DLMF:20.10.E5 (Q6841): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: hyperbolic cosecant function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosecant function / qualifier
 
Defining formula:

csch ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{csch}\NVar{z}}}

\csch@@{\NVar{z}}
Property / Symbols used: hyperbolic cosecant function / qualifier
 
xml-id: C4.S28.E5.m2adec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
Defining formula:

cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.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.m2addec
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.m3addec
Property / Symbols used
 
Property / Symbols used: Q11952 / rank
 
Normal rank
Property / Symbols used: Q11952 / qualifier
 
Defining formula:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q11952 / qualifier
 
xml-id: C20.S10.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q11953 / rank
 
Normal rank
Property / Symbols used: Q11953 / qualifier
 
Defining formula:

β 𝛽 {\displaystyle{\displaystyle\beta}}

\beta
Property / Symbols used: Q11953 / qualifier
 
xml-id: C20.S10.XMD2.m1adec
Property / Symbols used
 
Property / Symbols used: Q11954 / rank
 
Normal rank
Property / Symbols used: Q11954 / qualifier
 
Defining formula:

ℓ ℓ {\displaystyle{\displaystyle\ell}}

\ell
Property / Symbols used: Q11954 / qualifier
 
xml-id: C20.S10.XMD3.m1adec

Latest revision as of 14:43, 2 January 2020

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DLMF:20.10.E5
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e - s ⁢ t ⁢ θ 3 ⁡ ( ( 1 + β ) ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t = ∫ 0 ∞ e - s ⁢ t ⁢ θ 4 ⁡ ( β ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t = ℓ s ⁢ cosh ⁡ ( β ⁢ s ) ⁢ csch ⁡ ( ℓ ⁢ s ) . superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 3 1 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 4 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 ℓ 𝑠 𝛽 𝑠 ℓ 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-st}\theta_{3}\left(\frac{(1+% \beta)\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\int_{0}^{% \infty}e^{-st}\theta_{4}\left(\frac{\beta\pi}{2\ell}\middle|\frac{i\pi t}{\ell% ^{2}}\right)\mathrm{d}t=\frac{\ell}{\sqrt{s}}\cosh\left(\beta\sqrt{s}\right)% \operatorname{csch}\left(\ell\sqrt{s}\right).}}
    0 references
    DLMF id
    DLMF:20.10.E5
    0 references
    Symbols used
    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.m1addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    hyperbolic cosecant function
    Defining formula
    csch ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{csch}\NVar{z}}}
    xml-id
    C4.S28.E5.m2adec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
    0 references
    Q11952
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C20.S10.XMD1.m1adec
    0 references
    Q11953
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C20.S10.XMD2.m1adec
    0 references
    Q11954
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C20.S10.XMD3.m1adec
    0 references
     
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