DLMF:7.7.E14 (Q2384): Difference between revisions

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‎Changed an Item: Add constraint
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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.m2amdec

Revision as of 01:45, 2 January 2020

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DLMF:7.7.E14
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    Statements

    Defining formula
    g ⁡ ( z ) = ( 2 ⁢ π ) - 3 / 2 2 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ ζ - s ⁢ Γ ⁡ ( s ) ⁢ Γ ⁡ ( s + 1 2 ) ⁢ Γ ⁡ ( s + 1 4 ) ⁢ Γ ⁡ ( 3 4 - s ) ⁢ d s . Fresnel-auxilliary-function-g 𝑧 superscript 2 𝜋 3 2 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 superscript 𝜁 𝑠 Euler-Gamma 𝑠 Euler-Gamma 𝑠 1 2 Euler-Gamma 𝑠 1 4 Euler-Gamma 3 4 𝑠 𝑠 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i% }\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+% \tfrac{1}{2}\right)\*\Gamma\left(s+\tfrac{1}{4}\right)\Gamma\left(\tfrac{3}{4}% -s\right)\mathrm{d}s.}}
    0 references
    DLMF id
    DLMF:7.7.E14
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    auxiliary function for Fresnel integrals
    Defining formula
    g ⁡ ( z ) Fresnel-auxilliary-function-g 𝑧 {\displaystyle{\displaystyle\mathrm{g}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E11.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2amdec
    0 references
     
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