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DLMF:13.23.E14 - MaRDI portal

DLMF:13.23.E14 (Q5345)

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DLMF:13.23.E14
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    Statements

    DLMF defining formula
    f ⁢ ( x ) = 1 π ⁢ i ⁢ x ⁢ ∫ μ 1 - i ⁢ ∞ μ 1 + i ⁢ ∞ μ ⁢ g ⁢ ( μ ) ⁢ Γ ⁡ ( 1 2 + μ - κ ) ⁢ W κ , μ ⁡ ( x ) ⁢ d μ . 𝑓 𝑥 1 𝜋 imaginary-unit 𝑥 superscript subscript subscript 𝜇 1 imaginary-unit subscript 𝜇 1 imaginary-unit 𝜇 𝑔 𝜇 Euler-Gamma 1 2 𝜇 𝜅 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑥 𝜇 {\displaystyle{\displaystyle f(x)=\frac{1}{\pi\mathrm{i}\sqrt{x}}\int_{\mu_{1}% -\mathrm{i}\infty}^{\mu_{1}+\mathrm{i}\infty}\mu g(\mu)\Gamma\left(\tfrac{1}{2% }+\mu-\kappa\right)W_{\kappa,\mu}\left(x\right)\mathrm{d}\mu.}}
    0 references
    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2amdec
    0 references
    Whittaker confluent hypergeometric function
    DLMF defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1fdec
    0 references

    Identifiers

     
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