DLMF:10.66.E1 (Q3843): Difference between revisions

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

ber ν ⁡ ( x ) Kelvin-ber 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}% \right)}}

\Kelvinber{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Kelvin function / qualifier
 
xml-id: C10.S61.E1.m2adec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2adec
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.m2adec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5adec
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.m2adec
Property / Symbols used
 
Property / Symbols used: modified Bessel function of the first kind / rank
 
Normal rank
Property / Symbols used: modified Bessel function of the first kind / qualifier
 
test:

I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}

\modBesselI{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: modified Bessel function of the first kind / qualifier
 
xml-id: C10.S25.E2.m2adec
Property / Symbols used
 
Property / Symbols used: Q11428 / rank
 
Normal rank
Property / Symbols used: Q11428 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11428 / qualifier
 
xml-id: C10.S1.XMD3.m1dec
Property / Symbols used
 
Property / Symbols used: Q11435 / rank
 
Normal rank
Property / Symbols used: Q11435 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11435 / qualifier
 
xml-id: C10.S1.XMD4.m1dec
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.m1dec

Latest revision as of 13:24, 2 January 2020

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English
DLMF:10.66.E1
No description defined

    Statements

    test
    ber ν ⁡ x + i ⁢ bei ν ⁡ x = ∑ k = 0 ∞ e ( 3 ⁢ ν + k ) ⁢ π ⁢ i / 4 ⁢ x k ⁢ J ν + k ⁡ ( x ) 2 k / 2 ⁢ k ! = ∑ k = 0 ∞ e ( 3 ⁢ ν + 3 ⁢ k ) ⁢ π ⁢ i / 4 ⁢ x k ⁢ I ν + k ⁡ ( x ) 2 k / 2 ⁢ k ! . Kelvin-ber 𝜈 𝑥 𝑖 Kelvin-bei 𝜈 𝑥 superscript subscript 𝑘 0 superscript 𝑒 3 𝜈 𝑘 𝜋 𝑖 4 superscript 𝑥 𝑘 Bessel-J 𝜈 𝑘 𝑥 superscript 2 𝑘 2 𝑘 superscript subscript 𝑘 0 superscript 𝑒 3 𝜈 3 𝑘 𝜋 𝑖 4 superscript 𝑥 𝑘 modified-Bessel-first-kind 𝜈 𝑘 𝑥 superscript 2 𝑘 2 𝑘 {\displaystyle{\displaystyle\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu% }x=\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}J_{\nu+k}\left(x\right)}{2% ^{k/2}k!}=\sum_{k=0}^{\infty}\frac{e^{(3\nu+3k)\pi i/4}x^{k}I_{\nu+k}\left(x% \right)}{2^{k/2}k!}.}}
    0 references
    DLMF id
    DLMF:10.66.E1
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
    0 references
    Kelvin function
    test
    bei ν ⁡ ( x ) Kelvin-bei 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E1.m3adec
    0 references
    Kelvin function
    test
    ber ν ⁡ ( x ) Kelvin-ber 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    modified Bessel function of the first kind
    test
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
    0 references
    Q11428
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1dec
    0 references
    Q11435
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1dec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1dec
    0 references
     
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