DLMF:22.11.E13 (Q7037): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(2 intermediate revisions by the same user not shown)
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.m1ldec
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.m1ldec
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.m1ldec

Latest revision as of 15:15, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:22.11.E13
No description defined

    Statements

    Defining formula
    sn 2 ⁡ ( z , k ) = 1 k 2 ⁢ ( 1 - E ⁡ K ) - 2 ⁢ π 2 k 2 ⁢ K 2 ⁢ ∑ n = 1 ∞ n ⁢ q n 1 - q 2 ⁢ n ⁢ cos ⁡ ( 2 ⁢ n ⁢ ζ ) . Jacobi-elliptic-sn 2 𝑧 𝑘 1 superscript 𝑘 2 1 complete-elliptic-integral-second-kind-E 𝑘 𝐾 2 superscript 𝜋 2 superscript 𝑘 2 superscript 𝐾 2 superscript subscript 𝑛 1 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝜁 {\displaystyle{\displaystyle{\operatorname{sn}^{2}}\left(z,k\right)=\frac{1}{k% ^{2}}\left(1-\frac{E}{K}\right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}% \frac{nq^{n}}{1-q^{2n}}\cos\left(2n\zeta\right).}}
    0 references
    DLMF id
    DLMF:22.11.E13
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    sn ⁡ ( z , k ) Jacobi-elliptic-sn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aldec
    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.m1aldec
    0 references
    Q11601
    Defining formula
    E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2afdec
    0 references
    nome
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C22.S2.E1.m2aldec
    0 references
    Q11985
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C22.S1.XMD3.m1ldec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1ldec
    0 references
    Q11986
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C22.S2.XMD2.m1ldec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q7037&oldid=55855"