DLMF:22.11.E11 (Q7035): Difference between revisions

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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.m2aedec
Property / Symbols used
 
Property / Symbols used: nome / rank
 
Normal rank
Property / Symbols used: nome / qualifier
 
Defining formula:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: nome / qualifier
 
xml-id: C22.S2.E1.m2ajdec
Property / Symbols used
 
Property / Symbols used: secant function / rank
 
Normal rank
Property / Symbols used: secant function / qualifier
 
Defining formula:

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

\sec@@{\NVar{z}}
Property / Symbols used: secant function / qualifier
 
xml-id: C4.S14.E6.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11985 / rank
 
Normal rank
Property / Symbols used: Q11985 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11985 / qualifier
 
xml-id: C22.S1.XMD3.m1jdec
Property / Symbols used
 
Property / Symbols used: Q11984 / rank
 
Normal rank
Property / Symbols used: Q11984 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11984 / qualifier
 
xml-id: C22.S1.XMD4.m1jdec
Property / Symbols used
 
Property / Symbols used: Q11988 / rank
 
Normal rank
Property / Symbols used: Q11988 / qualifier
 
Defining formula:

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

k^{\prime}
Property / Symbols used: Q11988 / qualifier
 
xml-id: C22.S1.XMD5.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11986 / rank
 
Normal rank
Property / Symbols used: Q11986 / qualifier
 
Defining formula:

ζ 𝜁 {\displaystyle{\displaystyle\zeta}}

\zeta
Property / Symbols used: Q11986 / qualifier
 
xml-id: C22.S2.XMD2.m1jdec

Latest revision as of 15:14, 2 January 2020

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DLMF:22.11.E11
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    Statements

    Defining formula
    nc ⁡ ( z , k ) - π 2 ⁢ K ⁢ k ′ ⁢ sec ⁡ ζ = - 2 ⁢ π K ⁢ k ′ ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ q 2 ⁢ n + 1 ⁢ cos ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 + q 2 ⁢ n + 1 , Jacobi-elliptic-nc 𝑧 𝑘 𝜋 2 𝐾 superscript 𝑘 ′ 𝜁 2 𝜋 𝐾 superscript 𝑘 ′ superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 2 𝑛 1 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle\operatorname{nc}\left(z,k\right)-\frac{\pi}{2Kk^{% \prime}}\sec\zeta=-\frac{2\pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^% {2n+1}\cos\left((2n+1)\zeta\right)}{1+q^{2n+1}},}}
    0 references
    DLMF id
    DLMF:22.11.E11
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    nc ⁡ ( z , k ) Jacobi-elliptic-nc 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{nc}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m3adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
    0 references
    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1ajdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
    0 references
    nome
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C22.S2.E1.m2ajdec
    0 references
    secant function
    Defining formula
    sec ⁡ z 𝑧 {\displaystyle{\displaystyle\sec\NVar{z}}}
    xml-id
    C4.S14.E6.m2aadec
    0 references
    Q11985
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C22.S1.XMD3.m1jdec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1jdec
    0 references
    Q11988
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C22.S1.XMD5.m1bdec
    0 references
    Q11986
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C22.S2.XMD2.m1jdec
    0 references
     
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