DLMF:10.32.E11 (Q3531)

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DLMF:10.32.E11
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    Statements

    Defining formula
    K ν ⁡ ( x ⁢ z ) = Γ ⁡ ( ν + 1 2 ) ⁢ ( 2 ⁢ z ) ν π 1 2 ⁢ x ν ⁢ ∫ 0 ∞ cos ⁡ ( x ⁢ t ) ⁢ d t ( t 2 + z 2 ) ν + 1 2 , modified-Bessel-second-kind 𝜈 𝑥 𝑧 Euler-Gamma 𝜈 1 2 superscript 2 𝑧 𝜈 superscript 𝜋 1 2 superscript 𝑥 𝜈 superscript subscript 0 𝑥 𝑡 𝑡 superscript superscript 𝑡 2 superscript 𝑧 2 𝜈 1 2 {\displaystyle{\displaystyle K_{\nu}\left(xz\right)=\frac{\Gamma\left(\nu+% \frac{1}{2}\right)(2z)^{\nu}}{\pi^{\frac{1}{2}}x^{\nu}}\int_{0}^{\infty}\frac{% \cos\left(xt\right)\mathrm{d}t}{(t^{2}+z^{2})^{\nu+\frac{1}{2}}},}}
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    DLMF id
    DLMF:10.32.E11
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    constraint
    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
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    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}
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    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
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    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
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    | ph ⁡ z | < 1 2 ⁢ π ph 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2agdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
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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.m2afdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1addec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
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    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1bdec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1hdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1fdec
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