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DLMF:7.18.E14 - MaRDI portal

DLMF:7.18.E14 (Q3162)

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DLMF:7.18.E14
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    Statements

    DLMF defining formula
    i n ⁢ erfc ⁡ ( z ) ∼ 2 π ⁢ e - z 2 ( 2 ⁢ z ) n + 1 ⁢ ∑ m = 0 ∞ ( - 1 ) m ⁢ ( 2 ⁢ m + n ) ! n ! ⁢ m ! ⁢ ( 2 ⁢ z ) 2 ⁢ m , asymptotic-expansion repeated-integral-complementary-error-function 𝑛 𝑧 2 𝜋 superscript 𝑒 superscript 𝑧 2 superscript 2 𝑧 𝑛 1 superscript subscript 𝑚 0 superscript 1 𝑚 2 𝑚 𝑛 𝑛 𝑚 superscript 2 𝑧 2 𝑚 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)% \sim\frac{2}{\sqrt{\pi}}\frac{e^{-z^{2}}}{(2z)^{n+1}}\sum_{m=0}^{\infty}\frac{% (-1)^{m}(2m+n)!}{n!m!(2z)^{2m}},}}
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    constraint
    | ph ⁡ z | ≤ 3 4 ⁢ π - δ ( < 3 4 ⁢ π ) phase 𝑧 annotated 3 4 𝜋 𝛿 absent 3 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<% \tfrac{3}{4}\pi)}}
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    z → ∞ → 𝑧 {\displaystyle{\displaystyle z\to\infty}}
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    | ph ⁡ z | ≤ 3 4 ⁢ π - δ ( < 3 4 ⁢ π ) ph 𝑧 annotated 3 4 𝜋 𝛿 absent 3 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{3}{4}\pi-\delta(<% \tfrac{3}{4}\pi)}}
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    Symbols used
    Q10905
    DLMF defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
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    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
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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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    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
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    phase
    DLMF defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
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    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.m2andec
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    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1mdec
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    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C7.S1.XMD3.m1mdec
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    Identifiers

     
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