DLMF:19.7.E10 (Q6205): Difference between revisions

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Property / constraint
 

( k 2 - α 2 ) ⁢ ( k 2 - ω 2 ) = k 2 ⁢ ( k 2 - 1 ) superscript 𝑘 2 superscript 𝛼 2 superscript 𝑘 2 superscript 𝜔 2 superscript 𝑘 2 superscript 𝑘 2 1 {\displaystyle{\displaystyle(k^{2}-\alpha^{2})(k^{2}-\omega^{2})=k^{2}(k^{2}-1% )}}

(k^{2}-\alpha^{2})(k^{2}-\omega^{2})=k^{2}(k^{2}-1)
Property / constraint: ( k 2 - α 2 ) ⁢ ( k 2 - ω 2 ) = k 2 ⁢ ( k 2 - 1 ) superscript 𝑘 2 superscript 𝛼 2 superscript 𝑘 2 superscript 𝜔 2 superscript 𝑘 2 superscript 𝑘 2 1 {\displaystyle{\displaystyle(k^{2}-\alpha^{2})(k^{2}-\omega^{2})=k^{2}(k^{2}-1% )}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Carlson’s combination of inverse circular and inverse hyperbolic functions / rank
 
Normal rank
Property / Symbols used: Carlson’s combination of inverse circular and inverse hyperbolic functions / qualifier
 
test:

R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}

\CarlsonellintRC@{\NVar{x}}{\NVar{y}}
Property / Symbols used: Carlson’s combination of inverse circular and inverse hyperbolic functions / qualifier
 
xml-id: C19.S2.E17.m2abdec
Property / Symbols used
 
Property / Symbols used: Legendre’s incomplete elliptic integral of the first kind / rank
 
Normal rank
Property / Symbols used: Legendre’s incomplete elliptic integral of the first kind / qualifier
 
test:

F ⁡ ( ϕ , k ) elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}

\incellintFk@{\NVar{\phi}}{\NVar{k}}
Property / Symbols used: Legendre’s incomplete elliptic integral of the first kind / qualifier
 
xml-id: C19.S2.E4.m2aedec
Property / Symbols used
 
Property / Symbols used: Legendre’s incomplete elliptic integral of the third kind / rank
 
Normal rank
Property / Symbols used: Legendre’s incomplete elliptic integral of the third kind / qualifier
 
test:

Π ⁡ ( ϕ , α 2 , k ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}% \right)}}

\incellintPik@{\NVar{\phi}}{\NVar{\alpha}^{2}}{\NVar{k}}
Property / Symbols used: Legendre’s incomplete elliptic integral of the third kind / qualifier
 
xml-id: C19.S2.E7.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11807 / rank
 
Normal rank
Property / Symbols used: Q11807 / qualifier
 
test:

ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}

\phi
Property / Symbols used: Q11807 / qualifier
 
xml-id: C19.S1.XMD4.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11808 / rank
 
Normal rank
Property / Symbols used: Q11808 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11808 / qualifier
 
xml-id: C19.S1.XMD5.m1idec
Property / Symbols used
 
Property / Symbols used: Q11809 / rank
 
Normal rank
Property / Symbols used: Q11809 / qualifier
 
test:

α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}

\alpha^{2}
Property / Symbols used: Q11809 / qualifier
 
xml-id: C19.S1.XMD7.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11823 / rank
 
Normal rank
Property / Symbols used: Q11823 / qualifier
 
test:

c 𝑐 {\displaystyle{\displaystyle c}}

c
Property / Symbols used: Q11823 / qualifier
 
xml-id: C19.S7.XMD9.m1bdec

Latest revision as of 12:42, 2 January 2020

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DLMF:19.7.E10
No description defined

    Statements

    test
    ( 1 - α 2 ) ⁢ Π ⁡ ( ϕ , α 2 , k ) + ( 1 - ω 2 ) ⁢ Π ⁡ ( ϕ , ω 2 , k ) = F ⁡ ( ϕ , k ) + ( 1 - α 2 - ω 2 ) ⁢ c - k 2 ⁢ R C ⁡ ( c ⁢ ( c - 1 ) , ( c - α 2 ) ⁢ ( c - ω 2 ) ) , 1 superscript 𝛼 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 1 superscript 𝜔 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝜔 2 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 1 superscript 𝛼 2 superscript 𝜔 2 𝑐 superscript 𝑘 2 Carlson-integral-RC 𝑐 𝑐 1 𝑐 superscript 𝛼 2 𝑐 superscript 𝜔 2 {\displaystyle{\displaystyle(1-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(1% -\omega^{2})\Pi\left(\phi,\omega^{2},k\right)=F\left(\phi,k\right)+(1-\alpha^{% 2}-\omega^{2})\sqrt{c-k^{2}}\*R_{C}\left(c(c-1),(c-\alpha^{2})(c-\omega^{2})% \right),}}
    0 references
    DLMF id
    DLMF:19.7.E10
    0 references
    constraint
    ( k 2 - α 2 ) ⁢ ( k 2 - ω 2 ) = k 2 ⁢ ( k 2 - 1 ) superscript 𝑘 2 superscript 𝛼 2 superscript 𝑘 2 superscript 𝜔 2 superscript 𝑘 2 superscript 𝑘 2 1 {\displaystyle{\displaystyle(k^{2}-\alpha^{2})(k^{2}-\omega^{2})=k^{2}(k^{2}-1% )}}
    0 references
    Symbols used
    Carlson’s combination of inverse circular and inverse hyperbolic functions
    test
    R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}
    xml-id
    C19.S2.E17.m2abdec
    0 references
    Legendre’s incomplete elliptic integral of the first kind
    test
    F ⁡ ( ϕ , k ) elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E4.m2aedec
    0 references
    Legendre’s incomplete elliptic integral of the third kind
    test
    Π ⁡ ( ϕ , α 2 , k ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}% \right)}}
    xml-id
    C19.S2.E7.m2aedec
    0 references
    Q11807
    test
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1fdec
    0 references
    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1idec
    0 references
    Q11809
    test
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1fdec
    0 references
    Q11823
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C19.S7.XMD9.m1bdec
    0 references
     
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