DLMF:12.5.E4 (Q4118): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11320 / rank
 
Normal rank
Property / Symbols used: Q11320 / qualifier
 
Defining formula:

U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}

\paraU@{\NVar{a}}{\NVar{z}}
Property / Symbols used: Q11320 / qualifier
 
xml-id: C12.S2.SS1.p1.m1acdec

Revision as of 15:09, 2 January 2020

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DLMF:12.5.E4
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    Statements

    Defining formula
    U ⁡ ( a , z ) = 2 π ⁢ e 1 4 ⁢ z 2 ⁢ ∫ 0 ∞ t - a - 1 2 ⁢ e - 1 2 ⁢ t 2 ⁢ cos ⁡ ( z ⁢ t + ( 1 2 ⁢ a + 1 4 ) ⁢ π ) ⁢ d t , parabolic-U 𝑎 𝑧 2 𝜋 superscript 𝑒 1 4 superscript 𝑧 2 superscript subscript 0 superscript 𝑡 𝑎 1 2 superscript 𝑒 1 2 superscript 𝑡 2 𝑧 𝑡 1 2 𝑎 1 4 𝜋 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\sqrt{\frac{2}{\pi}}e^{\frac{1}% {4}z^{2}}\*\int_{0}^{\infty}t^{-a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}}\cos\left(% zt+\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)\pi\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:12.5.E4
    0 references
    constraint
    ℜ ⁡ a < 1 2 𝑎 1 2 {\displaystyle{\displaystyle\Re a<\tfrac{1}{2}}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1acdec
    0 references
     
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