DLMF:12.5.E9 (Q4123): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2aedec

Revision as of 15:10, 2 January 2020

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DLMF:12.5.E9
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    Statements

    Defining formula
    V ⁡ ( a , z ) = 2 π ⁢ e 1 4 ⁢ z 2 ⁢ z a - 1 2 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 2 - a ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( 1 2 - a - 2 ⁢ t ) ⁢ 2 t ⁢ z 2 ⁢ t ⁢ cos ⁡ ( π ⁢ t ) ⁢ d t , parabolic-V 𝑎 𝑧 2 𝜋 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 𝑎 1 2 2 𝜋 𝑖 Euler-Gamma 1 2 𝑎 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 Euler-Gamma 1 2 𝑎 2 𝑡 superscript 2 𝑡 superscript 𝑧 2 𝑡 𝜋 𝑡 𝑡 {\displaystyle{\displaystyle V\left(a,z\right)=\sqrt{\frac{2}{\pi}}\frac{e^{% \frac{1}{4}z^{2}}z^{a-\frac{1}{2}}}{2\pi i\Gamma\left(\frac{1}{2}-a\right)}\*% \int_{-i\infty}^{i\infty}\Gamma\left(t\right)\Gamma\left(\tfrac{1}{2}-a-2t% \right)2^{t}z^{2t}\cos\left(\pi t\right)\mathrm{d}t,}}
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    DLMF id
    DLMF:12.5.E9
    0 references
    constraint
    | ph ⁡ z | < 1 4 ⁢ π phase 𝑧 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{4}\pi}}
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    a ≠ 1 2 , 3 2 , 5 2 , … 𝑎 1 2 3 2 5 2 … {\displaystyle{\displaystyle a\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots}}
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    | ph ⁡ z | < 1 4 ⁢ π ph 𝑧 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{4}\pi}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
     
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