DLMF:10.22.E71 (Q3445): Difference between revisions

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Property / constraint
 

ℜ ⁡ μ > - 1 2 , ℜ ⁡ ν > - 1 , | b - c | < a < b + c , cos ⁡ ϕ = ( b 2 + c 2 - a 2 ) / ( 2 ⁢ b ⁢ c ) formulae-sequence formulae-sequence 𝜇 1 2 formulae-sequence 𝜈 1 𝑏 𝑐 𝑎 𝑏 𝑐 italic-ϕ superscript 𝑏 2 superscript 𝑐 2 superscript 𝑎 2 2 𝑏 𝑐 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-1,|b-c|<a<b+c,\cos% \phi=(b^{2}+c^{2}-a^{2})/(2bc)}}

\realpart@@{\mu}>-\tfrac{1}{2},\realpart@@{\nu}>-1,|b-c|<a<b+c,\cos@@{\phi}=(b^{2}+c^{2}-a^{2})/(2bc)
Property / constraint: ℜ ⁡ μ > - 1 2 , ℜ ⁡ ν > - 1 , | b - c | < a < b + c , cos ⁡ ϕ = ( b 2 + c 2 - a 2 ) / ( 2 ⁢ b ⁢ c ) formulae-sequence formulae-sequence 𝜇 1 2 formulae-sequence 𝜈 1 𝑏 𝑐 𝑎 𝑏 𝑐 italic-ϕ superscript 𝑏 2 superscript 𝑐 2 superscript 𝑎 2 2 𝑏 𝑐 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-1,|b-c|<a<b+c,\cos% \phi=(b^{2}+c^{2}-a^{2})/(2bc)}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Bessel function of the first kind / rank
 
Normal rank
Property / Symbols used: Bessel function of the first kind / qualifier
 
Defining formula:

J ν ⁡ ( 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
 
xml-id: C10.S2.E2.m2abddec
Property / Symbols used
 
Property / Symbols used: Ferrers function of the first kind / rank
 
Normal rank
Property / Symbols used: Ferrers function of the first kind / qualifier
 
Defining formula:

𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}

\FerrersP[\NVar{\mu}]{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Ferrers function of the first kind / qualifier
 
xml-id: C14.S3.E1.m2adec
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.m2aacdec
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.m2andec
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.m1abrdec
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.m3abrdec
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.m1aardec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2andec
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.m1awdec

Latest revision as of 13:27, 2 January 2020

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DLMF:10.22.E71
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ J μ ⁡ ( a ⁢ t ) ⁢ J ν ⁡ ( b ⁢ t ) ⁢ J ν ⁡ ( c ⁢ t ) ⁢ t 1 - μ ⁢ d t = ( b ⁢ c ) μ - 1 ⁢ ( sin ⁡ ϕ ) μ - 1 2 ( 2 ⁢ π ) 1 2 ⁢ a μ ⁢ 𝖯 ν - 1 2 1 2 - μ ⁡ ( cos ⁡ ϕ ) , superscript subscript 0 Bessel-J 𝜇 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 Bessel-J 𝜈 𝑐 𝑡 superscript 𝑡 1 𝜇 𝑡 superscript 𝑏 𝑐 𝜇 1 superscript italic-ϕ 𝜇 1 2 superscript 2 𝜋 1 2 superscript 𝑎 𝜇 Ferrers-Legendre-P-first-kind 1 2 𝜇 𝜈 1 2 italic-ϕ {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}% \left(bt\right)J_{\nu}\left(ct\right)t^{1-\mu}\mathrm{d}t=\frac{(bc)^{\mu-1}(% \sin\phi)^{\mu-\frac{1}{2}}}{(2\pi)^{\frac{1}{2}}a^{\mu}}\mathsf{P}^{\frac{1}{% 2}-\mu}_{\nu-\frac{1}{2}}(\cos\phi),}}
    0 references
    DLMF id
    DLMF:10.22.E71
    0 references
    constraint
    ℜ ⁡ μ > - 1 2 , ℜ ⁡ ν > - 1 , | b - c | < a < b + c , cos ⁡ ϕ = ( b 2 + c 2 - a 2 ) / ( 2 ⁢ b ⁢ c ) formulae-sequence formulae-sequence 𝜇 1 2 formulae-sequence 𝜈 1 𝑏 𝑐 𝑎 𝑏 𝑐 italic-ϕ superscript 𝑏 2 superscript 𝑐 2 superscript 𝑎 2 2 𝑏 𝑐 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-1,|b-c|<a<b+c,\cos% \phi=(b^{2}+c^{2}-a^{2})/(2bc)}}
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2abddec
    0 references
    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aacdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2andec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abrdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abrdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aardec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2andec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1awdec
    0 references
     
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