DLMF:33.16.E11 (Q9639): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(3 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: function (locally) / rank
 
Normal rank
Property / Symbols used: function (locally) / qualifier
 
Defining formula:

A ⁢ ( ϵ , ℓ ) 𝐴 italic-ϵ ℓ {\displaystyle{\displaystyle A(\epsilon,\ell)}}

A(\epsilon,\ell)
Property / Symbols used: function (locally) / qualifier
 
xml-id: C33.S14.XMD4.m1ddec
Property / Symbols used
 
Property / Symbols used: parameter (locally) / rank
 
Normal rank
Property / Symbols used: parameter (locally) / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: parameter (locally) / qualifier
 
xml-id: C33.S16.XMD2.m1cdec
Property / Symbols used
 
Property / Symbols used: Q12694 / rank
 
Normal rank
Property / Symbols used: Q12694 / qualifier
 
Defining formula:

ζ ℓ ⁢ ( ν , r ) subscript 𝜁 ℓ 𝜈 𝑟 {\displaystyle{\displaystyle\zeta_{\ell}(\nu,r)}}

\zeta_{\ell}(\nu,r)
Property / Symbols used: Q12694 / qualifier
 
xml-id: C33.S16.XMD3.m1bdec
Property / Symbols used
 
Property / Symbols used: Q12695 / rank
 
Normal rank
Property / Symbols used: Q12695 / qualifier
 
Defining formula:

ξ ℓ ⁢ ( ν , r ) subscript 𝜉 ℓ 𝜈 𝑟 {\displaystyle{\displaystyle\xi_{\ell}(\nu,r)}}

\xi_{\ell}(\nu,r)
Property / Symbols used: Q12695 / qualifier
 
xml-id: C33.S16.XMD4.m1bdec

Latest revision as of 14:35, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:33.16.E11
No description defined

    Statements

    Defining formula
    h ⁡ ( ϵ , ℓ ; r ) = ( - 1 ) ℓ ⁢ ν ℓ + 1 ⁢ A ⁢ ( ϵ , ℓ ) ⁢ ( sin ⁡ ( π ⁢ ν ) ⁢ ζ ℓ ⁢ ( ν , r ) Γ ⁡ ( ℓ + 1 + ν ) + cos ⁡ ( π ⁢ ν ) ⁢ Γ ⁡ ( ν - ℓ ) ⁢ ξ ℓ ⁢ ( ν , r ) π ) . Coulomb-h italic-ϵ ℓ 𝑟 superscript 1 ℓ superscript 𝜈 ℓ 1 𝐴 italic-ϵ ℓ 𝜋 𝜈 subscript 𝜁 ℓ 𝜈 𝑟 Euler-Gamma ℓ 1 𝜈 𝜋 𝜈 Euler-Gamma 𝜈 ℓ subscript 𝜉 ℓ 𝜈 𝑟 𝜋 {\displaystyle{\displaystyle h\left(\epsilon,\ell;r\right)=(-1)^{\ell}\nu^{% \ell+1}A(\epsilon,\ell)\left(\frac{\sin\left(\pi\nu\right)\zeta_{\ell}(\nu,r)}% {\Gamma\left(\ell+1+\nu\right)}+\frac{\cos\left(\pi\nu\right)\Gamma\left(\nu-% \ell\right)\xi_{\ell}(\nu,r)}{\pi}\right).}}
    0 references
    DLMF id
    DLMF:33.16.E11
    0 references
    Symbols used
    irregular Coulomb function
    Defining formula
    h ⁡ ( ϵ , ℓ ; r ) Coulomb-h italic-ϵ ℓ 𝑟 {\displaystyle{\displaystyle h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right% )}}
    xml-id
    C33.S14.E7.m2acdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.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.m2agdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q12666
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C33.S1.XMD2.m1hdec
    0 references
    Q12692
    Defining formula
    r 𝑟 {\displaystyle{\displaystyle r}}
    xml-id
    C33.S1.XMD3.m1fdec
    0 references
    Q12693
    Defining formula
    ϵ italic-ϵ {\displaystyle{\displaystyle\epsilon}}
    xml-id
    C33.S1.XMD6.m1gdec
    0 references
    function (locally)
    Defining formula
    A ⁢ ( ϵ , ℓ ) 𝐴 italic-ϵ ℓ {\displaystyle{\displaystyle A(\epsilon,\ell)}}
    xml-id
    C33.S14.XMD4.m1ddec
    0 references
    parameter (locally)
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C33.S16.XMD2.m1cdec
    0 references
    Q12694
    Defining formula
    ζ ℓ ⁢ ( ν , r ) subscript 𝜁 ℓ 𝜈 𝑟 {\displaystyle{\displaystyle\zeta_{\ell}(\nu,r)}}
    xml-id
    C33.S16.XMD3.m1bdec
    0 references
    Q12695
    Defining formula
    ξ ℓ ⁢ ( ν , r ) subscript 𝜉 ℓ 𝜈 𝑟 {\displaystyle{\displaystyle\xi_{\ell}(\nu,r)}}
    xml-id
    C33.S16.XMD4.m1bdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q9639&oldid=48801"