DLMF:10.66.E1 (Q3843): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Kelvin function / rank
 
Normal rank
Property / Symbols used: Kelvin function / qualifier
 
test:

bei ν ⁡ ( x ) Kelvin-bei 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}% \right)}}

\Kelvinbei{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Kelvin function / qualifier
 
xml-id: C10.S61.E1.m3adec

Revision as of 13:24, 2 January 2020

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DLMF:10.66.E1
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    Statements

    test
    ber ν ⁡ x + i ⁢ bei ν ⁡ x = ∑ k = 0 ∞ e ( 3 ⁢ ν + k ) ⁢ π ⁢ i / 4 ⁢ x k ⁢ J ν + k ⁡ ( x ) 2 k / 2 ⁢ k ! = ∑ k = 0 ∞ e ( 3 ⁢ ν + 3 ⁢ k ) ⁢ π ⁢ i / 4 ⁢ x k ⁢ I ν + k ⁡ ( x ) 2 k / 2 ⁢ k ! . Kelvin-ber 𝜈 𝑥 𝑖 Kelvin-bei 𝜈 𝑥 superscript subscript 𝑘 0 superscript 𝑒 3 𝜈 𝑘 𝜋 𝑖 4 superscript 𝑥 𝑘 Bessel-J 𝜈 𝑘 𝑥 superscript 2 𝑘 2 𝑘 superscript subscript 𝑘 0 superscript 𝑒 3 𝜈 3 𝑘 𝜋 𝑖 4 superscript 𝑥 𝑘 modified-Bessel-first-kind 𝜈 𝑘 𝑥 superscript 2 𝑘 2 𝑘 {\displaystyle{\displaystyle\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu% }x=\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}J_{\nu+k}\left(x\right)}{2% ^{k/2}k!}=\sum_{k=0}^{\infty}\frac{e^{(3\nu+3k)\pi i/4}x^{k}I_{\nu+k}\left(x% \right)}{2^{k/2}k!}.}}
    0 references
    DLMF id
    DLMF:10.66.E1
    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
    Kelvin function
    test
    bei ν ⁡ ( x ) Kelvin-bei 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E1.m3adec
    0 references
     
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