DLMF:13.4.E6 (Q4369): Difference between revisions

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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

Revision as of 14:46, 2 January 2020

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DLMF:13.4.E6
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    Statements

    test
    U ⁡ ( a , b , z ) = ( - 1 ) n ⁢ z 1 - b - n Γ ⁡ ( 1 + a - b ) ⁢ ∫ 0 ∞ 𝐌 ⁡ ( b - a , b , t ) ⁢ e - t ⁢ t b + n - 1 t + z ⁢ d t , Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 superscript 1 𝑛 superscript 𝑧 1 𝑏 𝑛 Euler-Gamma 1 𝑎 𝑏 superscript subscript 0 Kummer-confluent-hypergeometric-bold-M 𝑏 𝑎 𝑏 𝑡 superscript 𝑒 𝑡 superscript 𝑡 𝑏 𝑛 1 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{(-1)^{n}z^{1-b-n}}{% \Gamma\left(1+a-b\right)}\int_{0}^{\infty}\frac{{\mathbf{M}}\left(b-a,b,t% \right)e^{-t}t^{b+n-1}}{t+z}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.4.E6
    0 references
    constraint
    | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\pi}}
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    - ℜ ⁡ b < n < 1 + ℜ ⁡ ( a - b ) 𝑏 𝑛 1 𝑎 𝑏 {\displaystyle{\displaystyle-\Re b<n<1+\Re(a-b)}}
    0 references
    | ph ⁡ z | < π ph 𝑧 𝜋 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\pi}}
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    n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots}}
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    - ℜ ⁡ b < n < 1 + ℜ ⁡ ( a - b ) 𝑏 𝑛 1 𝑎 𝑏 {\displaystyle{\displaystyle-\Re b<n<1+\Re(a-b)}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    Kummer confluent hypergeometric function
    test
    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.m2abdec
    0 references
    Olver’s confluent hypergeometric function
    test
    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.m2acdec
    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
     
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