DLMF:22.12.E4 (Q7042): Difference between revisions

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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.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11987 / rank
 
Normal rank
Property / Symbols used: Q11987 / qualifier
 
Defining formula:

τ 𝜏 {\displaystyle{\displaystyle\tau}}

\tau
Property / Symbols used: Q11987 / qualifier
 
xml-id: C22.S1.XMD6.m1cdec

Latest revision as of 15:16, 2 January 2020

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DLMF:22.12.E4
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    Statements

    Defining formula
    2 ⁢ i ⁢ K ⁢ dn ⁡ ( 2 ⁢ K ⁢ t , k ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t - ( n + 1 2 ) ⁢ τ ) ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ ( lim M → ∞ ⁡ ∑ m = - M M 1 t - m - ( n + 1 2 ) ⁢ τ ) . 2 𝑖 𝐾 Jacobi-elliptic-dn 2 𝐾 𝑡 𝑘 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 𝑛 1 2 𝜏 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 subscript → 𝑀 superscript subscript 𝑚 𝑀 𝑀 1 𝑡 𝑚 𝑛 1 2 𝜏 {\displaystyle{\displaystyle 2iK\operatorname{dn}\left(2Kt,k\right)=\lim_{N\to% \infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi(t-(n+\frac{1}{2})\tau)% \right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\left(\lim_{M\to\infty}\sum_{% m=-M}^{M}\frac{1}{t-m-(n+\frac{1}{2})\tau}\right).}}
    0 references
    DLMF id
    DLMF:22.12.E4
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    dn ⁡ ( z , k ) Jacobi-elliptic-dn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    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.m1acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    tangent function
    Defining formula
    tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}
    xml-id
    C4.S14.E4.m2adec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1cdec
    0 references
    Q11987
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C22.S1.XMD6.m1cdec
    0 references
     
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