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DLMF:13.24.E1 - MaRDI portal

DLMF:13.24.E1 (Q5346)

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DLMF:13.24.E1
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    Statements

    DLMF defining formula
    M κ , μ ⁡ ( z ) = Γ ⁡ ( κ + μ ) ⁢ 2 2 ⁢ κ + 2 ⁢ μ ⁢ z 1 2 - κ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ ( 2 ⁢ κ + 2 ⁢ μ ) s ⁢ ( 2 ⁢ κ ) s ( 1 + 2 ⁢ μ ) s ⁢ s ! ⁢ ( κ + μ + s ) ⁢ I κ + μ + s ⁡ ( 1 2 ⁢ z ) , Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Euler-Gamma 𝜅 𝜇 superscript 2 2 𝜅 2 𝜇 superscript 𝑧 1 2 𝜅 superscript subscript 𝑠 0 superscript 1 𝑠 Pochhammer 2 𝜅 2 𝜇 𝑠 Pochhammer 2 𝜅 𝑠 Pochhammer 1 2 𝜇 𝑠 𝑠 𝜅 𝜇 𝑠 modified-Bessel-first-kind 𝜅 𝜇 𝑠 1 2 𝑧 {\displaystyle{\displaystyle M_{\kappa,\mu}\left(z\right)=\Gamma\left(\kappa+% \mu\right)2^{2\kappa+2\mu}z^{\frac{1}{2}-\kappa}\*\sum_{s=0}^{\infty}(-1)^{s}% \frac{{\left(2\kappa+2\mu\right)_{s}}{\left(2\kappa\right)_{s}}}{{\left(1+2\mu% \right)_{s}}s!}\*\left(\kappa+\mu+s\right)I_{\kappa+\mu+s}\left(\tfrac{1}{2}z% \right),}}
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    constraint
    2 ⁢ μ , κ + μ ≠ - 1 , - 2 , - 3 , … formulae-sequence 2 𝜇 𝜅 𝜇 1 2 3 … {\displaystyle{\displaystyle 2\mu,\kappa+\mu\neq-1,-2,-3,\dots}}
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    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
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    Whittaker confluent hypergeometric function
    DLMF defining formula
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2adec
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    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    modified Bessel function of the first kind
    DLMF defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
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    nonnegative integer
    DLMF defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C13.S1.XMD3.m1dec
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    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1dec
    0 references

    Identifiers

     
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