DLMF:13.10.E12 (Q4471): Difference between revisions

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

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2adec

Revision as of 16:02, 2 January 2020

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DLMF:13.10.E12
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    Statements

    Defining formula
    ∫ 0 ∞ cos ⁡ ( 2 ⁢ x ⁢ t ) ⁢ 𝐌 ⁡ ( a , b , - t 2 ) ⁢ d t = π 2 ⁢ Γ ⁡ ( a ) ⁢ x 2 ⁢ a - 1 ⁢ e - x 2 ⁢ U ⁡ ( b - 1 2 , a + 1 2 , x 2 ) , superscript subscript 0 2 𝑥 𝑡 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 superscript 𝑡 2 𝑡 𝜋 2 Euler-Gamma 𝑎 superscript 𝑥 2 𝑎 1 superscript 𝑒 superscript 𝑥 2 Kummer-confluent-hypergeometric-U 𝑏 1 2 𝑎 1 2 superscript 𝑥 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\cos\left(2xt\right){\mathbf{M}}% \left(a,b,-t^{2}\right)\mathrm{d}t=\frac{\sqrt{\pi}}{2\Gamma\left(a\right)}x^{% 2a-1}e^{-x^{2}}U\left(b-\tfrac{1}{2},a+\tfrac{1}{2},x^{2}\right),}}
    0 references
    DLMF id
    DLMF:13.10.E12
    0 references
    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    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.m2ahdec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2aedec
    0 references
    Olver’s confluent hypergeometric function
    Defining formula
    𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
     
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