DLMF:13.14.E13 (Q4502)

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DLMF:13.14.E13
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    Statements

    Defining formula
    ( - 1 ) m ⁢ W κ , μ ⁡ ( z ⁢ e 2 ⁢ m ⁢ π ⁢ i ) = - e 2 ⁢ κ ⁢ π ⁢ i ⁢ sin ⁡ ( 2 ⁢ m ⁢ μ ⁢ π ) + sin ⁡ ( ( 2 ⁢ m - 2 ) ⁢ μ ⁢ π ) sin ⁡ ( 2 ⁢ μ ⁢ π ) ⁢ W κ , μ ⁡ ( z ) - sin ⁡ ( 2 ⁢ m ⁢ μ ⁢ π ) ⁢ 2 ⁢ π ⁢ i ⁢ e κ ⁢ π ⁢ i sin ⁡ ( 2 ⁢ μ ⁢ π ) ⁢ Γ ⁡ ( 1 2 + μ - κ ) ⁢ Γ ⁡ ( 1 2 - μ - κ ) ⁢ W - κ , μ ⁡ ( z ⁢ e π ⁢ i ) . superscript 1 𝑚 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 2 𝑚 𝜋 imaginary-unit superscript 𝑒 2 𝜅 𝜋 imaginary-unit 2 𝑚 𝜇 𝜋 2 𝑚 2 𝜇 𝜋 2 𝜇 𝜋 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 2 𝑚 𝜇 𝜋 2 𝜋 imaginary-unit superscript 𝑒 𝜅 𝜋 imaginary-unit 2 𝜇 𝜋 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 𝜅 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 𝜋 imaginary-unit {\displaystyle{\displaystyle(-1)^{m}W_{\kappa,\mu}\left(ze^{2m\pi\mathrm{i}}% \right)=-\frac{e^{2\kappa\pi\mathrm{i}}\sin\left(2m\mu\pi\right)+\sin\left((2m% -2)\mu\pi\right)}{\sin\left(2\mu\pi\right)}W_{\kappa,\mu}\left(z\right)-\frac{% \sin\left(2m\mu\pi\right)2\pi\mathrm{i}e^{\kappa\pi\mathrm{i}}}{\sin\left(2\mu% \pi\right)\Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu% -\kappa\right)}W_{-\kappa,\mu}\left(ze^{\pi\mathrm{i}}\right).}}
    0 references
    DLMF id
    DLMF:13.14.E13
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2akdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q11561
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C13.S1.XMD1.m1bdec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1ldec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
     
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