alephpi/FormulaNet
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... | S \sim \tilde { \psi } Q _ { o } \tilde { \psi } + g _ { s } ^ { 1 / 2 } \tilde { \psi } ^ { 3 } + \tilde { \phi } Q _ { c } \tilde { \phi } + g _ { s } \tilde { \phi } ^ { 3 } + \tilde { \phi } B ( g _ { s } ^ { 1 / 2 } \tilde { \psi } ) + \cdots . |
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120... | e ^ { A } = e ^ { A _ { 0 } } \left( t _ { 0 } - \mathrm { s i g n } ( m ) t \right) ^ { - \frac { m } { 2 } } \; , \; \; \; \; \chi = \chi _ { 0 } \left( t _ { 0 } - \mathrm { s i g n } ( m ) t \right) ^ { m } \; , |
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... | g _ { J _ { 1 } \, J _ { 2 } } ^ { J } \bigl ( J _ { 1 } , M _ { 1 } ; J _ { 2 } , M _ { 2 } \vert J _ { 1 } , J _ { 2 } ; J , M _ { 1 } + M _ { 2 } \bigr ) \, \Bigl ( \xi _ { M _ { 1 } + M _ { 2 } } ^ { ( J ) } ( \sigma _ { 1 } ) + \mathrm { d e s c e n d a n t s } \Bigr ) \Bigr \} , |
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... | < \frac { 1 } { 2 } , m _ { s } | { \psi } _ { - } ^ { ( \frac { 1 } { 2 } ) } ( g ) > \equiv D _ { m _ { s } - \frac { 1 } { 2 } } ^ { ( \frac { 1 } { 2 } ) } ( g ) = < g , l + \frac { 1 } { 2 } | T _ { m _ { s } - } ^ { \frac { 1 } { 2 } } | g , l > . |
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120,... | \Delta { \cal A } = \frac { 1 } { 2 } ( 1 + g ) . |
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... | \frac 1 { \nabla ^ { 2 } } \, \delta _ { \Sigma } ^ { ( 2 ) } ( z - z _ { 0 } ) = - \frac 1 \pi \log { \cal E } ( z , z _ { 0 } ) |
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... | g x ( \alpha \cdot q , \xi ) \to \left\{ \begin{array} { c l } { { \mathrm { f i n i t e } , } } & { { \mathrm { f o r } \quad \pm \alpha _ { i } \in \Pi \quad ( \delta \leq 1 / h ) \quad \mathrm { a n d } \ \pm \alpha _ { h } \quad ( \delta = 1 / h ) , } } \\ { { 0 , } } & { { \mathrm { o t h e r w i s e , } } } \end{... |
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... | \left( A _ { 0 } , A _ { 1 } , \phi , \pi _ { 0 } , \pi _ { 1 } , \pi _ { \phi } \right) \longleftrightarrow \left( \xi _ { 1 } , . . . , \xi _ { 6 } \right) , |
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20... | \sigma _ { \pm } = \frac { \tau \pm \sigma } { 2 } \ . |
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... | { \cal R } = { \cal R } _ { r } \sin \eta , ~ ~ ~ \tau = { \cal R } _ { r } ( 1 - \cos \eta ) . |
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185,... | \stackrel { \mathrm { G } } { { \mathcal L } } \, : = \frac { 1 } { 2 m } \left[ ( D _ { \alpha } \overline { { { \Psi } } } ) D ^ { \alpha } \Psi - m ^ { 2 } \overline { { { \Psi } } } \Psi \right] \, . |
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... | \partial _ { \pm } = \frac { 1 } { 2 } \left( \partial _ { \tau } \pm \partial _ { \sigma } \right) |
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... | V _ { \mu } ^ { s } ( x , p ) = { \pi } ^ { - 4 } \partial _ { \nu } ^ { ( x ) } \int d ^ { 4 } q e ^ { 2 i p . q } { \bar { \psi } } ( x + q ) { \frac { \sigma _ { \mu \nu } } { 2 m } } \psi ( x - q ) . |
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9,... | \delta _ { { \hat { \xi } } _ { 2 } } L [ { \hat { \xi } } _ { 1 } ] = \left\{ L [ { \hat { \xi } } _ { 2 } ] , L [ { \hat { \xi } } _ { 1 } ] \right\} = L [ \{ { \hat { \xi } } _ { 1 } , { \hat { \xi } } _ { 2 } \} _ { \mathrm { \scriptsize ~ S D } } ] + K [ { \hat { \xi } } _ { 1 } , { \hat { \xi } } _ { 2 } ] |
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13... | \Delta = - D ^ { 2 } - \frac { \mathrm { i } } 2 \sigma _ { \mu \nu } F _ { \mu \nu } . |
