DLMF:10.32.E17 (Q3537)

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DLMF:10.32.E17
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    Statements

    test
    K μ ⁡ ( z ) ⁢ K ν ⁡ ( z ) = 2 ⁢ ∫ 0 ∞ K μ ± ν ⁡ ( 2 ⁢ z ⁢ cosh ⁡ t ) ⁢ cosh ⁡ ( ( μ ∓ ν ) ⁢ t ) ⁢ d t , modified-Bessel-second-kind 𝜇 𝑧 modified-Bessel-second-kind 𝜈 𝑧 2 superscript subscript 0 modified-Bessel-second-kind plus-or-minus 𝜇 𝜈 2 𝑧 𝑡 minus-or-plus 𝜇 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle K_{\mu}\left(z\right)K_{\nu}\left(z\right)=2\int_% {0}^{\infty}K_{\mu\pm\nu}\left(2z\cosh t\right)\cosh\left((\mu\mp\nu)t\right)% \mathrm{d}t,}}
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    DLMF id
    DLMF:10.32.E17
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    constraint
    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2andec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1apdec
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    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2afdec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3apdec
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    modified Bessel function of the second kind
    test
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2ajdec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ahdec
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    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1mdec
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