DLMF:19.14.E3 (Q6308): Difference between revisions

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Property / constraint
 

cos ⁡ ϕ = 1 - x 2 1 + x 2 italic-ϕ 1 superscript 𝑥 2 1 superscript 𝑥 2 {\displaystyle{\displaystyle\cos\phi=\dfrac{1-x^{2}}{1+x^{2}}}}

\cos\phi=\dfrac{1-x^{2}}{1+x^{2}}
Property / constraint: cos ⁡ ϕ = 1 - x 2 1 + x 2 italic-ϕ 1 superscript 𝑥 2 1 superscript 𝑥 2 {\displaystyle{\displaystyle\cos\phi=\dfrac{1-x^{2}}{1+x^{2}}}} / rank
 
Normal rank
Property / constraint
 

k 2 = 1 2 superscript 𝑘 2 1 2 {\displaystyle{\displaystyle k^{2}=\dfrac{1}{2}}}

k^{2}=\dfrac{1}{2}
Property / constraint: k 2 = 1 2 superscript 𝑘 2 1 2 {\displaystyle{\displaystyle k^{2}=\dfrac{1}{2}}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
Defining formula:

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: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

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: 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
 
Defining formula:

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.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3abdec
Property / Symbols used
 
Property / Symbols used: Q10897 / rank
 
Normal rank
Property / Symbols used: Q10897 / qualifier
 
Defining formula:

sign ⁡ x sign 𝑥 {\displaystyle{\displaystyle\operatorname{sign}\NVar{x}}}

\sign@@{\NVar{x}}
Property / Symbols used: Q10897 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r18.m6adec
Property / Symbols used
 
Property / Symbols used: Q11807 / rank
 
Normal rank
Property / Symbols used: Q11807 / qualifier
 
Defining formula:

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

\phi
Property / Symbols used: Q11807 / qualifier
 
xml-id: C19.S1.XMD4.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11808 / rank
 
Normal rank
Property / Symbols used: Q11808 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11808 / qualifier
 
xml-id: C19.S1.XMD5.m1bdec

Latest revision as of 13:51, 2 January 2020

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DLMF:19.14.E3
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    Statements

    Defining formula
    ∫ 0 x d t 1 + t 4 = sign ⁡ ( x ) 2 ⁢ F ⁡ ( ϕ , k ) , superscript subscript 0 𝑥 𝑡 1 superscript 𝑡 4 sign 𝑥 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{4}}}=% \frac{\operatorname{sign}\left(x\right)}{2}F\left(\phi,k\right),}}
    0 references
    DLMF id
    DLMF:19.14.E3
    0 references
    constraint
    cos ⁡ ϕ = 1 - x 2 1 + x 2 italic-ϕ 1 superscript 𝑥 2 1 superscript 𝑥 2 {\displaystyle{\displaystyle\cos\phi=\dfrac{1-x^{2}}{1+x^{2}}}}
    0 references
    cos ⁡ ϕ = 1 - x 2 1 + x 2 italic-ϕ 1 superscript 𝑥 2 1 superscript 𝑥 2 {\displaystyle{\displaystyle\cos\phi=\dfrac{1-x^{2}}{1+x^{2}}}}
    0 references
    k 2 = 1 2 superscript 𝑘 2 1 2 {\displaystyle{\displaystyle k^{2}=\dfrac{1}{2}}}
    0 references
    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Legendre’s incomplete elliptic integral of the first kind
    Defining formula
    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.m2abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q10897
    Defining formula
    sign ⁡ x sign 𝑥 {\displaystyle{\displaystyle\operatorname{sign}\NVar{x}}}
    xml-id
    introduction.Sx4.p2.t1.r18.m6adec
    0 references
    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1bdec
    0 references
    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1bdec
    0 references
     
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