DLMF:13.16.E7 (Q4558): Difference between revisions

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Property / constraint
 

- ℜ ⁡ ( 1 + 2 ⁢ μ ) < n < | ℜ ⁡ μ | + ℜ ⁡ κ < 1 2 1 2 𝜇 𝑛 𝜇 𝜅 1 2 {\displaystyle{\displaystyle-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<% \tfrac{1}{2}}}

-\realpart@@{(1+2\mu)}<n<\abs{\realpart@@{\mu}}+\realpart@@{\kappa}<\tfrac{1}{2}
Property / constraint: - ℜ ⁡ ( 1 + 2 ⁢ μ ) < n < | ℜ ⁡ μ | + ℜ ⁡ κ < 1 2 1 2 𝜇 𝑛 𝜇 𝜅 1 2 {\displaystyle{\displaystyle-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<% \tfrac{1}{2}}} / rank
 
Normal rank
Property / constraint
 

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

|\operatorname{ph}z|<\pi
Property / constraint: | ph ⁡ z | < π ph 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\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 ⁡ ( 1 + 2 ⁢ μ ) < n < | ℜ ⁡ μ | + ℜ ⁡ κ < 1 2 1 2 𝜇 𝑛 𝜇 𝜅 1 2 {\displaystyle{\displaystyle-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<% \tfrac{1}{2}}}

-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<\tfrac{1}{2}
Property / constraint: - ℜ ⁡ ( 1 + 2 ⁢ μ ) < n < | ℜ ⁡ μ | + ℜ ⁡ κ < 1 2 1 2 𝜇 𝑛 𝜇 𝜅 1 2 {\displaystyle{\displaystyle-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<% \tfrac{1}{2}}} / rank
 
Normal rank
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.m2afdec
Property / Symbols used
 
Property / Symbols used: Whittaker confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
Defining formula:

M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}

\WhittakerconfhyperM{\NVar{\kappa}}{\NVar{\mu}}@{\NVar{z}}
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
xml-id: C13.S14.E2.m2addec
Property / Symbols used
 
Property / Symbols used: Whittaker confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
Defining formula:

W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}

\WhittakerconfhyperW{\NVar{\kappa}}{\NVar{\mu}}@{\NVar{z}}
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
xml-id: C13.S14.E3.m2abdec
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
 
Defining formula:

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
 
Defining formula:

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

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1afdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3afdec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
Defining formula:

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
 
Defining formula:

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

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1afdec
Property / Symbols used
 
Property / Symbols used: Q11559 / rank
 
Normal rank
Property / Symbols used: Q11559 / qualifier
 
Defining formula:

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
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11557 / qualifier
 
xml-id: C13.S1.XMD6.m1fdec
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.m2afdec

Latest revision as of 16:11, 2 January 2020

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DLMF:13.16.E7
No description defined

    Statements

    Defining formula
    W κ , μ ⁡ ( z ) = ( - 1 ) n ⁢ e - 1 2 ⁢ z ⁢ z 1 2 - μ - n Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ ∫ 0 ∞ M - κ , μ ⁡ ( t ) ⁢ e - 1 2 ⁢ t ⁢ t n + μ - 1 2 t + z ⁢ d t , Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 1 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝜇 𝑛 Euler-Gamma 1 2 𝜇 Euler-Gamma 1 2 𝜇 𝜅 superscript subscript 0 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑡 superscript 𝑒 1 2 𝑡 superscript 𝑡 𝑛 𝜇 1 2 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(z\right)=\frac{(-1)^{n}e^{-% \frac{1}{2}z}z^{\frac{1}{2}-\mu-n}}{\Gamma\left(1+2\mu\right)\Gamma\left(\frac% {1}{2}-\mu-\kappa\right)}\*\int_{0}^{\infty}\frac{M_{-\kappa,\mu}\left(t\right% )e^{-\frac{1}{2}t}t^{n+\mu-\frac{1}{2}}}{t+z}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:13.16.E7
    0 references
    constraint
    | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
    0 references
    - ℜ ⁡ ( 1 + 2 ⁢ μ ) < n < | ℜ ⁡ μ | + ℜ ⁡ κ < 1 2 1 2 𝜇 𝑛 𝜇 𝜅 1 2 {\displaystyle{\displaystyle-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<% \tfrac{1}{2}}}
    0 references
    | ph ⁡ z | < π ph 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
    0 references
    n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots}}
    0 references
    - ℜ ⁡ ( 1 + 2 ⁢ μ ) < n < | ℜ ⁡ μ | + ℜ ⁡ κ < 1 2 1 2 𝜇 𝑛 𝜇 𝜅 1 2 {\displaystyle{\displaystyle-\Re(1+2\mu)<n<\left|\Re\mu\right|+\Re\kappa<% \tfrac{1}{2}}}
    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.m2afdec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2addec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
    0 references
    Q11559
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1dec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1fdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
     
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