯</mo></mover></mrow></mrow><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo><mi>:</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>ω</mi></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>i</mi><mi>ω</mi><mi>τ</mi></mrow></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>q</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mfrac></mrow><mo>=</mo><munder><mstyle displaystyle="true"><mo>∮</mo></mstyle><mrow data-mjx-texclass="ORD"><mi>C</mi></mrow></munder><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>ω</mi></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>i</mi><mi>ω</mi><mi>τ</mi></mrow></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>q</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mfrac></mrow><mo>=</mo><mn>2</mn><mi>π</mi><mi>i</mi><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>P</mi><mi>o</mi><mi>l</mi><mi>e</mi></mrow></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mi data-mjx-texclass="OP" mathvariant="normal">ℜ</mi><mo>⁡</mo><mi>s</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>i</mi><mi>ω</mi><mi>τ</mi></mrow></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>q</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mfrac></mrow></mstyle></mrow></math>
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