DLMF:19.4.E9 (Q6129): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
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Property / Symbols used
 
Property / Symbols used: Legendre’s incomplete elliptic integral of the second kind / rank
 
Normal rank
Property / Symbols used: Legendre’s incomplete elliptic integral of the second kind / qualifier
 
Defining formula:

E ( ϕ , k ) elliptic-integral-second-kind-E italic-ϕ 𝑘 {\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
 
xml-id: C19.S2.E5.m2acdec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: Q11807 / rank
 
Normal rank
Property / Symbols used: Q11807 / qualifier
 
Defining formula:

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

\phi
Property / Symbols used: Q11807 / qualifier
 
xml-id: C19.S1.XMD4.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11808 / rank
 
Normal rank
Property / Symbols used: Q11808 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11808 / qualifier
 
xml-id: C19.S1.XMD5.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11815 / rank
 
Normal rank
Property / Symbols used: Q11815 / qualifier
 
Defining formula:

k superscript 𝑘 {\displaystyle{\displaystyle k^{\prime}}}

k^{\prime}
Property / Symbols used: Q11815 / qualifier
 
xml-id: C19.S1.XMD6.m1gdec
Property / Symbols used
 
Property / Symbols used: Q11817 / rank
 
Normal rank
Property / Symbols used: Q11817 / qualifier
 
Defining formula:

D k subscript 𝐷 𝑘 {\displaystyle{\displaystyle D_{k}}}

D_{k}
Property / Symbols used: Q11817 / qualifier
 
xml-id: C19.S4.XMD1.m1adec

Latest revision as of 13:38, 2 January 2020

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DLMF:19.4.E9
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    Statements

    ( k k 2 D k 2 + k 2 D k + k ) E ( ϕ , k ) = k sin ϕ cos ϕ 1 - k 2 sin 2 ϕ . 𝑘 superscript superscript 𝑘 2 superscript subscript 𝐷 𝑘 2 superscript superscript 𝑘 2 subscript 𝐷 𝑘 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 𝑘 italic-ϕ italic-ϕ 1 superscript 𝑘 2 2 italic-ϕ {\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+% k)E\left(\phi,k\right)=\frac{k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}.}}
    0 references
    DLMF:19.4.E9
    0 references
    cos z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    C4.S14.E2.m2acdec
    0 references
    E ( ϕ , k ) elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle E\left(\NVar{\phi},\NVar{k}\right)}}
    C19.S2.E5.m2acdec
    0 references
    sin z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    C4.S14.E1.m2acdec
    0 references
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    C19.S1.XMD4.m1ddec
    0 references
    k 𝑘 {\displaystyle{\displaystyle k}}
    C19.S1.XMD5.m1hdec
    0 references
    k superscript 𝑘 {\displaystyle{\displaystyle k^{\prime}}}
    C19.S1.XMD6.m1gdec
    0 references
    D k subscript 𝐷 𝑘 {\displaystyle{\displaystyle D_{k}}}
    C19.S4.XMD1.m1adec
    0 references