DLMF:13.10.E14 (Q4473): 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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- 1 < β„œ ⁑ Ξ½ < 2 ⁒ β„œ ⁑ ( b - a ) - 1 2 1 𝜈 2 𝑏 π‘Ž 1 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}}}

-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}
Property / constraint: - 1 < β„œ ⁑ Ξ½ < 2 ⁒ β„œ ⁑ ( b - a ) - 1 2 1 𝜈 2 𝑏 π‘Ž 1 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}}} / 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.m2aadec
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.m2aidec
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.m2afdec
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.m2aidec
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.m1amdec
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.m2aidec
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.m3amdec
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.m1bdec
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.m1akdec

Latest revision as of 15:02, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:13.10.E14
No description defined

    Statements

    test
    ∫ 0 ∞ e - t ⁒ t 1 2 ⁒ Ξ½ ⁒ 𝐌 ⁑ ( a , b , t ) ⁒ J Ξ½ ⁑ ( 2 ⁒ x ⁒ t ) ⁒ d t = x 1 2 ⁒ Ξ½ ⁒ e - x Ξ“ ⁑ ( b - a ) ⁒ U ⁑ ( a , a - b + Ξ½ + 2 , x ) , superscript subscript 0 superscript 𝑒 𝑑 superscript 𝑑 1 2 𝜈 Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑑 Bessel-J 𝜈 2 π‘₯ 𝑑 𝑑 superscript π‘₯ 1 2 𝜈 superscript 𝑒 π‘₯ Euler-Gamma 𝑏 π‘Ž Kummer-confluent-hypergeometric-U π‘Ž π‘Ž 𝑏 𝜈 2 π‘₯ {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}\nu}{\mathbf{% M}}\left(a,b,t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d}t=\frac{x^{\frac{% 1}{2}\nu}e^{-x}}{\Gamma\left(b-a\right)}U\left(a,a-b+\nu+2,x\right),}}
    0 references
    DLMF id
    DLMF:13.10.E14
    0 references
    constraint
    - 1 < β„œ ⁑ Ξ½ < 2 ⁒ β„œ ⁑ ( b - a ) - 1 2 1 𝜈 2 𝑏 π‘Ž 1 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}}}
    0 references
    x > 0 π‘₯ 0 {\displaystyle{\displaystyle x>0}}
    0 references
    - 1 < β„œ ⁑ Ξ½ < 2 ⁒ β„œ ⁑ ( b - a ) - 1 2 1 𝜈 2 𝑏 π‘Ž 1 2 {\displaystyle{\displaystyle-1<\Re\nu<2\Re(b-a)-\tfrac{1}{2}}}
    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.m2aadec
    0 references
    gamma function
    test
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aidec
    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.m2afdec
    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.m2aidec
    0 references
    Q10770
    test
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
    0 references
    Q11566
    test
    x π‘₯ {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1bdec
    0 references
    Q10811
    test
    β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1akdec
    0 references
     
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