DLMF:19.25.E10 (Q6514): Difference between revisions

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Property / constraint
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c > k 2 𝑐 superscript π‘˜ 2 {\displaystyle{\displaystyle c>k^{2}}}

c>k^{2}
Property / constraint: c > k 2 𝑐 superscript π‘˜ 2 {\displaystyle{\displaystyle c>k^{2}}} / rank
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Normal rank
Property / Symbols used
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Property / Symbols used: elliptic integral symmetric in only two variables / rank
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Normal rank
Property / Symbols used: elliptic integral symmetric in only two variables / qualifier
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test:

R D ⁑ ( x , y , z ) Carlson-integral-RD π‘₯ 𝑦 𝑧 {\displaystyle{\displaystyle R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}

\CarlsonsymellintRD@{\NVar{x}}{\NVar{y}}{\NVar{z}}
Property / Symbols used: elliptic integral symmetric in only two variables / qualifier
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xml-id: C19.S16.E5.m2acdec
Property / Symbols used
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Property / Symbols used: symmetric elliptic integral of first kind / rank
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Normal rank
Property / Symbols used: symmetric elliptic integral of first kind / qualifier
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test:

R F ⁑ ( x , y , z ) Carlson-integral-RF π‘₯ 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}

\CarlsonsymellintRF@{\NVar{x}}{\NVar{y}}{\NVar{z}}
Property / Symbols used: symmetric elliptic integral of first kind / qualifier
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xml-id: C19.S16.E1.m2addec
Property / Symbols used
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Property / Symbols used: Legendre’s incomplete elliptic integral of the second kind / rank
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Normal rank
Property / Symbols used: Legendre’s incomplete elliptic integral of the second kind / qualifier
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test:

E ⁑ ( Ο• , k ) elliptic-integral-second-kind-E italic-Ο• π‘˜ {\displaystyle{\displaystyle E\left(\NVar{\phi},\NVar{k}\right)}}

\incellintEk@{\NVar{\phi}}{\NVar{k}}
Property / Symbols used: Legendre’s incomplete elliptic integral of the second kind / qualifier
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xml-id: C19.S2.E5.m2acdec
Property / Symbols used
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Property / Symbols used: Q11807 / rank
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Normal rank
Property / Symbols used: Q11807 / qualifier
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test:

Ο• italic-Ο• {\displaystyle{\displaystyle\phi}}

\phi
Property / Symbols used: Q11807 / qualifier
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xml-id: C19.S1.XMD4.m1edec
Property / Symbols used
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Property / Symbols used: Q11808 / rank
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Normal rank
Property / Symbols used: Q11808 / qualifier
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test:

k π‘˜ {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11808 / qualifier
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xml-id: C19.S1.XMD5.m1idec
Property / Symbols used
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Property / Symbols used: Q11815 / rank
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Normal rank
Property / Symbols used: Q11815 / qualifier
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test:

k β€² superscript π‘˜ β€² {\displaystyle{\displaystyle k^{\prime}}}

k^{\prime}
Property / Symbols used: Q11815 / qualifier
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xml-id: C19.S1.XMD6.m1cdec

Latest revision as of 13:06, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:19.25.E10
No description defined

    Statements

    test
    E ⁑ ( Ο• , k ) = k β€² 2 ⁒ R F ⁑ ( c - 1 , c - k 2 , c ) + 1 3 ⁒ k 2 ⁒ k β€² 2 ⁒ R D ⁑ ( c - 1 , c , c - k 2 ) + k 2 ⁒ ( c - 1 ) / ( c ⁒ ( c - k 2 ) ) , elliptic-integral-second-kind-E italic-Ο• π‘˜ superscript superscript π‘˜ β€² 2 Carlson-integral-RF 𝑐 1 𝑐 superscript π‘˜ 2 𝑐 1 3 superscript π‘˜ 2 superscript superscript π‘˜ β€² 2 Carlson-integral-RD 𝑐 1 𝑐 𝑐 superscript π‘˜ 2 superscript π‘˜ 2 𝑐 1 𝑐 𝑐 superscript π‘˜ 2 {\displaystyle{\displaystyle E\left(\phi,k\right)={k^{\prime}}^{2}R_{F}\left(c% -1,c-k^{2},c\right)+\tfrac{1}{3}k^{2}{k^{\prime}}^{2}R_{D}\left(c-1,c,c-k^{2}% \right)+k^{2}\sqrt{(c-1)/(c(c-k^{2}))},}}
    0 references
    DLMF id
    DLMF:19.25.E10
    0 references
    constraint
    c > k 2 𝑐 superscript π‘˜ 2 {\displaystyle{\displaystyle c>k^{2}}}
    0 references
    Symbols used
    elliptic integral symmetric in only two variables
    test
    R D ⁑ ( x , y , z ) Carlson-integral-RD π‘₯ 𝑦 𝑧 {\displaystyle{\displaystyle R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E5.m2acdec
    0 references
    symmetric elliptic integral of first kind
    test
    R F ⁑ ( x , y , z ) Carlson-integral-RF π‘₯ 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E1.m2addec
    0 references
    Legendre’s incomplete elliptic integral of the second kind
    test
    E ⁑ ( Ο• , k ) elliptic-integral-second-kind-E italic-Ο• π‘˜ {\displaystyle{\displaystyle E\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E5.m2acdec
    0 references
    Q11807
    test
    Ο• italic-Ο• {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1edec
    0 references
    Q11808
    test
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1idec
    0 references
    Q11815
    test
    k β€² superscript π‘˜ β€² {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1cdec
    0 references
     
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