DLMF:23.8.E7 (Q7289): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12059 / rank
 
Normal rank
Property / Symbols used: Q12059 / qualifier
 
Defining formula:

η j subscript 𝜂 𝑗 {\displaystyle{\displaystyle\eta_{j}}}

\eta_{j}
Property / Symbols used: Q12059 / qualifier
 
xml-id: C23.S2.XMD6.m1cdec

Latest revision as of 15:57, 2 January 2020

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DLMF:23.8.E7
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    Statements

    Defining formula
    σ ⁡ ( z ) = 2 ⁢ ω 1 π ⁢ exp ⁡ ( η 1 ⁢ z 2 2 ⁢ ω 1 ) ⁢ sin ⁡ ( π ⁢ z 2 ⁢ ω 1 ) ⁢ ∏ n = 1 ∞ sin ⁡ ( π ⁢ ( 2 ⁢ n ⁢ ω 3 + z ) / ( 2 ⁢ ω 1 ) ) ⁢ sin ⁡ ( π ⁢ ( 2 ⁢ n ⁢ ω 3 - z ) / ( 2 ⁢ ω 1 ) ) sin 2 ⁡ ( π ⁢ n ⁢ ω 3 / ω 1 ) . Weierstrass-sigma-on-lattice 𝑧 𝕃 2 subscript 𝜔 1 𝜋 subscript 𝜂 1 superscript 𝑧 2 2 subscript 𝜔 1 𝜋 𝑧 2 subscript 𝜔 1 superscript subscript product 𝑛 1 𝜋 2 𝑛 subscript 𝜔 3 𝑧 2 subscript 𝜔 1 𝜋 2 𝑛 subscript 𝜔 3 𝑧 2 subscript 𝜔 1 2 𝜋 𝑛 subscript 𝜔 3 subscript 𝜔 1 {\displaystyle{\displaystyle\sigma\left(z\right)=\frac{2\omega_{1}}{\pi}\exp% \left(\frac{\eta_{1}z^{2}}{2\omega_{1}}\right)\sin\left(\frac{\pi z}{2\omega_{% 1}}\right)\prod_{n=1}^{\infty}\frac{\sin\left(\pi(2n\omega_{3}+z)/(2\omega_{1}% )\right)\sin\left(\pi(2n\omega_{3}-z)/(2\omega_{1})\right)}{{\sin^{2}}\left(% \pi n\omega_{3}/\omega_{1}\right)}.}}
    0 references
    DLMF id
    DLMF:23.8.E7
    0 references
    Symbols used
    DLMF:23.2.E6
    Defining formula
    σ ⁡ ( z ) Weierstrass-sigma-on-lattice 𝑧 𝕃 {\displaystyle{\displaystyle\sigma\left(\NVar{z}\right)}}
    xml-id
    C23.S2.E6.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.m2afdec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    Q12055
    Defining formula
    𝕃 𝕃 {\displaystyle{\displaystyle\mathbb{L}}}
    xml-id
    C23.S1.XMD1.m1edec
    0 references
    Q12060
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C23.S1.XMD3.m1fdec
    0 references
    Q12056
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C23.S1.XMD7.m1edec
    0 references
    Q12049
    Defining formula
    ω 1 subscript 𝜔 1 {\displaystyle{\displaystyle\omega_{1}}}
    xml-id
    C23.S1.XMD8.m1fdec
    0 references
    Q12059
    Defining formula
    η j subscript 𝜂 𝑗 {\displaystyle{\displaystyle\eta_{j}}}
    xml-id
    C23.S2.XMD6.m1cdec
    0 references
     
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