DLMF:22.12.E13 (Q7051): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: tangent function / rank
 
Normal rank
Property / Symbols used: tangent function / qualifier
 
Defining formula:

tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}

\tan@@{\NVar{z}}
Property / Symbols used: tangent function / qualifier
 
xml-id: C4.S14.E4.m2acdec

Revision as of 15:18, 2 January 2020

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DLMF:22.12.E13
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    Statements

    Defining formula
    2 ⁢ K ⁢ cs ⁡ ( 2 ⁢ K ⁢ t , k ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t - n ⁢ τ ) ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ ( lim M → ∞ ⁡ ∑ m = - M M 1 t - m - n ⁢ τ ) . 2 𝐾 Jacobi-elliptic-cs 2 𝐾 𝑡 𝑘 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 𝑛 𝜏 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 subscript → 𝑀 superscript subscript 𝑚 𝑀 𝑀 1 𝑡 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle 2K\operatorname{cs}\left(2Kt,k\right)=\lim_{N\to% \infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi(t-n\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\tau}\right).}}
    0 references
    DLMF id
    DLMF:22.12.E13
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    cs ⁡ ( z , k ) Jacobi-elliptic-cs 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cs}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E9.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.m2akdec
    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
    tangent function
    Defining formula
    tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}
    xml-id
    C4.S14.E4.m2acdec
    0 references
     
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