DLMF:10.22.E50 (Q3424): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: Bessel function of the second kind / qualifier
 
Defining formula:

Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}

\BesselY{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the second kind / qualifier
 
xml-id: C10.S2.E3.m2agdec
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.m2aodec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2asdec
Property / Symbols used
 
Property / Symbols used: cosecant function / rank
 
Normal rank
Property / Symbols used: cosecant function / qualifier
 
Defining formula:

csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}

\csc@@{\NVar{z}}
Property / Symbols used: cosecant function / qualifier
 
xml-id: C4.S14.E5.m2aadec
Property / Symbols used
 
Property / Symbols used: cotangent function / rank
 
Normal rank
Property / Symbols used: cotangent function / qualifier
 
Defining formula:

cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}

\cot@@{\NVar{z}}
Property / Symbols used: cotangent function / qualifier
 
xml-id: C4.S14.E7.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aawdec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

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

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2abdec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aawdec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aaedec
Property / Symbols used
 
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / rank
 
Normal rank
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
 
Defining formula:

𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}

\hyperOlverF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
 
xml-id: C15.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1aidec

Latest revision as of 13:22, 2 January 2020

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DLMF:10.22.E50
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    Statements

    Defining formula
    ∫ 0 ∞ t μ - 1 ⁢ e - a ⁢ t ⁢ Y ν ⁡ ( b ⁢ t ) ⁢ d t = cot ⁡ ( ν ⁢ π ) ⁢ ( 1 2 ⁢ b ) ν ⁢ Γ ⁡ ( μ + ν ) ( a 2 + b 2 ) 1 2 ⁢ ( μ + ν ) ⁢ 𝐅 ⁡ ( μ + ν 2 , 1 - μ + ν 2 ; ν + 1 ; b 2 a 2 + b 2 ) - csc ⁡ ( ν ⁢ π ) ⁢ ( 1 2 ⁢ b ) - ν ⁢ Γ ⁡ ( μ - ν ) ( a 2 + b 2 ) 1 2 ⁢ ( μ - ν ) ⁢ 𝐅 ⁡ ( μ - ν 2 , 1 - μ - ν 2 ; 1 - ν ; b 2 a 2 + b 2 ) , superscript subscript 0 superscript 𝑡 𝜇 1 superscript 𝑒 𝑎 𝑡 Bessel-Y-Weber 𝜈 𝑏 𝑡 𝑡 𝜈 𝜋 superscript 1 2 𝑏 𝜈 Euler-Gamma 𝜇 𝜈 superscript superscript 𝑎 2 superscript 𝑏 2 1 2 𝜇 𝜈 scaled-hypergeometric-bold-F 𝜇 𝜈 2 1 𝜇 𝜈 2 𝜈 1 superscript 𝑏 2 superscript 𝑎 2 superscript 𝑏 2 𝜈 𝜋 superscript 1 2 𝑏 𝜈 Euler-Gamma 𝜇 𝜈 superscript superscript 𝑎 2 superscript 𝑏 2 1 2 𝜇 𝜈 scaled-hypergeometric-bold-F 𝜇 𝜈 2 1 𝜇 𝜈 2 1 𝜈 superscript 𝑏 2 superscript 𝑎 2 superscript 𝑏 2 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}e^{-at}Y_{\nu}\left(bt% \right)\mathrm{d}t=\cot\left(\nu\pi\right)\frac{(\tfrac{1}{2}b)^{\nu}\Gamma% \left(\mu+\nu\right)}{(a^{2}+b^{2})^{\frac{1}{2}(\mu+\nu)}}\*\mathbf{F}\left(% \frac{\mu+\nu}{2},\frac{1-\mu+\nu}{2};\nu+1;\frac{b^{2}}{a^{2}+b^{2}}\right)-% \csc\left(\nu\pi\right)\frac{(\tfrac{1}{2}b)^{-\nu}\Gamma\left(\mu-\nu\right)}% {(a^{2}+b^{2})^{\frac{1}{2}(\mu-\nu)}}\*\mathbf{F}\left(\frac{\mu-\nu}{2},% \frac{1-\mu-\nu}{2};1-\nu;\frac{b^{2}}{a^{2}+b^{2}}\right),}}
    0 references
    DLMF id
    DLMF:10.22.E50
    0 references
    constraint
    ℜ ⁡ μ > | ℜ ⁡ ν | , ℜ ⁡ ( a ± i ⁢ b ) > 0 formulae-sequence 𝜇 𝜈 plus-or-minus 𝑎 𝑖 𝑏 0 {\displaystyle{\displaystyle\Re\mu>|\Re\nu|,\Re(a\pm ib)>0}}
    0 references
    Symbols used
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2agdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aodec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2asdec
    0 references
    cosecant function
    Defining formula
    csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}
    xml-id
    C4.S14.E5.m2aadec
    0 references
    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aawdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aawdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aaedec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2aadec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1aidec
    0 references
     
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