DLMF:10.14.E6 (Q3143): Difference between revisions

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Property / constraint
 

ν > 0 , 0 < x ≤ 1 formulae-sequence 𝜈 0 0 𝑥 1 {\displaystyle{\displaystyle\nu>0,0<x\leq 1}}

\nu>0,0<x\leq 1
Property / constraint: ν > 0 , 0 < x ≤ 1 formulae-sequence 𝜈 0 0 𝑥 1 {\displaystyle{\displaystyle\nu>0,0<x\leq 1}} / 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.m2aedec
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.m2adec
Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
Defining formula:

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

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11435 / rank
 
Normal rank
Property / Symbols used: Q11435 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11435 / qualifier
 
xml-id: C10.S1.XMD4.m1bdec
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.m1ddec

Latest revision as of 15:23, 2 January 2020

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DLMF:10.14.E6
No description defined

    Statements

    Defining formula
    | J ν ′ ⁡ ( ν ⁢ x ) | ≤ ( 1 + x 2 ) 1 4 x ⁢ ( 2 ⁢ π ⁢ ν ) 1 2 ⁢ x ν ⁢ exp ⁡ ( ν ⁢ ( 1 - x 2 ) 1 2 ) ( 1 + ( 1 - x 2 ) 1 2 ) ν , diffop Bessel-J 𝜈 1 𝜈 𝑥 superscript 1 superscript 𝑥 2 1 4 𝑥 superscript 2 𝜋 𝜈 1 2 superscript 𝑥 𝜈 𝜈 superscript 1 superscript 𝑥 2 1 2 superscript 1 superscript 1 superscript 𝑥 2 1 2 𝜈 {\displaystyle{\displaystyle|J_{\nu}'\left(\nu x\right)|\leq\frac{(1+x^{2})^{% \frac{1}{4}}}{x(2\pi\nu)^{\frac{1}{2}}}\frac{x^{\nu}\exp\left(\nu(1-x^{2})^{% \frac{1}{2}}\right)}{\left(1+(1-x^{2})^{\frac{1}{2}}\right)^{\nu}},}}
    0 references
    DLMF id
    DLMF:10.14.E6
    0 references
    constraint
    ν > 0 , 0 < x ≤ 1 formulae-sequence 𝜈 0 0 𝑥 1 {\displaystyle{\displaystyle\nu>0,0<x\leq 1}}
    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.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2aadec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1bdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1ddec
    0 references
     
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