DLMF:11.5.E7 (Q3974): Difference between revisions

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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.m1afdec
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.m3afdec
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
 
Defining formula:

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.m2adec
Property / Symbols used
 
Property / Symbols used: modified Struve function / rank
 
Normal rank
Property / Symbols used: modified Struve function / qualifier
 
Defining formula:

𝐋 ν ⁡ ( z ) modified-Struve-L 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modStruveL{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Struve function / qualifier
 
xml-id: C11.S2.E2.m2aadec
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.m1aedec
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.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11494 / rank
 
Normal rank
Property / Symbols used: Q11494 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11494 / qualifier
 
xml-id: C11.S1.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q11492 / rank
 
Normal rank
Property / Symbols used: Q11492 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11492 / qualifier
 
xml-id: C11.S1.XMD3.m1ddec

Latest revision as of 14:45, 2 January 2020

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English
DLMF:11.5.E7
No description defined

    Statements

    Defining formula
    I - ν ⁡ ( x ) - 𝐋 ν ⁡ ( x ) = 2 ⁢ ( 1 2 ⁢ x ) ν π ⁢ Γ ⁡ ( ν + 1 2 ) ⁢ ∫ 0 ∞ ( 1 + t 2 ) ν - 1 2 ⁢ sin ⁡ ( x ⁢ t ) ⁢ d t , modified-Bessel-first-kind 𝜈 𝑥 modified-Struve-L 𝜈 𝑥 2 superscript 1 2 𝑥 𝜈 𝜋 Euler-Gamma 𝜈 1 2 superscript subscript 0 superscript 1 superscript 𝑡 2 𝜈 1 2 𝑥 𝑡 𝑡 {\displaystyle{\displaystyle I_{-\nu}\left(x\right)-\mathbf{L}_{\nu}\left(x% \right)=\frac{2(\tfrac{1}{2}x)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}% \right)}\int_{0}^{\infty}(1+t^{2})^{\nu-\frac{1}{2}}\sin\left(xt\right)\mathrm% {d}t,}}
    0 references
    DLMF id
    DLMF:11.5.E7
    0 references
    constraint
    ℜ ⁡ ν < 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu<\tfrac{1}{2}}}
    0 references
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
    0 references
    ℜ ⁡ ν < 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu<\tfrac{1}{2}}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
    0 references
    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
    0 references
    modified Struve function
    Defining formula
    𝐋 ν ⁡ ( z ) modified-Struve-L 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C11.S2.E2.m2aadec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    Q11494
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C11.S1.XMD1.m1dec
    0 references
    Q11492
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C11.S1.XMD3.m1ddec
    0 references
     
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