DLMF:19.30.E6 (Q6680): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: cosecant function / rank
 
Normal rank
Property / Symbols used: cosecant function / qualifier
 
test:

csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}

\csc@@{\NVar{z}}
Property / Symbols used: cosecant function / qualifier
 
xml-id: C4.S14.E5.m2aadec
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.m2adec
Property / Symbols used
 
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / rank
 
Normal rank
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / qualifier
 
test:

∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}

\pderiv{\NVar{f}}{\NVar{x}}
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / qualifier
 
xml-id: C1.S5.E3.m4adec
Property / Symbols used
 
Property / Symbols used: partial differential of $$x$$ / rank
 
Normal rank
Property / Symbols used: partial differential of $$x$$ / qualifier
 
test:

∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}

\pdiff{\NVar{x}}
Property / Symbols used: partial differential of $$x$$ / qualifier
 
xml-id: C1.S5.E3.m2adec
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.m1ddec
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.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11915 / rank
 
Normal rank
Property / Symbols used: Q11915 / qualifier
 
test:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q11915 / qualifier
 
xml-id: C19.S30.XMD1.m1bdec

Latest revision as of 13:23, 2 January 2020

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DLMF:19.30.E6
No description defined

    Statements

    test
    ∂ ⁡ s ∂ ⁡ ( 1 / k ) = a 2 - b 2 ⁢ F ⁡ ( ϕ , k ) = a 2 - b 2 ⁢ R F ⁡ ( c - 1 , c - k 2 , c ) , partial-derivative 𝑠 1 𝑘 superscript 𝑎 2 superscript 𝑏 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 superscript 𝑎 2 superscript 𝑏 2 Carlson-integral-RF 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle\frac{\partial s}{\partial(1/k)}=\sqrt{a^{2}-b^{2}% }F\left(\phi,k\right)=\sqrt{a^{2}-b^{2}}R_{F}\left(c-1,c-k^{2},c\right),}}
    0 references
    DLMF id
    DLMF:19.30.E6
    0 references
    constraint
    c = csc 2 ⁡ ϕ 𝑐 2 italic-ϕ {\displaystyle{\displaystyle c={\csc^{2}}\phi}}
    0 references
    k 2 = ( a 2 - b 2 ) / ( a 2 + λ ) superscript 𝑘 2 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑎 2 𝜆 {\displaystyle{\displaystyle k^{2}=(a^{2}-b^{2})/(a^{2}+\lambda)}}
    0 references
    c = csc 2 ⁢ ϕ 𝑐 2 italic-ϕ {\displaystyle{\displaystyle c={{\csc^{2}}}\phi}}
    0 references
    Symbols used
    symmetric elliptic integral of first kind
    test
    R F ⁡ ( x , y , z ) Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E1.m2aadec
    0 references
    cosecant function
    test
    csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}
    xml-id
    C4.S14.E5.m2aadec
    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.m2adec
    0 references
    partial derivative of $$f$$ with respect to $$x$$
    test
    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4adec
    0 references
    partial differential of $$x$$
    test
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2adec
    0 references
    Q11807
    test
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1ddec
    0 references
    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1ddec
    0 references
    Q11915
    test
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C19.S30.XMD1.m1bdec
    0 references
     
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