DLMF:13.8.E12 (Q4437): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2agdec
Property / Symbols used
 
Property / Symbols used: Olver’s confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Olver’s confluent hypergeometric function / qualifier
 
Defining formula:

𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}

\OlverconfhyperM@{\NVar{a}}{\NVar{b}}{\NVar{z}}
Property / Symbols used: Olver’s confluent hypergeometric function / qualifier
 
xml-id: C13.S2.E3.m2adec
Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
Defining formula:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11abdec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

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

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2agdec
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
 
Defining formula:

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: Q11558 / rank
 
Normal rank
Property / Symbols used: Q11558 / qualifier
 
Defining formula:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q11558 / qualifier
 
xml-id: C13.S1.XMD3.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11557 / rank
 
Normal rank
Property / Symbols used: Q11557 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11557 / qualifier
 
xml-id: C13.S1.XMD6.m1fdec

Latest revision as of 15:58, 2 January 2020

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DLMF:13.8.E12
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    Statements

    Defining formula
    𝐌 ⁡ ( a , b , z ) ∼ ( z / a ) ( 1 - b ) / 2 ⁢ e z / 2 ⁢ Γ ⁡ ( 1 + a - b ) Γ ⁡ ( a ) ⁢ ( I b - 1 ⁡ ( 2 ⁢ a ⁢ z ) ⁢ ∑ s = 0 ∞ p s ⁢ ( z ) a s - z / a ⁢ I b ⁡ ( 2 ⁢ a ⁢ z ) ⁢ ∑ s = 0 ∞ q s ⁢ ( z ) a s ) , asymptotic-expansion Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 superscript 𝑧 𝑎 1 𝑏 2 superscript 𝑒 𝑧 2 Euler-Gamma 1 𝑎 𝑏 Euler-Gamma 𝑎 modified-Bessel-first-kind 𝑏 1 2 𝑎 𝑧 superscript subscript 𝑠 0 subscript 𝑝 𝑠 𝑧 superscript 𝑎 𝑠 𝑧 𝑎 modified-Bessel-first-kind 𝑏 2 𝑎 𝑧 superscript subscript 𝑠 0 subscript 𝑞 𝑠 𝑧 superscript 𝑎 𝑠 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,z\right)\sim\left(z/a\right)% ^{(1-b)/2}\frac{e^{z/2}\Gamma\left(1+a-b\right)}{\Gamma\left(a\right)}\*\left(% I_{b-1}\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{p_{s}(z)}{a^{s}}-\sqrt{% z/a}I_{b}\left(2\sqrt{az}\right)\sum_{s=0}^{\infty}\frac{q_{s}(z)}{a^{s}}% \right),}}
    0 references
    DLMF id
    DLMF:13.8.E12
    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.m2agdec
    0 references
    Olver’s confluent hypergeometric function
    Defining formula
    𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2adec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    modified Bessel function of the first kind
    Defining formula
    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
    Q11558
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C13.S1.XMD3.m1ddec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1fdec
    0 references
     
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