DLMF:13.4.E5 (Q4368): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
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.m2acdec

Revision as of 15:46, 2 January 2020

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DLMF:13.4.E5
No description defined

    Statements

    Defining formula
    U ⁡ ( a , b , z ) = z 1 - a Γ ⁡ ( a ) ⁢ Γ ⁡ ( 1 + a - b ) ⁢ ∫ 0 ∞ U ⁡ ( b - a , b , t ) ⁢ e - t ⁢ t a - 1 t + z ⁢ d t , Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 𝑧 1 𝑎 Euler-Gamma 𝑎 Euler-Gamma 1 𝑎 𝑏 superscript subscript 0 Kummer-confluent-hypergeometric-U 𝑏 𝑎 𝑏 𝑡 superscript 𝑒 𝑡 superscript 𝑡 𝑎 1 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{z^{1-a}}{\Gamma\left(a% \right)\Gamma\left(1+a-b\right)}\int_{0}^{\infty}\frac{U\left(b-a,b,t\right)e^% {-t}t^{a-1}}{t+z}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.4.E5
    0 references
    constraint
    | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}{z}|<\pi}}
    0 references
    ℜ ⁡ a > max ⁡ ( ℜ ⁡ b - 1 , 0 ) 𝑎 𝑏 1 0 {\displaystyle{\displaystyle\Re a>\max\left(\Re b-1,0\right)}}
    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.m2addec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
     
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