DLMF:10.65.E4 (Q3836): 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:

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

\Kelvinkei{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Kelvin function / qualifier
 
xml-id: C10.S61.E2.m3adec
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.m2abdec
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
test:

ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

\digamma@{\NVar{z}}
Property / Symbols used: psi (or digamma) function / qualifier
 
xml-id: C5.S2.E2.m2aadec
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.m5acdec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11429 / rank
 
Normal rank
Property / Symbols used: Q11429 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11429 / qualifier
 
xml-id: C10.S1.XMD2.m1adec
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.m1bdec
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.m1cdec

Latest revision as of 13:23, 2 January 2020

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DLMF:10.65.E4
No description defined

    Statements

    test
    kei n ⁡ x = - 1 2 ⁢ ( 1 2 ⁢ x ) - n ⁢ ∑ k = 0 n - 1 ( n - k - 1 ) ! k ! ⁢ sin ⁡ ( 3 4 ⁢ n ⁢ π + 1 2 ⁢ k ⁢ π ) ⁢ ( 1 4 ⁢ x 2 ) k - ln ⁡ ( 1 2 ⁢ x ) ⁢ bei n ⁡ x - 1 4 ⁢ π ⁢ ber n ⁡ x + 1 2 ⁢ ( 1 2 ⁢ x ) n ⁢ ∑ k = 0 ∞ ψ ⁡ ( k + 1 ) + ψ ⁡ ( n + k + 1 ) k ! ⁢ ( n + k ) ! ⁢ sin ⁡ ( 3 4 ⁢ n ⁢ π + 1 2 ⁢ k ⁢ π ) ⁢ ( 1 4 ⁢ x 2 ) k . Kelvin-kei 𝑛 𝑥 1 2 superscript 1 2 𝑥 𝑛 superscript subscript 𝑘 0 𝑛 1 𝑛 𝑘 1 𝑘 3 4 𝑛 𝜋 1 2 𝑘 𝜋 superscript 1 4 superscript 𝑥 2 𝑘 1 2 𝑥 Kelvin-bei 𝑛 𝑥 1 4 𝜋 Kelvin-ber 𝑛 𝑥 1 2 superscript 1 2 𝑥 𝑛 superscript subscript 𝑘 0 digamma 𝑘 1 digamma 𝑛 𝑘 1 𝑘 𝑛 𝑘 3 4 𝑛 𝜋 1 2 𝑘 𝜋 superscript 1 4 superscript 𝑥 2 𝑘 {\displaystyle{\displaystyle\operatorname{kei}_{n}x=-\tfrac{1}{2}(\tfrac{1}{2}% x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\sin\left(\tfrac{3}{4}n\pi+\tfrac{1}% {2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-\ln\left(\tfrac{1}{2}x\right)% \operatorname{bei}_{n}x-\tfrac{1}{4}\pi\operatorname{ber}_{n}x+\tfrac{1}{2}(% \tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\psi\left(k+1\right)+\psi\left(n+k+% 1\right)}{k!(n+k)!}\sin\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1% }{4}x^{2})^{k}.}}
    0 references
    DLMF id
    DLMF:10.65.E4
    0 references
    Symbols used
    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.m3acdec
    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.m2acdec
    0 references
    Kelvin function
    test
    kei ν ⁡ ( x ) Kelvin-kei 𝜈 𝑥 {\displaystyle{\displaystyle\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C10.S61.E2.m3adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q11429
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1adec
    0 references
    Q11428
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1bdec
    0 references
    Q11435
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1cdec
    0 references
     
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