DLMF:10.32.E14 (Q3534): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
test:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2ahdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1amdec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2ahdec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10818 / rank
 
Normal rank
Property / Symbols used: Q10818 / qualifier
 
test:

ℤ {\displaystyle{\displaystyle\mathbb{Z}}}

\Integers
Property / Symbols used: Q10818 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r20.m2adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3amdec
Property / Symbols used
 
Property / Symbols used: modified Bessel function of the second kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
test:

K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselK{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the second kind / qualifier
 
xml-id: C10.S25.E3.m2ahdec
Property / Symbols used
 
Property / Symbols used: Q11245 / rank
 
Normal rank
Property / Symbols used: Q11245 / qualifier
 
test:

∉ {\displaystyle{\displaystyle\notin}}

\notin
Property / Symbols used: Q11245 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r11.m2adec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
test:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1agdec
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1kdec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1idec

Latest revision as of 12:44, 2 January 2020

No description defined
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DLMF:10.32.E14
No description defined

    Statements

    test
    K ν ⁡ ( z ) = 1 2 ⁢ π 2 ⁢ i ⁢ ( π 2 ⁢ z ) 1 2 ⁢ e - z ⁢ cos ⁡ ( ν ⁢ π ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( 1 2 - t - ν ) ⁢ Γ ⁡ ( 1 2 - t + ν ) ⁢ ( 2 ⁢ z ) t ⁢ d t , modified-Bessel-second-kind 𝜈 𝑧 1 2 superscript 𝜋 2 𝑖 superscript 𝜋 2 𝑧 1 2 superscript 𝑒 𝑧 𝜈 𝜋 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 Euler-Gamma 1 2 𝑡 𝜈 Euler-Gamma 1 2 𝑡 𝜈 superscript 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle K_{\nu}\left(z\right)=\frac{1}{2\pi^{2}i}\left(% \frac{\pi}{2z}\right)^{\frac{1}{2}}e^{-z}\cos\left(\nu\pi\right)\*\int_{-i% \infty}^{i\infty}\Gamma\left(t\right)\Gamma\left(\tfrac{1}{2}-t-\nu\right)% \Gamma\left(\tfrac{1}{2}-t+\nu\right)(2z)^{t}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.32.E14
    0 references
    constraint
    ν - 1 2 ∉ ℤ , | ph ⁡ z | < 3 2 ⁢ π formulae-sequence 𝜈 1 2 phase 𝑧 3 2 𝜋 {\displaystyle{\displaystyle\nu-\tfrac{1}{2}\notin\mathbb{Z},|\operatorname{ph% }z|<\tfrac{3}{2}\pi}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aldec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2ahdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q10818
    test
    ℤ {\displaystyle{\displaystyle\mathbb{Z}}}
    xml-id
    introduction.Sx4.p2.t1.r20.m2adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
    0 references
    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.m2ahdec
    0 references
    Q11245
    test
    ∉ {\displaystyle{\displaystyle\notin}}
    xml-id
    introduction.Sx4.p1.t1.r11.m2adec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1agdec
    0 references
    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1kdec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1idec
    0 references
     
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