DLMF:13.10.E16 (Q4475): Difference between revisions

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Property / constraint
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x > 0 π‘₯ 0 {\displaystyle{\displaystyle x>0}}

x>0
Property / constraint: x > 0 π‘₯ 0 {\displaystyle{\displaystyle x>0}} / rank
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Normal rank
Property / constraint
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max ⁑ ( β„œ ⁑ b - 2 , - 1 ) < β„œ ⁑ Ξ½ 𝑏 2 1 𝜈 {\displaystyle{\displaystyle\max\left(\Re b-2,-1\right)<\Re\nu}}

\max\left(\Re b-2,-1\right)<\Re\nu
Property / constraint: max ⁑ ( β„œ ⁑ b - 2 , - 1 ) < β„œ ⁑ Ξ½ 𝑏 2 1 𝜈 {\displaystyle{\displaystyle\max\left(\Re b-2,-1\right)<\Re\nu}} / rank
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Normal rank
Property / Symbols used
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Property / Symbols used: Bessel function of the first kind / rank
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Normal rank
Property / Symbols used: Bessel function of the first kind / qualifier
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test:

J Ξ½ ⁑ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}

\BesselJ{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the first kind / qualifier
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xml-id: C10.S2.E2.m2acdec
Property / Symbols used
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Property / Symbols used: gamma function / rank
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Normal rank
Property / Symbols used: gamma function / qualifier
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test:

Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
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xml-id: C5.S2.E1.m2akdec
Property / Symbols used
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Property / Symbols used: Kummer confluent hypergeometric function / rank
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Normal rank
Property / Symbols used: Kummer confluent hypergeometric function / qualifier
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test:

U ⁑ ( a , b , z ) Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}

\KummerconfhyperU@{\NVar{a}}{\NVar{b}}{\NVar{z}}
Property / Symbols used: Kummer confluent hypergeometric function / qualifier
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xml-id: C13.S2.E6.m2ahdec
Property / Symbols used
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Property / Symbols used: Olver’s confluent hypergeometric function / rank
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Normal rank
Property / Symbols used: Olver’s confluent hypergeometric function / qualifier
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test:

𝐌 ⁑ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}

\OlverconfhyperM@{\NVar{a}}{\NVar{b}}{\NVar{z}}
Property / Symbols used: Olver’s confluent hypergeometric function / qualifier
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xml-id: C13.S2.E3.m2ajdec
Property / Symbols used
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Property / Symbols used: Q10770 / rank
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Normal rank
Property / Symbols used: Q10770 / qualifier
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test:

d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
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xml-id: C1.S4.SS4.m1aodec
Property / Symbols used
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Property / Symbols used: base of natural logarithm / rank
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Normal rank
Property / Symbols used: base of natural logarithm / qualifier
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test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
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xml-id: C4.S2.E11.m2ajdec
Property / Symbols used
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Property / Symbols used: Q10811 / rank
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Normal rank
Property / Symbols used: Q10811 / qualifier
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test:

β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
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xml-id: C1.S9.E2.m1amdec
Property / Symbols used
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Property / Symbols used: Q11566 / rank
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Normal rank
Property / Symbols used: Q11566 / qualifier
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test:

x π‘₯ {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11566 / qualifier
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xml-id: C13.S1.XMD4.m1ddec
Property / Symbols used
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Property / Symbols used: Q10771 / rank
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Normal rank
Property / Symbols used: Q10771 / qualifier
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test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
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xml-id: C1.S4.SS4.m3aodec

Latest revision as of 15:03, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:13.10.E16
No description defined

    Statements

    test
    ∫ 0 ∞ e - t ⁒ t 1 2 ⁒ Ξ½ ⁒ U ⁑ ( a , b , t ) ⁒ J Ξ½ ⁑ ( 2 ⁒ x ⁒ t ) ⁒ d t = Ξ“ ⁑ ( Ξ½ - b + 2 ) ⁒ x 1 2 ⁒ Ξ½ ⁒ e - x ⁒ 𝐌 ⁑ ( a , a - b + Ξ½ + 2 , x ) , superscript subscript 0 superscript 𝑒 𝑑 superscript 𝑑 1 2 𝜈 Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑑 Bessel-J 𝜈 2 π‘₯ 𝑑 𝑑 Euler-Gamma 𝜈 𝑏 2 superscript π‘₯ 1 2 𝜈 superscript 𝑒 π‘₯ Kummer-confluent-hypergeometric-bold-M π‘Ž π‘Ž 𝑏 𝜈 2 π‘₯ {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}U\left(a,% b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\Gamma\left(\nu-b+2\right)% x^{\frac{1}{2}\nu}e^{-x}{\mathbf{M}}\left(a,a-b+\nu+2,x\right),}}
    0 references
    DLMF id
    DLMF:13.10.E16
    0 references
    constraint
    max ⁑ ( β„œ ⁑ b - 2 , - 1 ) < β„œ ⁑ Ξ½ 𝑏 2 1 𝜈 {\displaystyle{\displaystyle\max\left(\Re b-2,-1\right)<\Re\nu}}
    0 references
    x > 0 π‘₯ 0 {\displaystyle{\displaystyle x>0}}
    0 references
    max ⁑ ( β„œ ⁑ b - 2 , - 1 ) < β„œ ⁑ Ξ½ 𝑏 2 1 𝜈 {\displaystyle{\displaystyle\max\left(\Re b-2,-1\right)<\Re\nu}}
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J Ξ½ ⁑ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2acdec
    0 references
    gamma function
    test
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2akdec
    0 references
    Kummer confluent hypergeometric function
    test
    U ⁑ ( a , b , z ) Kummer-confluent-hypergeometric-U π‘Ž 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2ahdec
    0 references
    Olver’s confluent hypergeometric function
    test
    𝐌 ⁑ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2ajdec
    0 references
    Q10770
    test
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aodec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ajdec
    0 references
    Q10811
    test
    β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1amdec
    0 references
    Q11566
    test
    x π‘₯ {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1ddec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aodec
    0 references
     
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