DLMF:12.7.E6 (Q4129)

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DLMF:12.7.E6
No description defined

    Statements

    Defining formula
    U ⁡ ( n + 1 2 , z ) = D - n - 1 ⁡ ( z ) = π 2 ⁢ ( - 1 ) n n ! ⁢ e - 1 4 ⁢ z 2 ⁢ d n ( e 1 2 ⁢ z 2 ⁢ erfc ⁡ ( z / 2 ) ) d z n , parabolic-U 𝑛 1 2 𝑧 Whittaker-D 𝑛 1 𝑧 𝜋 2 superscript 1 𝑛 𝑛 superscript 𝑒 1 4 superscript 𝑧 2 derivative superscript 𝑒 1 2 superscript 𝑧 2 complementary-error-function 𝑧 2 𝑧 𝑛 {\displaystyle{\displaystyle U\left(n+\tfrac{1}{2},z\right)=D_{-n-1}\left(z% \right)=\sqrt{\frac{\pi}{2}}\frac{(-1)^{n}}{n!}e^{-\frac{1}{4}z^{2}}\frac{{% \mathrm{d}}^{n}\left(e^{\frac{1}{2}z^{2}}\operatorname{erfc}\left(z/\sqrt{2}% \right)\right)}{{\mathrm{d}z}^{n}},}}
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    DLMF id
    DLMF:12.7.E6
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    constraint
    n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots}}
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    Symbols used
    Q11515
    Defining formula
    D ν ⁡ ( z ) Whittaker-D 𝜈 𝑧 {\displaystyle{\displaystyle D_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C12.S1.p3.m6acdec
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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.m2acdec
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    complementary error function
    Defining formula
    erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    xml-id
    C7.S2.E2.m2aadec
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    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1acdec
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    Q11513
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C12.S1.XMD3.m1edec
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    Q11518
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C12.S1.XMD4.m1bdec
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