DLMF:10.65.E9 (Q3842)

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DLMF:10.65.E9
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    Statements

    Defining formula
    ( ber ν ′ ⁡ x ) 2 + ( bei ν ′ ⁡ x ) 2 = ( 1 2 ⁢ x ) 2 ⁢ ν - 2 ⁢ ∑ k = 0 ∞ 2 ⁢ k 2 + 2 ⁢ ν ⁢ k + 1 4 ⁢ ν 2 Γ ⁡ ( ν + k + 1 ) ⁢ Γ ⁡ ( ν + 2 ⁢ k + 1 ) ⁢ ( 1 4 ⁢ x 2 ) 2 ⁢ k k ! . superscript diffop Kelvin-ber 𝜈 1 𝑥 2 superscript diffop Kelvin-bei 𝜈 1 𝑥 2 superscript 1 2 𝑥 2 𝜈 2 superscript subscript 𝑘 0 2 superscript 𝑘 2 2 𝜈 𝑘 1 4 superscript 𝜈 2 Euler-Gamma 𝜈 𝑘 1 Euler-Gamma 𝜈 2 𝑘 1 superscript 1 4 superscript 𝑥 2 2 𝑘 𝑘 {\displaystyle{\displaystyle\left(\operatorname{ber}_{\nu}'x\right)^{2}+\left(% \operatorname{bei}_{\nu}'x\right)^{2}=(\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}^{% \infty}\frac{2k^{2}+2\nu k+\frac{1}{4}\nu^{2}}{\Gamma\left(\nu+k+1\right)% \Gamma\left(\nu+2k+1\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}.}}
    0 references
    DLMF id
    DLMF:10.65.E9
    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
    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.m3ahdec
    0 references
    Kelvin function
    Defining formula
    ber ν ⁡ ( x ) Kelvin-ber 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E1.m2ahdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5ahdec
    0 references
    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1gdec
    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.m1ddec
    0 references
     
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