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142... | \frac { d \Phi } { d r } = - \frac { \sqrt { 3 } } { 4 } \ell ( r ) ^ { \Lambda \Sigma } \, \mathrm { I m } { \cal N } _ { \Lambda \Sigma | \Gamma \Delta } \, f _ { ~ ~ A B } ^ { \Gamma \Delta } \, \frac { e ^ { U } } { r ^ { 2 } } \, . |
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23... | \delta \phi = 2 \alpha \frac { H _ { e } ^ { 2 } } { m ^ { 2 } ( \phi _ { 0 } ) } ( \cos ( m ( \phi _ { 0 } ) ( \tau - \tau _ { e } ) ) - ( \frac { \tau _ { e } } { \tau } ) ^ { 2 } ) - \alpha \frac { H _ { e } ^ { 3 } } { m ( \phi _ { 0 } ) ^ { 3 } } \sin ( m ( \phi _ { 0 } ) ( \tau - \tau _ { e } ) ) |
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... | r = \frac { \alpha } { \beta } \vert \sin \beta \left( \sigma - \sigma _ { 0 } \right) \vert |
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... | \begin{array} { l l } { { f ( \psi ) = 1 + \frac { 1 } { R ^ { 2 } } \sum _ { r = 1 } ^ { N } m _ { r } ^ { 2 } ( \psi _ { r } ) ^ { 2 } + O ( \psi ^ { 3 } ) } } & { { \mathrm { w i t h ~ \frac { m _ { r } ^ { 2 } } { R ^ { 2 } } ~ \equiv ~ \frac { 1 } { 2 } ~ \partial _ { r } ~ \partial _ { r } ~ f ~ | _ { \ p s i ~ =... |
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... | H ( \omega , 0 ) = \frac { \phi _ { 1 } f _ { 2 } \partial _ { r } ( \phi _ { 2 } ^ { ( - ) } / A _ { - } ) - ( \phi _ { 2 } ^ { ( - ) } / A _ { - } ) f _ { 1 } \partial _ { r } \phi _ { 1 } } { \phi _ { 1 } f _ { 2 } \partial _ { r } ( \phi _ { 2 } ^ { ( + ) } / A _ { + } ) - ( \phi _ { 2 } ^ { ( + ) } / A _ { + } ) f... |
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... | \frac { d \phi ^ { i } } { d t } = - g ^ { i j } \frac { \partial W } { \partial \phi ^ { j } } \ , |
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3... | m _ { h } = \frac { g _ { s m } ^ { 2 } | \psi ( 0 ) | ^ { 2 } } { M _ { G } ^ { 2 } } = \frac { M _ { G } } { \sqrt { \pi } c _ { f } \alpha _ { s } \sqrt { N _ { s d } } } |
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52,... | e _ { \mu } ^ { f } = \left( { \frac { 1 } { 2 } } \left( e _ { \mu } ^ { [ + 2 ] } + e _ { \mu } ^ { [ - 2 ] } \right) , { \frac { 1 } { 2 } } \left( e _ { \mu } ^ { [ + 2 ] } - e _ { \mu } ^ { [ - 2 ] } \right) \right) |
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2... | L = { \frac { 1 } { 2 } } ( \dot { x } ^ { 2 } + \dot { y } ^ { 2 } ) - \lambda \biggl [ y \biggl ( x ^ { 2 } + ( y - { \frac { 1 } { 2 } } ) ^ { 2 } - 1 \biggr ) \biggr ] ^ { 2 } . |
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219... | \int \mathrm { d } ^ { 3 } { \bf r } \; \left| \, \psi _ { j } \left( { \bf r } \right) \right| ^ { 2 } = \frac { 1 } { 2 \vert E _ { j } \vert } \, , \quad j = 1 , \ldots , N \, , |
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... | { \cal H } = { \frac { 1 } { 2 } } ( \pi _ { \phi } - e A _ { 1 } ) ^ { 2 } + { \frac { 1 } { 2 } } \pi _ { 1 } ^ { 2 } + { \frac { 1 } { 2 } } \phi ^ { 2 } + \pi _ { 1 } A _ { 0 } ^ { \prime } + e \phi ^ { \prime } A _ { 0 } + { \frac { 1 } { 2 } } a e ^ { 2 } ( A _ { 0 } ^ { 2 } - A _ { 1 } ^ { 2 } ) . |
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19,... | \Phi ( \omega ) \sim { \frac { 2 \omega ^ { D - 2 } } { ( 4 \pi ) ^ { ( D - 1 ) / 2 } } } \sum _ { n = 0 } ^ { \infty } { \frac { c _ { n } } { \Gamma \left( { \frac { D - 1 } { 2 } } - n \right) } } \omega ^ { - 2 n } , |
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... | T ^ { 0 } = \frac { 1 } { 4 \pi \kappa _ { B } k } \frac { ( d - 2 k - 1 ) } { r _ { + } } . |
