DLMF:19.2.E7 (Q6088): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aedec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aedec
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.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
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.m1dec

Latest revision as of 12:35, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:19.2.E7
No description defined

    Statements

    test
    Π ⁡ ( ϕ , α 2 , k ) = ∫ 0 ϕ d θ 1 - k 2 ⁢ sin 2 ⁡ θ ⁢ ( 1 - α 2 ⁢ sin 2 ⁡ θ ) = ∫ 0 sin ⁡ ϕ d t 1 - t 2 ⁢ 1 - k 2 ⁢ t 2 ⁢ ( 1 - α 2 ⁢ t 2 ) . elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 superscript subscript 0 italic-ϕ 𝜃 1 superscript 𝑘 2 2 𝜃 1 superscript 𝛼 2 2 𝜃 superscript subscript 0 italic-ϕ 𝑡 1 superscript 𝑡 2 1 superscript 𝑘 2 superscript 𝑡 2 1 superscript 𝛼 2 superscript 𝑡 2 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)=\int_{0}^{\phi}% \frac{\mathrm{d}\theta}{\sqrt{1-k^{2}{\sin^{2}}\theta}(1-\alpha^{2}{\sin^{2}}% \theta)}=\int_{0}^{\sin\phi}\frac{\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2% }}(1-\alpha^{2}t^{2})}.}}
    0 references
    DLMF id
    DLMF:19.2.E7
    0 references
    Symbols used
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    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.m1cdec
    0 references
    Q11808
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1cdec
    0 references
    Q11809
    test
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1dec
    0 references
     
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