DLMF:19.9.E16 (Q6259): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11600 / rank
 
Normal rank
Property / Symbols used: Q11600 / qualifier
 
test:

K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}

\compellintKk@{\NVar{k}}
Property / Symbols used: Q11600 / qualifier
 
xml-id: C19.S2.E8.m1agdec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
test:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.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.m2acdec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2aedec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

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
 
test:

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

\phi
Property / Symbols used: Q11807 / qualifier
 
xml-id: C19.S1.XMD4.m1edec
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.m1ndec
Property / Symbols used
 
Property / Symbols used: Q11815 / rank
 
Normal rank
Property / Symbols used: Q11815 / qualifier
 
test:

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

k^{\prime}
Property / Symbols used: Q11815 / qualifier
 
xml-id: C19.S1.XMD6.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11839 / rank
 
Normal rank
Property / Symbols used: Q11839 / qualifier
 
test:

Δ Δ {\displaystyle{\displaystyle\Delta}}

\Delta
Property / Symbols used: Q11839 / qualifier
 
xml-id: C19.S9.XMD2.m1edec

Latest revision as of 12:46, 2 January 2020

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English
DLMF:19.9.E16
No description defined

    Statements

    test
    F ⁡ ( ϕ , k ) = 2 π ⁢ K ⁡ ( k ′ ) ⁢ ln ⁡ ( 4 Δ + cos ⁡ ϕ ) - θ ⁢ Δ 2 , elliptic-integral-first-kind-F italic-ϕ 𝑘 2 𝜋 complete-elliptic-integral-first-kind-K superscript 𝑘 ′ 4 Δ italic-ϕ 𝜃 superscript Δ 2 {\displaystyle{\displaystyle F\left(\phi,k\right)=\frac{2}{\pi}K\left(k^{% \prime}\right)\ln\left(\frac{4}{\Delta+\cos\phi}\right)-\theta\Delta^{2},}}
    0 references
    DLMF id
    DLMF:19.9.E16
    0 references
    constraint
    ( sin ⁡ ϕ ) / 8 < θ < ( ln ⁡ 2 ) / ( k 2 ⁢ sin ⁡ ϕ ) italic-ϕ 8 𝜃 2 superscript 𝑘 2 italic-ϕ {\displaystyle{\displaystyle(\sin\phi)/8<\theta<(\ln 2)/(k^{2}\sin\phi)}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    Q11600
    test
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1agdec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.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.m2acdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aedec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
    Q11807
    test
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1edec
    0 references
    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1ndec
    0 references
    Q11815
    test
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1hdec
    0 references
    Q11839
    test
    Δ Δ {\displaystyle{\displaystyle\Delta}}
    xml-id
    C19.S9.XMD2.m1edec
    0 references
     
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