DLMF:19.14.E4 (Q6309): Difference between revisions

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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.m1cdec
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.m1cdec

Latest revision as of 12:51, 2 January 2020

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DLMF:19.14.E4
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    Statements

    test
    ∫ y x d t ( a 1 + b 1 ⁢ t 2 ) ⁢ ( a 2 + b 2 ⁢ t 2 ) = 1 γ - α ⁢ F ⁡ ( ϕ , k ) , superscript subscript 𝑦 𝑥 𝑡 subscript 𝑎 1 subscript 𝑏 1 superscript 𝑡 2 subscript 𝑎 2 subscript 𝑏 2 superscript 𝑡 2 1 𝛾 𝛼 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle{\int_{y}^{x}\frac{\mathrm{d}t}{\sqrt{(a_{1}+b_{1}% t^{2})(a_{2}+b_{2}t^{2})}}=\frac{1}{\sqrt{\gamma-\alpha}}F\left(\phi,k\right)}% ,}}
    0 references
    DLMF id
    DLMF:19.14.E4
    0 references
    constraint
    k 2 = ( γ - β ) / ( γ - α ) superscript 𝑘 2 𝛾 𝛽 𝛾 𝛼 {\displaystyle{\displaystyle k^{2}=\ifrac{(\gamma-\beta)}{(\gamma-\alpha)}}}
    0 references
    Symbols used
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
    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.m2acdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
    Q11807
    test
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1cdec
    0 references
    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1cdec
    0 references
     
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