DLMF:19.30.E11 (Q6686): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: symmetric elliptic integral of first kind / rank
 
Normal rank
Property / Symbols used: symmetric elliptic integral of first kind / qualifier
 
test:

R F ⁡ ( x , y , z ) Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}

\CarlsonsymellintRF@{\NVar{x}}{\NVar{y}}{\NVar{z}}
Property / Symbols used: symmetric elliptic integral of first kind / qualifier
 
xml-id: C19.S16.E1.m2abdec
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.m1abdec
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.m3abdec
Property / Symbols used
 
Property / Symbols used: secant function / rank
 
Normal rank
Property / Symbols used: secant function / qualifier
 
test:

sec ⁡ z 𝑧 {\displaystyle{\displaystyle\sec\NVar{z}}}

\sec@@{\NVar{z}}
Property / Symbols used: secant function / qualifier
 
xml-id: C4.S14.E6.m2adec
Property / Symbols used
 
Property / Symbols used: Q11918 / rank
 
Normal rank
Property / Symbols used: Q11918 / qualifier
 
test:

r 𝑟 {\displaystyle{\displaystyle r}}

r
Property / Symbols used: Q11918 / qualifier
 
xml-id: C19.S30.XMD4.m1adec
Property / Symbols used
 
Property / Symbols used: Q11919 / rank
 
Normal rank
Property / Symbols used: Q11919 / qualifier
 
test:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11919 / qualifier
 
xml-id: C19.S30.XMD5.m1dec
Property / Symbols used
 
Property / Symbols used: Q11920 / rank
 
Normal rank
Property / Symbols used: Q11920 / qualifier
 
test:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q11920 / qualifier
 
xml-id: C19.S30.XMD6.m1dec

Latest revision as of 13:23, 2 January 2020

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DLMF:19.30.E11
No description defined

    Statements

    test
    s = 2 ⁢ a 2 ⁢ ∫ 0 r d t 4 ⁢ a 4 - t 4 = 2 ⁢ a 2 ⁢ R F ⁡ ( q - 1 , q , q + 1 ) , 𝑠 2 superscript 𝑎 2 superscript subscript 0 𝑟 𝑡 4 superscript 𝑎 4 superscript 𝑡 4 2 superscript 𝑎 2 Carlson-integral-RF 𝑞 1 𝑞 𝑞 1 {\displaystyle{\displaystyle s=2a^{2}\int_{0}^{r}\frac{\mathrm{d}t}{\sqrt{4a^{% 4}-t^{4}}}=\sqrt{2a^{2}}R_{F}\left(q-1,q,q+1\right),}}
    0 references
    DLMF id
    DLMF:19.30.E11
    0 references
    constraint
    q = 2 ⁢ a 2 / r 2 = sec ⁡ ( 2 ⁢ θ ) 𝑞 2 superscript 𝑎 2 superscript 𝑟 2 2 𝜃 {\displaystyle{\displaystyle q=2a^{2}/r^{2}=\sec\left(2\theta\right)}}
    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.m2abdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    secant function
    test
    sec ⁡ z 𝑧 {\displaystyle{\displaystyle\sec\NVar{z}}}
    xml-id
    C4.S14.E6.m2adec
    0 references
    Q11918
    test
    r 𝑟 {\displaystyle{\displaystyle r}}
    xml-id
    C19.S30.XMD4.m1adec
    0 references
    Q11919
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C19.S30.XMD5.m1dec
    0 references
    Q11920
    test
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C19.S30.XMD6.m1dec
    0 references
     
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