DLMF:10.32.E13 (Q3533)

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DLMF:10.32.E13
No description defined

    Statements

    Defining formula
    K ν ⁡ ( z ) = ( 1 2 ⁢ z ) ν 4 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( t - ν ) ⁢ ( 1 2 ⁢ z ) - 2 ⁢ t ⁢ d t , modified-Bessel-second-kind 𝜈 𝑧 superscript 1 2 𝑧 𝜈 4 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 Euler-Gamma 𝑡 Euler-Gamma 𝑡 𝜈 superscript 1 2 𝑧 2 𝑡 𝑡 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}% {4\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(t\right)\Gamma\left(t-\nu% \right)(\tfrac{1}{2}z)^{-2t}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.32.E13
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    constraint
    c > max ⁡ ( ℜ ⁡ ν , 0 ) , | ph ⁡ z | < 1 2 ⁢ π formulae-sequence 𝑐 𝜈 0 phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle c>\max(\Re\nu,0),|\operatorname{ph}z|<\frac{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.m2acdec
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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.m2akdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aldec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aldec
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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.m2agdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1afdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1jdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1hdec
    0 references
     
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