DLMF:13.14.E8 (Q4497): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10759 / rank
 
Normal rank
Property / Symbols used: Q10759 / qualifier
 
test:

( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}

\Pochhammersym{\NVar{a}}{\NVar{n}}
Property / Symbols used: Q10759 / qualifier
 
xml-id: C5.S2.SS3.m1abdec
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.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.m2afdec
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.m5abdec
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.m2adec
Property / Symbols used
 
Property / Symbols used: Whittaker confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
test:

W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}

\WhittakerconfhyperW{\NVar{\kappa}}{\NVar{\mu}}@{\NVar{z}}
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
xml-id: C13.S14.E3.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11557 / rank
 
Normal rank
Property / Symbols used: Q11557 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11557 / qualifier
 
xml-id: C13.S1.XMD6.m1gdec
Property / Symbols used
 
Property / Symbols used: Q11559 / rank
 
Normal rank
Property / Symbols used: Q11559 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11559 / qualifier
 
xml-id: C13.S1.XMD2.m1bdec

Latest revision as of 15:05, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:13.14.E8
No description defined

    Statements

    test
    W κ , ± 1 2 ⁢ n ⁡ ( z ) = ( - 1 ) n ⁢ e - 1 2 ⁢ z ⁢ z 1 2 ⁢ n + 1 2 n ! ⁢ Γ ⁡ ( 1 2 - 1 2 ⁢ n - κ ) ⁢ ( ∑ k = 1 n n ! ⁢ ( k - 1 ) ! ( n - k ) ! ⁢ ( κ + 1 2 - 1 2 ⁢ n ) k ⁢ z - k - ∑ k = 0 ∞ ( 1 2 ⁢ n + 1 2 - κ ) k ( n + 1 ) k ⁢ k ! ⁢ z k ⁢ ( ln ⁡ z + ψ ⁡ ( 1 2 ⁢ n + 1 2 - κ + k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + 1 + k ) ) ) , Whittaker-confluent-hypergeometric-W 𝜅 plus-or-minus 1 2 𝑛 𝑧 superscript 1 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝑛 1 2 𝑛 Euler-Gamma 1 2 1 2 𝑛 𝜅 superscript subscript 𝑘 1 𝑛 𝑛 𝑘 1 𝑛 𝑘 Pochhammer 𝜅 1 2 1 2 𝑛 𝑘 superscript 𝑧 𝑘 superscript subscript 𝑘 0 Pochhammer 1 2 𝑛 1 2 𝜅 𝑘 Pochhammer 𝑛 1 𝑘 𝑘 superscript 𝑧 𝑘 𝑧 digamma 1 2 𝑛 1 2 𝜅 𝑘 digamma 1 𝑘 digamma 𝑛 1 𝑘 {\displaystyle{\displaystyle W_{\kappa,\pm\frac{1}{2}n}\left(z\right)=\frac{(-% 1)^{n}e^{-\frac{1}{2}z}z^{\frac{1}{2}n+\frac{1}{2}}}{n!\Gamma\left(\frac{1}{2}% -\frac{1}{2}n-\kappa\right)}\left(\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(% \kappa+\frac{1}{2}-\frac{1}{2}n\right)_{k}}}z^{-k}-\sum_{k=0}^{\infty}\frac{{% \left(\frac{1}{2}n+\frac{1}{2}-\kappa\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^% {k}\left(\ln z+\psi\left(\tfrac{1}{2}n+\tfrac{1}{2}-\kappa+k\right)-\psi\left(% 1+k\right)-\psi\left(n+1+k\right)\right)\right),}}
    0 references
    DLMF id
    DLMF:13.14.E8
    0 references
    constraint
    κ - 1 2 ⁢ n - 1 2 ≠ 0 , 1 , 2 , … 𝜅 1 2 𝑛 1 2 0 1 2 … {\displaystyle{\displaystyle\kappa-\frac{1}{2}n-\frac{1}{2}\neq 0,1,2,\dots}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Whittaker confluent hypergeometric function
    test
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2abdec
    0 references
    Q11557
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1gdec
    0 references
    Q11559
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1bdec
    0 references
     
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