DLMF:10.32.E16 (Q3536): Difference between revisions

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Property / constraint
 

ℜ ⁡ ( μ ± ν ) > - 1 plus-or-minus 𝜇 𝜈 1 {\displaystyle{\displaystyle\Re(\mu\pm\nu)>-1}}

\Re(\mu\pm\nu)>-1
Property / constraint: ℜ ⁡ ( μ ± ν ) > - 1 plus-or-minus 𝜇 𝜈 1 {\displaystyle{\displaystyle\Re(\mu\pm\nu)>-1}} / rank
 
Normal rank
Property / constraint
 

x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}

x>0
Property / constraint: x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}} / 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
 
test:

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.m2adec
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.m1aodec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

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

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2aidec
Property / Symbols used
 
Property / Symbols used: hyperbolic sine function / rank
 
Normal rank
Property / Symbols used: hyperbolic sine function / qualifier
 
test:

sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}

\sinh@@{\NVar{z}}
Property / Symbols used: hyperbolic sine function / qualifier
 
xml-id: C4.S28.E1.m2acdec
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.m3aodec
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.m2afdec
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.m2aidec
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.m1afdec
Property / Symbols used
 
Property / Symbols used: Q11435 / rank
 
Normal rank
Property / Symbols used: Q11435 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11435 / qualifier
 
xml-id: C10.S1.XMD4.m1cdec
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.m1kdec

Latest revision as of 12:45, 2 January 2020

No description defined
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English
DLMF:10.32.E16
No description defined

    Statements

    test
    I μ ⁡ ( x ) ⁢ K ν ⁡ ( x ) = ∫ 0 ∞ J μ ± ν ⁡ ( 2 ⁢ x ⁢ sinh ⁡ t ) ⁢ e ( - μ ± ν ) ⁢ t ⁢ d t , modified-Bessel-first-kind 𝜇 𝑥 modified-Bessel-second-kind 𝜈 𝑥 superscript subscript 0 Bessel-J plus-or-minus 𝜇 𝜈 2 𝑥 𝑡 superscript 𝑒 plus-or-minus 𝜇 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle I_{\mu}\left(x\right)K_{\nu}\left(x\right)=\int_{% 0}^{\infty}J_{\mu\pm\nu}\left(2x\sinh t\right)e^{(-\mu\pm\nu)t}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.32.E16
    0 references
    constraint
    ℜ ⁡ ( μ ∓ ν ) > - 1 2 minus-or-plus 𝜇 𝜈 1 2 {\displaystyle{\displaystyle\Re(\mu\mp\nu)>-\tfrac{1}{2}}}
    0 references
    ℜ ⁡ ( μ ± ν ) > - 1 plus-or-minus 𝜇 𝜈 1 {\displaystyle{\displaystyle\Re(\mu\pm\nu)>-1}}
    0 references
    ℜ ⁡ ( μ ∓ ν ) > - 1 2 minus-or-plus 𝜇 𝜈 1 2 {\displaystyle{\displaystyle\Re(\mu\mp\nu)>-\tfrac{1}{2}}}
    0 references
    ℜ ⁡ ( μ ± ν ) > - 1 plus-or-minus 𝜇 𝜈 1 {\displaystyle{\displaystyle\Re(\mu\pm\nu)>-1}}
    0 references
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aodec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
    0 references
    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2acdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aodec
    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.m2afdec
    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.m2aidec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
    0 references
    Q11435
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1cdec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1kdec
    0 references
     
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