DLMF:10.68.E9 (Q3875): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: modulus of Bessel functions / rank
 
Normal rank
Property / Symbols used: modulus of Bessel functions / qualifier
 
Defining formula:

M ν ⁡ ( x ) modulus-Bessel-M 𝜈 𝑥 {\displaystyle{\displaystyle M_{\NVar{\nu}}\left(\NVar{x}\right)}}

\HankelmodM{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: modulus of Bessel functions / qualifier
 
xml-id: C10.S18.E1.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11444 / rank
 
Normal rank
Property / Symbols used: Q11444 / qualifier
 
Defining formula:

θ ν ⁡ ( x ) phase-Bessel-Theta 𝜈 𝑥 {\displaystyle{\displaystyle\theta_{\NVar{\nu}}\left(\NVar{x}\right)}}

\Hankelphasetheta{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Q11444 / qualifier
 
xml-id: C10.S18.E3.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: 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.m1hdec
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.m1gdec

Latest revision as of 14:29, 2 January 2020

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DLMF:10.68.E9
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    Statements

    Defining formula
    bei ν ′ ⁡ x = 1 2 ⁢ M ν + 1 ⁢ sin ⁡ ( θ ν + 1 - 1 4 ⁢ π ) - 1 2 ⁢ M ν - 1 ⁢ sin ⁡ ( θ ν - 1 - 1 4 ⁢ π ) = ( ν / x ) ⁢ M ν ⁢ sin ⁡ θ ν + M ν + 1 ⁢ sin ⁡ ( θ ν + 1 - 1 4 ⁢ π ) = - ( ν / x ) ⁢ M ν ⁢ sin ⁡ θ ν - M ν - 1 ⁢ sin ⁡ ( θ ν - 1 - 1 4 ⁢ π ) . diffop Kelvin-bei 𝜈 1 𝑥 1 2 modulus-Bessel-M 𝜈 1 phase-Bessel-Theta 𝜈 1 1 4 𝜋 1 2 modulus-Bessel-M 𝜈 1 phase-Bessel-Theta 𝜈 1 1 4 𝜋 𝜈 𝑥 modulus-Bessel-M 𝜈 phase-Bessel-Theta 𝜈 modulus-Bessel-M 𝜈 1 phase-Bessel-Theta 𝜈 1 1 4 𝜋 𝜈 𝑥 modulus-Bessel-M 𝜈 phase-Bessel-Theta 𝜈 modulus-Bessel-M 𝜈 1 phase-Bessel-Theta 𝜈 1 1 4 𝜋 {\displaystyle{\displaystyle\operatorname{bei}_{\nu}'x=\tfrac{1}{2}M_{\nu+1}% \sin\left(\theta_{\nu+1}-\tfrac{1}{4}\pi\right)-\tfrac{1}{2}M_{\nu-1}\sin\left% (\theta_{\nu-1}-\tfrac{1}{4}\pi\right)=(\nu/x)M_{\nu}\sin\theta_{\nu}+M_{\nu+1% }\sin\left(\theta_{\nu+1}-\tfrac{1}{4}\pi\right)=-(\nu/x)M_{\nu}\sin\theta_{% \nu}-M_{\nu-1}\sin\left(\theta_{\nu-1}-\tfrac{1}{4}\pi\right).}}
    0 references
    DLMF id
    DLMF:10.68.E9
    0 references
    Symbols used
    Kelvin function
    Defining formula
    bei ν ⁡ ( x ) Kelvin-bei 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E1.m3addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    modulus of Bessel functions
    Defining formula
    M ν ⁡ ( x ) modulus-Bessel-M 𝜈 𝑥 {\displaystyle{\displaystyle M_{\NVar{\nu}}\left(\NVar{x}\right)}}
    xml-id
    C10.S18.E1.m2aedec
    0 references
    Q11444
    Defining formula
    θ ν ⁡ ( x ) phase-Bessel-Theta 𝜈 𝑥 {\displaystyle{\displaystyle\theta_{\NVar{\nu}}\left(\NVar{x}\right)}}
    xml-id
    C10.S18.E3.m1aedec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1hdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1gdec
    0 references
     
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