DLMF:23.15.E9 (Q7339): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11946 / rank
 
Normal rank
Property / Symbols used: Q11946 / qualifier
 
Defining formula:

θ j ⁡ ( z | τ ) Jacobi-theta-tau 𝑗 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}% \right)}}

\Jacobithetatau{\NVar{j}}@{\NVar{z}}{\NVar{\tau}}
Property / Symbols used: Q11946 / qualifier
 
xml-id: C20.S2.SS1.m1adec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2adec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2adec
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.m1edec
Property / Symbols used
 
Property / Symbols used: Q12068 / rank
 
Normal rank
Property / Symbols used: Q12068 / qualifier
 
Defining formula:

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

\tau
Property / Symbols used: Q12068 / qualifier
 
xml-id: C23.S1.XMD9.m1ddec

Latest revision as of 01:19, 2 January 2020

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DLMF:23.15.E9
No description defined

    Statements

    Defining formula
    η ⁡ ( τ ) = ( 1 2 ⁢ θ 1 ′ ⁡ ( 0 , q ) ) 1 / 3 = e i ⁢ π ⁢ τ / 12 ⁢ θ 3 ⁡ ( 1 2 ⁢ π ⁢ ( 1 + τ ) | 3 ⁢ τ ) . Dedekind-modular-Eta 𝜏 superscript 1 2 diffop Jacobi-theta 1 1 0 𝑞 1 3 superscript 𝑒 𝑖 𝜋 𝜏 12 Jacobi-theta-tau 3 1 2 𝜋 1 𝜏 3 𝜏 {\displaystyle{\displaystyle\eta\left(\tau\right)=\left(\tfrac{1}{2}\theta_{1}% '\left(0,q\right)\right)^{1/3}=e^{i\pi\tau/12}\theta_{3}\left(\tfrac{1}{2}\pi(% 1+\tau)\middle|3\tau\right).}}
    0 references
    DLMF id
    DLMF:23.15.E9
    0 references
    Symbols used
    Dedekind’s eta function (or Dedekind modular function)
    Defining formula
    η ⁡ ( τ ) Dedekind-modular-Eta 𝜏 {\displaystyle{\displaystyle\eta\left(\NVar{\tau}\right)}}
    xml-id
    C27.S14.E12.m2adec
    0 references
    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.m2addec
    0 references
    Q11946
    Defining formula
    θ j ⁡ ( z | τ ) Jacobi-theta-tau 𝑗 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}% \right)}}
    xml-id
    C20.S2.SS1.m1adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q12066
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C23.S1.XMD10.m1edec
    0 references
    Q12068
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C23.S1.XMD9.m1ddec
    0 references
     
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