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DLMF:12.7.E7 - MaRDI portal

DLMF:12.7.E7 (Q4827)

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DLMF:12.7.E7
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    Statements

    DLMF defining formula
    U ⁡ ( n + 1 2 , z ) = e 1 4 ⁢ z 2 ⁢ 𝐻ℎ n ⁡ ( z ) = π ⁢  2 1 2 ⁢ ( n - 1 ) ⁢ e 1 4 ⁢ z 2 ⁢ i n ⁢ erfc ⁡ ( z / 2 ) , parabolic-U 𝑛 1 2 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Fishers-Hh 𝑛 𝑧 𝜋 superscript  2 1 2 𝑛 1 superscript 𝑒 1 4 superscript 𝑧 2 repeated-integral-complementary-error-function 𝑛 𝑧 2 {\displaystyle{\displaystyle U\left(n+\tfrac{1}{2},z\right)=e^{\frac{1}{4}z^{2% }}\mathit{Hh}_{n}\left(z\right)=\sqrt{\pi}\,2^{\frac{1}{2}(n-1)}e^{\frac{1}{4}% z^{2}}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z/\sqrt{2}\right),}}
    0 references
    constraint
    n = - 1 , 0 , 1 , … 𝑛 1 0 1 … {\displaystyle{\displaystyle n=-1,0,1,\dots}}
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    Symbols used
    probability function
    DLMF defining formula
    𝐻ℎ n ⁡ ( z ) Fishers-Hh 𝑛 𝑧 {\displaystyle{\displaystyle\mathit{Hh}_{\NVar{n}}\left(\NVar{z}\right)}}
    xml-id
    C7.S18.E12.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
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    Q11291
    DLMF defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1addec
    0 references
    repeated integrals of the complementary error function
    DLMF defining formula
    i n ⁢ erfc ⁡ ( z ) repeated-integral-complementary-error-function 𝑛 𝑧 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{\NVar{n}}\mathrm{erfc}}\left(% \NVar{z}\right)}}
    xml-id
    C7.S18.E2.m2adec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C12.S1.XMD3.m1fdec
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    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C12.S1.XMD4.m1cdec
    0 references

    Identifiers

     
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