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243... | \int _ { \beta } \Omega ^ { ( 1 , 0 ) } = \frac { \sqrt { 6 } } { \pi } ( \pm \sqrt { B ^ { \prime } } ) ^ { - 1 / 2 } \, ( \pm k _ { \mp } ^ { 2 } ) ^ { - 1 / 2 } \, \, { \bf K } ( k _ { \mp } ^ { - 2 } ) |
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1... | 2 \kappa _ { 1 1 } { } ^ { 2 } T _ { 3 } { \tilde { T } } _ { 6 } = 2 \pi n |
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... | V ( \Phi ) = \frac { m ^ { 2 } } { 2 } | \Phi | ^ { 2 } + \frac { \lambda } { 4 ! } | \Phi | ^ { 4 } , |
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... | ( c _ { 0 } ^ { ( 1 ) } + c _ { 0 } ^ { ( 2 ) } + c _ { 0 } ^ { ( 3 ) } ) | V _ { 3 } \rangle = 0 , \ \ c _ { 0 } | I \star A \rangle = | I \star ( c _ { 0 } A ) \rangle , \ \ \forall A |
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120,... | e ^ { i \theta } = { \frac { p - q \tau ^ { * } } { | p - q \tau ^ { * } | } } , |
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9... | S [ \vec { r } ] \ = \ \int d ^ { D } \! x \ { \frac { 1 } { 2 } } ( \nabla _ { \! x } \vec { r } ) ^ { 2 } \ + \b \ \int d ^ { D } \! x \int d ^ { D } \! y \ \delta ^ { d } ( \vec { r } ( x ) - \vec { r } ( y ) ) \ . |
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... | \begin{array} { l l } { { { \cal L } _ { K } = } } & { { \frac { 1 } { 4 } F _ { \mu \nu } F ^ { \mu \nu } - ( 1 - g ^ { 2 } \zeta \eta ^ { 2 } ) \tilde { G } _ { \mu } \tilde { G } ^ { \mu } + 2 ( m + g h \zeta \eta ^ { 2 } ) A _ { \mu } \tilde { G } ^ { \mu } + } } \\ { { } } & { { - 2 h \zeta \eta ^ { 2 } \Sigma \pa... |
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... | - \frac { R ( P ) } { 2 } | \Phi ( P ) | ^ { 2 } - \frac { 1 } { 2 } | \Phi ( P ) | ^ { 4 } \ge 0 |
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120,... | \Gamma ( s + 2 n - 1 ) \zeta _ { R } ( s ) \zeta ( \frac { s + 2 n - 1 } { 2 } | \L _ { 3 } ) \; \beta ^ { - s } \: , \nonumber |
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120... | \sigma _ { a b } = m ^ { 2 } \delta _ { a b } + v _ { a b } , |
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120... | \mathcal { F } _ { \mu \nu } ^ { a } = \left( \begin{array} { c c c c c c c } { { 0 } } & { { 0 } } & { { 0 } } & { { 0 } } & { { 0 } } & { { 0 } } & { { 0 } } \\ { { 0 } } & { { 0 } } & { { \mathcal { F } _ { 1 2 } ^ { a } } } & { { 0 } } & { { 0 } } & { { 0 } } & { { 0 } } \\ { { 0 } } & { { - \mathcal { F } _ { 1 2 ... |
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17... | w ( m ) = w _ { 0 } ( m \sqrt { \alpha \prime } ) ^ { - a } e ^ { b m \sqrt { \alpha \prime } } |
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16... | Z _ { 3 } = \prod _ { i = 1 } ^ { 3 } e ^ { \frac { i \pi } { 4 } \mathrm { s i g n } ( p _ { i } q _ { i } ) } \exp \left( \frac { i \pi } { 2 K } \frac { r _ { i } } { p _ { i } } \right) |
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120... | d s ^ { 2 } = a ( \Sigma ) \left( - d \Sigma ^ { 2 } + d \chi ^ { 2 } + \sinh ^ { 2 } \chi d \varphi ^ { 2 } \right) , |
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... | S _ { 2 } ( \phi , \phi ) = - \int d ^ { D } X \sqrt { - G } \left\{ ( D _ { M } \phi ) ^ { \dagger } D ^ { M } \phi + \frac { 1 } { 2 } \phi \frac { \partial ^ { 2 } U } { \partial \Phi ^ { 2 } } \phi + \frac { 1 } { 2 } \phi ^ { * } \frac { \partial ^ { 2 } U } { \partial { \Phi ^ { * } } ^ { 2 } } \phi ^ { * } + \ph... |
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... | \frac { 1 } { \epsilon } \approx \ln ( \frac { 1 } { H _ { 5 } \delta } ) . |
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120... | Z _ { c y l } ^ { k / 4 } = \sum _ { j = 0 } ^ { k / 2 } \ \chi _ { j } ( \frac { i t } { 2 } ) . |
