DLMF:14.20.E3 (Q4924)

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DLMF:14.20.E3
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    Statements

    Defining formula
    𝖰 ^ - 1 2 + i ⁢ τ - μ ⁡ ( x ) = π ⁢ e - τ ⁢ π ⁢ sin ⁡ ( μ ⁢ π ) ⁢ sinh ⁡ ( τ ⁢ π ) 2 ⁢ ( cosh 2 ⁡ ( τ ⁢ π ) - sin 2 ⁡ ( μ ⁢ π ) ) ⁢ 𝖯 - 1 2 + i ⁢ τ - μ ⁡ ( x ) + π ⁢ ( e - τ ⁢ π ⁢ cos 2 ⁡ ( μ ⁢ π ) + sinh ⁡ ( τ ⁢ π ) ) 2 ⁢ ( cosh 2 ⁡ ( τ ⁢ π ) - sin 2 ⁡ ( μ ⁢ π ) ) ⁢ 𝖯 - 1 2 + i ⁢ τ - μ ⁡ ( - x ) . Ferrers-conical-legendre-Q-hat 𝜇 1 2 imaginary-unit 𝜏 𝑥 𝜋 superscript 𝑒 𝜏 𝜋 𝜇 𝜋 𝜏 𝜋 2 2 𝜏 𝜋 2 𝜇 𝜋 Ferrers-Legendre-P-first-kind 𝜇 1 2 imaginary-unit 𝜏 𝑥 𝜋 superscript 𝑒 𝜏 𝜋 2 𝜇 𝜋 𝜏 𝜋 2 2 𝜏 𝜋 2 𝜇 𝜋 Ferrers-Legendre-P-first-kind 𝜇 1 2 imaginary-unit 𝜏 𝑥 {\displaystyle{\displaystyle\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{% i}\tau}\left(x\right)=\frac{\pi e^{-\tau\pi}\sin\left(\mu\pi\right)\sinh\left(% \tau\pi\right)}{2({\cosh^{2}}\left(\tau\pi\right)-{\sin^{2}}\left(\mu\pi\right% ))}\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)+\frac{\pi(e^{% -\tau\pi}{\cos^{2}}\left(\mu\pi\right)+\sinh\left(\tau\pi\right))}{2({\cosh^{2% }}\left(\tau\pi\right)-{\sin^{2}}\left(\mu\pi\right))}\mathsf{P}^{-\mu}_{-% \frac{1}{2}+\mathrm{i}\tau}\left(-x\right).}}
    0 references
    DLMF id
    DLMF:14.20.E3
    0 references
    Symbols used
    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2aadec
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    conical function
    Defining formula
    𝖰 ^ - 1 2 + i ⁢ τ - μ ⁡ ( x ) Ferrers-conical-legendre-Q-hat 𝜇 1 2 𝑖 𝜏 𝑥 {\displaystyle{\displaystyle\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1% }{2}+i\tau}}\left(\NVar{x}\right)}}
    xml-id
    C14.S20.E2.m2aadec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1bdec
    0 references
    Q11630
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C14.S1.XMD3.m1bdec
    0 references
     
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