DLMF:23.11.E2 (Q7321): Difference between revisions

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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.m2aadec
Property / Symbols used
 
Property / Symbols used: hyperbolic sine function / rank
 
Normal rank
Property / Symbols used: hyperbolic sine function / qualifier
 
Defining formula:

sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}

\sinh@@{\NVar{z}}
Property / Symbols used: hyperbolic sine function / qualifier
 
xml-id: C4.S28.E1.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.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2adec
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.m1dec
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.m1dec
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.m1adec
Property / Symbols used
 
Property / Symbols used: Q12082 / rank
 
Normal rank
Property / Symbols used: Q12082 / qualifier
 
Defining formula:

f j ⁢ ( s , τ ) subscript 𝑓 𝑗 𝑠 𝜏 {\displaystyle{\displaystyle f_{j}(s,\tau)}}

f_{j}(s,\tau)
Property / Symbols used: Q12082 / qualifier
 
xml-id: C23.S11.XMD1.m1adec

Latest revision as of 16:01, 2 January 2020

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DLMF:23.11.E2
No description defined

    Statements

    Defining formula
    ℘ ⁡ ( z ) = 1 z 2 + 8 ⁢ ∫ 0 ∞ s ⁢ ( e - s ⁢ sinh 2 ⁡ ( 1 2 ⁢ z ⁢ s ) ⁢ f 1 ⁢ ( s , τ ) + e i ⁢ τ ⁢ s ⁢ sin 2 ⁡ ( 1 2 ⁢ z ⁢ s ) ⁢ f 2 ⁢ ( s , τ ) ) ⁢ d s , Weierstrass-P-on-lattice 𝑧 𝕃 1 superscript 𝑧 2 8 superscript subscript 0 𝑠 superscript 𝑒 𝑠 2 1 2 𝑧 𝑠 subscript 𝑓 1 𝑠 𝜏 superscript 𝑒 𝑖 𝜏 𝑠 2 1 2 𝑧 𝑠 subscript 𝑓 2 𝑠 𝜏 𝑠 {\displaystyle{\displaystyle\wp\left(z\right)=\frac{1}{z^{2}}+8\int_{0}^{% \infty}s\left(e^{-s}{\sinh^{2}}\left(\tfrac{1}{2}zs\right)f_{1}(s,\tau)+e^{i% \tau s}{\sin^{2}}\left(\tfrac{1}{2}zs\right)f_{2}(s,\tau)\right)\mathrm{d}s,}}
    0 references
    DLMF id
    DLMF:23.11.E2
    0 references
    Symbols used
    DLMF:23.2.E4
    Defining formula
    ℘ ⁡ ( z ) Weierstrass-P-on-lattice 𝑧 𝕃 {\displaystyle{\displaystyle\wp\left(\NVar{z}\right)}}
    xml-id
    C23.S2.E4.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q12055
    Defining formula
    𝕃 𝕃 {\displaystyle{\displaystyle\mathbb{L}}}
    xml-id
    C23.S1.XMD1.m1dec
    0 references
    Q12056
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C23.S1.XMD7.m1dec
    0 references
    Q12068
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C23.S1.XMD9.m1adec
    0 references
    Q12082
    Defining formula
    f j ⁢ ( s , τ ) subscript 𝑓 𝑗 𝑠 𝜏 {\displaystyle{\displaystyle f_{j}(s,\tau)}}
    xml-id
    C23.S11.XMD1.m1adec
    0 references
     
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