DLMF:15.12.E5 (Q5165): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11682 / rank
 
Normal rank
Property / Symbols used: Q11682 / qualifier
 
Defining formula:

ζ 𝜁 {\displaystyle{\displaystyle\zeta}}

\zeta
Property / Symbols used: Q11682 / qualifier
 
xml-id: C15.S12.XMD5.m1dec

Latest revision as of 14:24, 2 January 2020

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DLMF:15.12.E5
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    Statements

    Defining formula
    𝐅 ⁡ ( a + λ , b - λ c ; 1 2 - 1 2 ⁢ z ) = 2 ( a + b - 1 ) / 2 ⁢ ( z + 1 ) ( c - a - b - 1 ) / 2 ( z - 1 ) c / 2 ⁢ ζ ⁢ sinh ⁡ ζ ⁢ ( λ + 1 2 ⁢ a - 1 2 ⁢ b ) 1 - c ⁢ ( I c - 1 ⁡ ( ( λ + 1 2 ⁢ a - 1 2 ⁢ b ) ⁢ ζ ) ⁢ ( 1 + O ⁡ ( λ - 2 ) ) + I c - 2 ⁡ ( ( λ + 1 2 ⁢ a - 1 2 ⁢ b ) ⁢ ζ ) 2 ⁢ λ + a - b ⁢ ( ( c - 1 2 ) ⁢ ( c - 3 2 ) ⁢ ( 1 ζ - coth ⁡ ζ ) + 1 2 ⁢ ( 2 ⁢ c - a - b - 1 ) ⁢ ( a + b - 1 ) ⁢ tanh ⁡ ( 1 2 ⁢ ζ ) + O ⁡ ( λ - 2 ) ) ) , scaled-hypergeometric-bold-F 𝑎 𝜆 𝑏 𝜆 𝑐 1 2 1 2 𝑧 superscript 2 𝑎 𝑏 1 2 superscript 𝑧 1 𝑐 𝑎 𝑏 1 2 superscript 𝑧 1 𝑐 2 𝜁 𝜁 superscript 𝜆 1 2 𝑎 1 2 𝑏 1 𝑐 modified-Bessel-first-kind 𝑐 1 𝜆 1 2 𝑎 1 2 𝑏 𝜁 1 Big-O superscript 𝜆 2 modified-Bessel-first-kind 𝑐 2 𝜆 1 2 𝑎 1 2 𝑏 𝜁 2 𝜆 𝑎 𝑏 𝑐 1 2 𝑐 3 2 1 𝜁 hyperbolic-cotangent 𝜁 1 2 2 𝑐 𝑎 𝑏 1 𝑎 𝑏 1 1 2 𝜁 Big-O superscript 𝜆 2 {\displaystyle{\displaystyle\mathbf{F}\left({a+\lambda,b-\lambda\atop c};% \tfrac{1}{2}-\tfrac{1}{2}z\right)=2^{(a+b-1)/2}\frac{(z+1)^{(c-a-b-1)/2}}{(z-1% )^{c/2}}\sqrt{\zeta\sinh\zeta}\left(\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b\right)% ^{1-c}\left(I_{c-1}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)(1+O% (\lambda^{-2}))+\frac{I_{c-2}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta% \right)}{2\lambda+a-b}\left(\left(c-\tfrac{1}{2}\right)\left(c-\tfrac{3}{2}% \right)\left(\frac{1}{\zeta}-\coth\zeta\right)+\tfrac{1}{2}(2c-a-b-1)(a+b-1)% \tanh\left(\tfrac{1}{2}\zeta\right)+O(\lambda^{-2})\right)\right),}}
    0 references
    DLMF id
    DLMF:15.12.E5
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aadec
    0 references
    hyperbolic cotangent function
    Defining formula
    coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}
    xml-id
    C4.S28.E7.m2adec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    hyperbolic tangent function
    Defining formula
    tanh ⁡ z 𝑧 {\displaystyle{\displaystyle\tanh\NVar{z}}}
    xml-id
    C4.S28.E4.m2adec
    0 references
    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
    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 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2adec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1ddec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1cdec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1cdec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1cdec
    0 references
    Q11682
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C15.S12.XMD5.m1dec
    0 references
     
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