DLMF:15.12.E5 (Q5165)

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DLMF:15.12.E5
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    Statements

    Defining formula
    𝐅 ⁑ ( a + Ξ» , b - Ξ» c ; 1 2 - 1 2 ⁒ z ) = 2 ( a + b - 1 ) / 2 ⁒ ( z + 1 ) ( c - a - b - 1 ) / 2 ( z - 1 ) c / 2 ⁒ ΞΆ ⁒ sinh ⁑ ΞΆ ⁒ ( Ξ» + 1 2 ⁒ a - 1 2 ⁒ b ) 1 - c ⁒ ( I c - 1 ⁑ ( ( Ξ» + 1 2 ⁒ a - 1 2 ⁒ b ) ⁒ ΞΆ ) ⁒ ( 1 + O ⁑ ( Ξ» - 2 ) ) + I c - 2 ⁑ ( ( Ξ» + 1 2 ⁒ a - 1 2 ⁒ b ) ⁒ ΞΆ ) 2 ⁒ Ξ» + a - b ⁒ ( ( c - 1 2 ) ⁒ ( c - 3 2 ) ⁒ ( 1 ΞΆ - coth ⁑ ΞΆ ) + 1 2 ⁒ ( 2 ⁒ c - a - b - 1 ) ⁒ ( a + b - 1 ) ⁒ tanh ⁑ ( 1 2 ⁒ ΞΆ ) + O ⁑ ( Ξ» - 2 ) ) ) , scaled-hypergeometric-bold-F π‘Ž πœ† 𝑏 πœ† 𝑐 1 2 1 2 𝑧 superscript 2 π‘Ž 𝑏 1 2 superscript 𝑧 1 𝑐 π‘Ž 𝑏 1 2 superscript 𝑧 1 𝑐 2 𝜁 𝜁 superscript πœ† 1 2 π‘Ž 1 2 𝑏 1 𝑐 modified-Bessel-first-kind 𝑐 1 πœ† 1 2 π‘Ž 1 2 𝑏 𝜁 1 Big-O superscript πœ† 2 modified-Bessel-first-kind 𝑐 2 πœ† 1 2 π‘Ž 1 2 𝑏 𝜁 2 πœ† π‘Ž 𝑏 𝑐 1 2 𝑐 3 2 1 𝜁 hyperbolic-cotangent 𝜁 1 2 2 𝑐 π‘Ž 𝑏 1 π‘Ž 𝑏 1 1 2 𝜁 Big-O superscript πœ† 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 π‘₯ {\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 Ξ½ ⁑ ( 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 π‘Ž 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 π‘Ž {\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 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C15.S12.XMD5.m1dec
    0 references
     
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