DLMF:7.18.E9 (Q2459)

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DLMF:7.18.E9
No description defined

    Statements

    Defining formula
    i n ⁢ erfc ⁡ ( z ) = e - z 2 ⁢ ( 1 2 n ⁢ Γ ⁡ ( 1 2 ⁢ n + 1 ) ⁢ M ⁡ ( 1 2 ⁢ n + 1 2 , 1 2 , z 2 ) - z 2 n - 1 ⁢ Γ ⁡ ( 1 2 ⁢ n + 1 2 ) ⁢ M ⁡ ( 1 2 ⁢ n + 1 , 3 2 , z 2 ) ) , repeated-integral-complementary-error-function 𝑛 𝑧 superscript 𝑒 superscript 𝑧 2 1 superscript 2 𝑛 Euler-Gamma 1 2 𝑛 1 Kummer-confluent-hypergeometric-M 1 2 𝑛 1 2 1 2 superscript 𝑧 2 𝑧 superscript 2 𝑛 1 Euler-Gamma 1 2 𝑛 1 2 Kummer-confluent-hypergeometric-M 1 2 𝑛 1 3 2 superscript 𝑧 2 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)% =e^{-z^{2}}\left(\frac{1}{2^{n}\Gamma\left(\tfrac{1}{2}n+1\right)}M\left(% \tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)-\frac{z}{2^{n-1}\Gamma% \left(\tfrac{1}{2}n+\tfrac{1}{2}\right)}M\left(\tfrac{1}{2}n+1,\tfrac{3}{2},z^% {2}\right)\right),}}
    0 references
    DLMF id
    DLMF:7.18.E9
    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
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
    Defining formula
    M ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E2.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    repeated integrals of the complementary error function
    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.m2aidec
    0 references
    Q11312
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1hdec
    0 references
    Q11314
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C7.S1.XMD3.m1hdec
    0 references
     
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