DLMF:10.32.E10 (Q3530)

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DLMF:10.32.E10
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    Statements

    Defining formula
    K ν ⁡ ( z ) = 1 2 ⁢ ( 1 2 ⁢ z ) ν ⁢ ∫ 0 ∞ exp ⁡ ( - t - z 2 4 ⁢ t ) ⁢ d t t ν + 1 , modified-Bessel-second-kind 𝜈 𝑧 1 2 superscript 1 2 𝑧 𝜈 superscript subscript 0 𝑡 superscript 𝑧 2 4 𝑡 𝑡 superscript 𝑡 𝜈 1 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)% ^{\nu}\int_{0}^{\infty}\exp\left(-t-\frac{z^{2}}{4t}\right)\frac{\mathrm{d}t}{% t^{\nu+1}},}}
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    DLMF id
    DLMF:10.32.E10
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    constraint
    | ph ⁡ z | < 1 4 ⁢ π phase 𝑧 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{4}\pi}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aidec
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    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2aedec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1acdec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1gdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1edec
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