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26... | F _ { i j } = \sum _ { k = 1 } ^ { [ \frac { n } { 2 } ] } H _ { k } ( \delta _ { i } ^ { k } \delta _ { j } ^ { n + 1 - k } - \delta _ { j } ^ { k } \delta _ { i } ^ { n + 1 - k } ) |
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... | S [ h _ { g } , X ] = \frac { 1 } { 4 \pi \alpha ^ { \prime } } \int \sqrt { h } h ^ { a b } \partial _ { a } X ^ { \mu } \partial _ { b } X ^ { \mu } |
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16... | \delta \sigma \propto \omega ^ { 3 } R ^ { 6 } \ln ( \omega R ) , |
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... | { \cal R } _ { a b c d } = \frac 1 2 ( G _ { a d } G _ { b c } - G _ { a c } G _ { b d } ) { \cal R } + G _ { a c } { \cal R } |
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115,... | ( \tilde { X } ^ { I } ; \tilde { F } _ { I } ) | _ { \mathrm { 1 - l o o p } } = ( 1 , T ^ { a } \eta _ { a b } T ^ { b } , i T ^ { a } ; i S , i S T ^ { a } \eta _ { a b } T ^ { b } + 2 i h ( T ^ { a } ) - i T ^ { a } { \frac { \partial h } { \partial T ^ { a } } } , - S T ^ { a } + { \frac { \partial h } { \partial ... |
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... | C _ { i _ { p } \, j _ { q } } ^ { k _ { s } } = \left\{ \begin{array} { l l l } { { 0 , } } & { { \mathrm { i f } \ p + q \neq s } } \\ { { c _ { i _ { p } j _ { q } } ^ { k _ { s } } , } } & { { \mathrm { i f } \ p + q = s } } \end{array} \right. \quad \begin{array} { l } { { p = 0 , 1 , \ldots , n } } \\ { { i _ { p... |
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... | P _ { \pm } \equiv - i \nabla _ { \pm } \equiv - i ( \partial _ { \pm } - i A _ { \pm } ) , |
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... | U ( a , b , z ) = \frac { 1 } { \Gamma ( a ) } \, \int _ { 0 } ^ { \infty } d t \, t ^ { a - 1 } ~ ( 1 + t ) ^ { b - a - 1 } ~ e ^ { - z t } |
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... | p _ { d } = p _ { d } ( n , L ) = ( n + \frac 1 { 2 } ) \frac { \pi } { L } , \, \, \, \, \, n = 0 , 1 , 2 , 3 , . . . |
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1... | ( \Delta \otimes 1 ) \Delta ( P _ { \mu } ) = \Delta \cdot ( \Delta \otimes 1 ) ( P _ { \mu } ) \, , |
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1... | \chi _ { i } \left( - \ { \frac { 1 } { \tau } } \right) \ = \ \sum _ { j } \ S _ { i j } \ \chi _ { j } ( \tau ) \ \qquad \chi _ { i } ( \tau + 1 ) \ = \ \sum _ { j } \ T _ { i j } \ \chi _ { j } ( \tau ) \ , |
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13,... | S ( \phi ) = \int d x \left( \frac { 1 } { 2 } ( \partial _ { \mu } \phi ) ^ { 2 } - \frac { \mu ^ { 2 } } { 2 } \phi ^ { 2 } - \lambda \, \phi ^ { 4 } \right) . |
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... | \widetilde { G } ( x _ { 1 } , x _ { 2 } ) = \omega ( A \longrightarrow f ^ { c ( x _ { 1 } , x _ { 2 } ) } ) G ( A \mid x _ { 1 } , x _ { 2 } ) |
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... | \frac { \partial } { \partial t } \left( \begin{array} { c c } { { b } } \\ { { { \overline { { b } } } } } \end{array} \right) = \left( \begin{array} { c c } { { B } } \\ { { { \overline { { B } } } } } \end{array} \right) , |
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... | \Phi ^ { - } = - \frac { \partial ^ { I } } { \partial ^ { + } } \Phi ^ { I } + \frac { d - 3 } { \hat { \partial } ^ { + } } \Phi ^ { z } \, . |
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... | A _ { 4 } = { \frac { \mathrm { i } } { 2 } } { \frac { \nabla \rho } { \rho } } \cdot \vec { \tau } |
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