DLMF:15.9.E17 (Q5107)

From DRMF
Revision as of 14:10, 2 January 2020 by Admin (talk | contribs) (β€ŽChanged an Item: Add constraint)
(diff) ← Older revision | Latest revision (diff) | Newer revision β†’ (diff)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:15.9.E17
No description defined

    Statements

    Defining formula
    𝐅 ⁑ ( a , a + 1 2 c ; z ) = 2 c - 1 ⁒ z ( 1 - c ) / 2 ⁒ ( 1 - z ) - a + ( ( c - 1 ) / 2 ) ⁒ P 2 ⁒ a - c 1 - c ⁑ ( 1 1 - z ) , scaled-hypergeometric-bold-F π‘Ž π‘Ž 1 2 𝑐 𝑧 superscript 2 𝑐 1 superscript 𝑧 1 𝑐 2 superscript 1 𝑧 π‘Ž 𝑐 1 2 Legendre-P-first-kind 1 𝑐 2 π‘Ž 𝑐 1 1 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,a+\tfrac{1}{2}\atop c};z\right)% =2^{c-1}z^{\ifrac{(1-c)}{2}}(1-z)^{-a+(\ifrac{(c-1)}{2})}\*P^{1-c}_{2a-c}\left% (\frac{1}{\sqrt{1-z}}\right),}}
    0 references
    DLMF id
    DLMF:15.9.E17
    0 references
    constraint
    | ph ⁑ z | < Ο€ phase 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
    0 references
    | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
    0 references
    Symbols used
    Q11481
    Defining formula
    P Ξ½ ΞΌ ⁑ ( z ) Legendre-P-first-kind πœ‡ 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aadec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁑ ( a , b ; c ; z ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2aadec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1bdec
    0 references
    Q11645
    Defining formula
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1adec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1adec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5107&oldid=44325"