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

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imported>Admin
 
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Property / constraint
 

| ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\pi}}

\abs{\phase@@{z}}<\pi
Property / constraint: | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\pi}} / rank
 
Normal rank
Property / constraint
 

- ℜ ⁡ b < n < 1 + ℜ ⁡ ( a - b ) 𝑏 𝑛 1 𝑎 𝑏 {\displaystyle{\displaystyle-\Re b<n<1+\Re(a-b)}}

-\realpart@@{b}<n<1+\realpart@@{(a-b)}
Property / constraint: - ℜ ⁡ b < n < 1 + ℜ ⁡ ( a - b ) 𝑏 𝑛 1 𝑎 𝑏 {\displaystyle{\displaystyle-\Re b<n<1+\Re(a-b)}} / rank
 
Normal rank
Property / constraint
 

| ph ⁡ z | < π ph 𝑧 𝜋 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\pi}}

\left|\operatorname{ph}z\right|<\pi
Property / constraint: | ph ⁡ z | < π ph 𝑧 𝜋 {\displaystyle{\displaystyle\left|\operatorname{ph}z\right|<\pi}} / rank
 
Normal rank
Property / constraint
 

n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots}}

n=0,1,2,\dots
Property / constraint: n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots}} / rank
 
Normal rank
Property / constraint
 

- ℜ ⁡ b < n < 1 + ℜ ⁡ ( a - b ) 𝑏 𝑛 1 𝑎 𝑏 {\displaystyle{\displaystyle-\Re b<n<1+\Re(a-b)}}

-\Re b<n<1+\Re(a-b)
Property / constraint: - ℜ ⁡ b < n < 1 + ℜ ⁡ ( a - b ) 𝑏 𝑛 1 𝑎 𝑏 {\displaystyle{\displaystyle-\Re b<n<1+\Re(a-b)}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
test:

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.m2aedec
Property / Symbols used
 
Property / Symbols used: Kummer confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Kummer confluent hypergeometric function / qualifier
 
test:

U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}

\KummerconfhyperU@{\NVar{a}}{\NVar{b}}{\NVar{z}}
Property / Symbols used: Kummer confluent hypergeometric function / qualifier
 
xml-id: C13.S2.E6.m2abdec
Property / Symbols used
 
Property / Symbols used: Olver’s confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Olver’s confluent hypergeometric function / qualifier
 
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% )}}

\OlverconfhyperM@{\NVar{a}}{\NVar{b}}{\NVar{z}}
Property / Symbols used: Olver’s confluent hypergeometric function / qualifier
 
xml-id: C13.S2.E3.m2acdec
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
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aedec
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.m2addec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aedec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
test:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1abdec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aedec
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.m1dec
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.m1edec

Latest revision as of 14:46, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:13.4.E6
No description defined

    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}}
    0 references
    - 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}}
    0 references
    n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots}}
    0 references
    - 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
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
    0 references
    Q11559
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1dec
    0 references
    Q11557
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1edec
    0 references
     
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