DLMF:13.14.E12 (Q4501)

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DLMF:13.14.E12
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    Statements

    Defining formula
    W κ , μ ⁡ ( z ⁢ e 2 ⁢ m ⁢ π ⁢ i ) = ( - 1 ) m + 1 ⁢ 2 ⁢ π ⁢ i ⁢ sin ⁡ ( 2 ⁢ π ⁢ μ ⁢ m ) Γ ⁡ ( 1 2 - μ - κ ) ⁢ Γ ⁡ ( 1 + 2 ⁢ μ ) ⁢ sin ⁡ ( 2 ⁢ π ⁢ μ ) ⁢ M κ , μ ⁡ ( z ) + ( - 1 ) m ⁢ e - 2 ⁢ m ⁢ μ ⁢ π ⁢ i ⁢ W κ , μ ⁡ ( z ) . Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 superscript 𝑒 2 𝑚 𝜋 imaginary-unit superscript 1 𝑚 1 2 𝜋 imaginary-unit 2 𝜋 𝜇 𝑚 Euler-Gamma 1 2 𝜇 𝜅 Euler-Gamma 1 2 𝜇 2 𝜋 𝜇 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 superscript 1 𝑚 superscript 𝑒 2 𝑚 𝜇 𝜋 imaginary-unit Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\kappa,\mu}\left(ze^{2m\pi\mathrm{i}}\right)=% \frac{(-1)^{m+1}2\pi\mathrm{i}\sin\left(2\pi\mu m\right)}{\Gamma\left(\frac{1}% {2}-\mu-\kappa\right)\Gamma\left(1+2\mu\right)\sin\left(2\pi\mu\right)}M_{% \kappa,\mu}\left(z\right)+(-1)^{m}e^{-2m\mu\pi\mathrm{i}}W_{\kappa,\mu}\left(z% \right).}}
    0 references
    DLMF id
    DLMF:13.14.E12
    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.m2abdec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2afdec
    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.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ajdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
     
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