DLMF:8.7.E5 (Q2531): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11329 / rank
 
Normal rank
Property / Symbols used: Q11329 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11329 / qualifier
 
xml-id: C8.S1.XMD6.m1bdec

Latest revision as of 13:46, 2 January 2020

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DLMF:8.7.E5
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    Statements

    Defining formula
    γ * ⁡ ( a , z ) = e - 1 2 ⁢ z ⁢ ∑ n = 0 ∞ ( 1 - a ) n Γ ⁡ ( n + a + 1 ) ⁢ ( 2 ⁢ n + 1 ) ⁢ 𝗂 n ( 1 ) ⁡ ( 1 2 ⁢ z ) . incomplete-gamma-star 𝑎 𝑧 superscript 𝑒 1 2 𝑧 superscript subscript 𝑛 0 Pochhammer 1 𝑎 𝑛 Euler-Gamma 𝑛 𝑎 1 2 𝑛 1 spherical-Bessel-I-1 𝑛 1 2 𝑧 {\displaystyle{\displaystyle\gamma^{*}\left(a,z\right)=e^{-\frac{1}{2}z}\sum_{% n=0}^{\infty}\frac{{\left(1-a\right)_{n}}}{\Gamma\left(n+a+1\right)}{\left(2n+% 1\right)}{\mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}z\right).}}
    0 references
    DLMF id
    DLMF:8.7.E5
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    incomplete gamma function
    Defining formula
    γ * ⁡ ( a , z ) incomplete-gamma-star 𝑎 𝑧 {\displaystyle{\displaystyle\gamma^{*}\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E6.m2aadec
    0 references
    modified spherical Bessel function
    Defining formula
    𝗂 n ( 1 ) ⁡ ( z ) spherical-Bessel-I-1 𝑛 𝑧 {\displaystyle{\displaystyle{\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S47.E7.m2adec
    0 references
    Q11323
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1bdec
    0 references
    Q11324
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1ddec
    0 references
    Q11329
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C8.S1.XMD6.m1bdec
    0 references
     
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