DLMF:10.32.E19 (Q3539): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1odec

Revision as of 13:45, 2 January 2020

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DLMF:10.32.E19
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    Statements

    Defining formula
    K μ ⁡ ( z ) ⁢ K ν ⁡ ( z ) = 1 8 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ Γ ⁡ ( t + 1 2 ⁢ μ + 1 2 ⁢ ν ) ⁢ Γ ⁡ ( t + 1 2 ⁢ μ - 1 2 ⁢ ν ) ⁢ Γ ⁡ ( t - 1 2 ⁢ μ + 1 2 ⁢ ν ) ⁢ Γ ⁡ ( t - 1 2 ⁢ μ - 1 2 ⁢ ν ) Γ ⁡ ( 2 ⁢ t ) ⁢ ( 1 2 ⁢ z ) - 2 ⁢ t ⁢ d t , modified-Bessel-second-kind 𝜇 𝑧 modified-Bessel-second-kind 𝜈 𝑧 1 8 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 Euler-Gamma 𝑡 1 2 𝜇 1 2 𝜈 Euler-Gamma 𝑡 1 2 𝜇 1 2 𝜈 Euler-Gamma 𝑡 1 2 𝜇 1 2 𝜈 Euler-Gamma 𝑡 1 2 𝜇 1 2 𝜈 Euler-Gamma 2 𝑡 superscript 1 2 𝑧 2 𝑡 𝑡 {\displaystyle{\displaystyle K_{\mu}\left(z\right)K_{\nu}\left(z\right)=\frac{% 1}{8\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma\left(t+\frac{1}{2}\mu+\frac% {1}{2}\nu\right)\Gamma\left(t+\frac{1}{2}\mu-\frac{1}{2}\nu\right)\Gamma\left(% t-\frac{1}{2}\mu+\frac{1}{2}\nu\right)\Gamma\left(t-\frac{1}{2}\mu-\frac{1}{2}% \nu\right)}{\Gamma\left(2t\right)}(\tfrac{1}{2}z)^{-2t}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.32.E19
    0 references
    constraint
    c > 1 2 ⁢ ( | ℜ ⁡ μ | + | ℜ ⁡ ν | ) , | ph ⁡ z | < 1 2 ⁢ π formulae-sequence 𝑐 1 2 𝜇 𝜈 phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle c>\tfrac{1}{2}(|\Re\mu|+|\Re\nu|),|\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.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2apdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ardec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ardec
    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.m2aldec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ajdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1agdec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1odec
    0 references
     
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