DLMF:23.6.E15 (Q7256): Difference between revisions

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Property / constraint
 

j = 1 , 2 , 3 𝑗 1 2 3 {\displaystyle{\displaystyle j=1,2,3}}

j=1,2,3
Property / constraint: j = 1 , 2 , 3 𝑗 1 2 3 {\displaystyle{\displaystyle j=1,2,3}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q11945 / rank
 
Normal rank
Property / Symbols used: Q11945 / qualifier
 
Defining formula:

θ j ⁡ ( z , q ) Jacobi-theta 𝑗 𝑧 𝑞 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)}}

\Jacobithetaq{\NVar{j}}@{\NVar{z}}{\NVar{q}}
Property / Symbols used: Q11945 / qualifier
 
xml-id: C20.S2.SS1.m2amdec
Property / Symbols used
 
Property / Symbols used: DLMF:23.2.E6 / rank
 
Normal rank
Property / Symbols used: DLMF:23.2.E6 / qualifier
 
Defining formula:

σ ⁡ ( z ) Weierstrass-sigma-on-lattice 𝑧 𝕃 {\displaystyle{\displaystyle\sigma\left(\NVar{z}\right)}}

\Weierstrasssigmalatt@@{\NVar{z}}{\NVar{\mathbb{L}}}
Property / Symbols used: DLMF:23.2.E6 / qualifier
 
xml-id: C23.S2.E6.m2addec
Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
Defining formula:

exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2addec
Property / Symbols used
 
Property / Symbols used: Q12055 / rank
 
Normal rank
Property / Symbols used: Q12055 / qualifier
 
Defining formula:

𝕃 𝕃 {\displaystyle{\displaystyle\mathbb{L}}}

\mathbb{L}
Property / Symbols used: Q12055 / qualifier
 
xml-id: C23.S1.XMD1.m1ldec
Property / Symbols used
 
Property / Symbols used: Q12066 / rank
 
Normal rank
Property / Symbols used: Q12066 / qualifier
 
Defining formula:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q12066 / qualifier
 
xml-id: C23.S1.XMD10.m1ndec
Property / Symbols used
 
Property / Symbols used: Q12056 / rank
 
Normal rank
Property / Symbols used: Q12056 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q12056 / qualifier
 
xml-id: C23.S1.XMD7.m1fdec
Property / Symbols used
 
Property / Symbols used: Q12049 / rank
 
Normal rank
Property / Symbols used: Q12049 / qualifier
 
Defining formula:

ω 1 subscript 𝜔 1 {\displaystyle{\displaystyle\omega_{1}}}

\omega_{1}
Property / Symbols used: Q12049 / qualifier
 
xml-id: C23.S1.XMD8.m1ndec
Property / Symbols used
 
Property / Symbols used: Q12067 / rank
 
Normal rank
Property / Symbols used: Q12067 / qualifier
 
Defining formula:

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

\eta_{j}
Property / Symbols used: Q12067 / qualifier
 
xml-id: C23.S6.XMD1.m1fdec
Property / Symbols used
 
Property / Symbols used: Q12069 / rank
 
Normal rank
Property / Symbols used: Q12069 / qualifier
 
Defining formula:

u 𝑢 {\displaystyle{\displaystyle u}}

u
Property / Symbols used: Q12069 / qualifier
 
xml-id: C23.S6.XMD2.m1bdec

Latest revision as of 15:49, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:23.6.E15
No description defined

    Statements

    Defining formula
    σ ⁡ ( u + ω j ) σ ⁡ ( ω j ) = exp ⁡ ( η j ⁢ u + η j ⁢ u 2 2 ⁢ ω 1 ) ⁢ θ j + 1 ⁡ ( z , q ) θ j + 1 ⁡ ( 0 , q ) , Weierstrass-sigma-on-lattice 𝑢 subscript 𝜔 𝑗 𝕃 Weierstrass-sigma-on-lattice subscript 𝜔 𝑗 𝕃 subscript 𝜂 𝑗 𝑢 subscript 𝜂 𝑗 superscript 𝑢 2 2 subscript 𝜔 1 Jacobi-theta 𝑗 1 𝑧 𝑞 Jacobi-theta 𝑗 1 0 𝑞 {\displaystyle{\displaystyle\frac{\sigma\left(u+\omega_{j}\right)}{\sigma\left% (\omega_{j}\right)}=\exp\left(\eta_{j}u+\frac{\eta_{j}u^{2}}{2\omega_{1}}% \right)\frac{\theta_{j+1}\left(z,q\right)}{\theta_{j+1}\left(0,q\right)},}}
    0 references
    DLMF id
    DLMF:23.6.E15
    0 references
    constraint
    j = 1 , 2 , 3 𝑗 1 2 3 {\displaystyle{\displaystyle j=1,2,3}}
    0 references
    Symbols used
    Q11945
    Defining formula
    θ j ⁡ ( z , q ) Jacobi-theta 𝑗 𝑧 𝑞 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)}}
    xml-id
    C20.S2.SS1.m2amdec
    0 references
    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.m2addec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2addec
    0 references
    Q12055
    Defining formula
    𝕃 𝕃 {\displaystyle{\displaystyle\mathbb{L}}}
    xml-id
    C23.S1.XMD1.m1ldec
    0 references
    Q12066
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C23.S1.XMD10.m1ndec
    0 references
    Q12056
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C23.S1.XMD7.m1fdec
    0 references
    Q12049
    Defining formula
    ω 1 subscript 𝜔 1 {\displaystyle{\displaystyle\omega_{1}}}
    xml-id
    C23.S1.XMD8.m1ndec
    0 references
    Q12067
    Defining formula
    η j subscript 𝜂 𝑗 {\displaystyle{\displaystyle\eta_{j}}}
    xml-id
    C23.S6.XMD1.m1fdec
    0 references
    Q12069
    Defining formula
    u 𝑢 {\displaystyle{\displaystyle u}}
    xml-id
    C23.S6.XMD2.m1bdec
    0 references
     
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