DLMF:10.32.E17 (Q3537): Difference between revisions

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‎Changed an Item: Add constraint
 
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1ldec

Latest revision as of 13:45, 2 January 2020

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DLMF:10.32.E17
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    Statements

    Defining formula
    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,}}
    0 references
    DLMF id
    DLMF:10.32.E17
    0 references
    constraint
    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2andec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1apdec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2afdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3apdec
    0 references
    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.m2ajdec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ahdec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1mdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1ldec
    0 references
     
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