DLMF:10.22.E53 (Q3427): Difference between revisions

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Property / constraint
 

| ℜ ⁡ ν | < 1 2 𝜈 1 2 {\displaystyle{\displaystyle|\Re\nu|<\tfrac{1}{2}}}

|\realpart@@{\nu}|<\tfrac{1}{2}
Property / constraint: | ℜ ⁡ ν | < 1 2 𝜈 1 2 {\displaystyle{\displaystyle|\Re\nu|<\tfrac{1}{2}}} / rank
 
Normal rank
Property / constraint
 

ℜ ⁡ ( p 2 ) > 0 superscript 𝑝 2 0 {\displaystyle{\displaystyle\Re(p^{2})>0}}

\realpart@@{(p^{2})}>0
Property / constraint: ℜ ⁡ ( p 2 ) > 0 superscript 𝑝 2 0 {\displaystyle{\displaystyle\Re(p^{2})>0}} / rank
 
Normal rank
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
 
test:

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.m2ahdec
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
 
test:

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.m2audec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

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

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aazdec
Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
test:

exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aazdec
Property / Symbols used
 
Property / Symbols used: modified Bessel function of the first kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the first kind / qualifier
 
test:

I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselI{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the first kind / qualifier
 
xml-id: C10.S25.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: modified Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
test:

K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselK{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
xml-id: C10.S25.E3.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

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

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aahdec
Property / Symbols used
 
Property / Symbols used: secant function / rank
 
Normal rank
Property / Symbols used: secant function / qualifier
 
test:

sec ⁡ z 𝑧 {\displaystyle{\displaystyle\sec\NVar{z}}}

\sec@@{\NVar{z}}
Property / Symbols used: secant function / qualifier
 
xml-id: C4.S14.E6.m2abdec
Property / Symbols used
 
Property / Symbols used: tangent function / rank
 
Normal rank
Property / Symbols used: tangent function / qualifier
 
test:

tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}

\tan@@{\NVar{z}}
Property / Symbols used: tangent function / qualifier
 
xml-id: C4.S14.E4.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1aldec

Latest revision as of 12:23, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:10.22.E53
No description defined

    Statements

    test
    ∫ 0 ∞ Y 2 ⁢ ν ⁡ ( b ⁢ t ) ⁢ exp ⁡ ( - p 2 ⁢ t 2 ) ⁢ d t = - π 2 ⁢ p ⁢ exp ⁡ ( - b 2 8 ⁢ p 2 ) ⁢ ( I ν ⁡ ( b 2 8 ⁢ p 2 ) ⁢ tan ⁡ ( ν ⁢ π ) + 1 π ⁢ K ν ⁡ ( b 2 8 ⁢ p 2 ) ⁢ sec ⁡ ( ν ⁢ π ) ) , superscript subscript 0 Bessel-Y-Weber 2 𝜈 𝑏 𝑡 superscript 𝑝 2 superscript 𝑡 2 𝑡 𝜋 2 𝑝 superscript 𝑏 2 8 superscript 𝑝 2 modified-Bessel-first-kind 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 𝜈 𝜋 1 𝜋 modified-Bessel-second-kind 𝜈 superscript 𝑏 2 8 superscript 𝑝 2 𝜈 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}Y_{2\nu}\left(bt\right)\exp\left(% -p^{2}t^{2}\right)\mathrm{d}t=-\frac{\sqrt{\pi}}{2p}\exp\left(-\frac{b^{2}}{8p% ^{2}}\right)\left(I_{\nu}\left(\frac{b^{2}}{8p^{2}}\right)\tan\left(\nu\pi% \right)+\frac{1}{\pi}K_{\nu}\left(\frac{b^{2}}{8p^{2}}\right)\sec\left(\nu\pi% \right)\right),}}
    0 references
    DLMF id
    DLMF:10.22.E53
    0 references
    constraint
    | ℜ ⁡ ν | < 1 2 𝜈 1 2 {\displaystyle{\displaystyle|\Re\nu|<\tfrac{1}{2}}}
    0 references
    ℜ ⁡ ( p 2 ) > 0 superscript 𝑝 2 0 {\displaystyle{\displaystyle\Re(p^{2})>0}}
    0 references
    Symbols used
    Bessel function of the second kind
    test
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2ahdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2audec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aazdec
    0 references
    exponential function
    test
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2abdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aazdec
    0 references
    modified Bessel function of the first kind
    test
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2abdec
    0 references
    modified Bessel function of the second kind
    test
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2abdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aahdec
    0 references
    secant function
    test
    sec ⁡ z 𝑧 {\displaystyle{\displaystyle\sec\NVar{z}}}
    xml-id
    C4.S14.E6.m2abdec
    0 references
    tangent function
    test
    tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}
    xml-id
    C4.S14.E4.m2abdec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1aldec
    0 references
     
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