DLMF:18.17.E8 (Q5749)

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DLMF:18.17.E8
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    Statements

    Defining formula
    ( H n ⁡ ( x ) ) 2 + 2 n ⁢ ( n ! ) 2 ⁢ e x 2 ⁢ ( V ⁡ ( - n - 1 2 , 2 1 2 ⁢ x ) ) 2 = 2 n + 3 2 ⁢ n ! ⁢ e x 2 π ⁢ ∫ 0 ∞ e - ( 2 ⁢ n + 1 ) ⁢ t + x 2 ⁢ tanh ⁡ t ( sinh ⁡ 2 ⁢ t ) 1 2 ⁢ d t . superscript Hermite-polynomial-H 𝑛 𝑥 2 superscript 2 𝑛 superscript 𝑛 2 superscript 𝑒 superscript 𝑥 2 superscript parabolic-V 𝑛 1 2 superscript 2 1 2 𝑥 2 superscript 2 𝑛 3 2 𝑛 superscript 𝑒 superscript 𝑥 2 𝜋 superscript subscript 0 superscript 𝑒 2 𝑛 1 𝑡 superscript 𝑥 2 𝑡 superscript 2 𝑡 1 2 𝑡 {\displaystyle{\displaystyle\left(H_{n}\left(x\right)\right)^{2}+2^{n}(n!)^{2}% e^{x^{2}}\left(V\left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right)\right)^{2}=\frac% {2^{n+\frac{3}{2}}n!\,e^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}% \tanh t}}{(\sinh 2t)^{\frac{1}{2}}}\mathrm{d}t.}}
    0 references
    DLMF id
    DLMF:18.17.E8
    0 references
    Symbols used
    Q10922
    Defining formula
    H n ⁡ ( x ) Hermite-polynomial-H 𝑛 𝑥 {\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r13.m2abdec
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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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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
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    hyperbolic tangent function
    Defining formula
    tanh ⁡ z 𝑧 {\displaystyle{\displaystyle\tanh\NVar{z}}}
    xml-id
    C4.S28.E4.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
    0 references
    Q11517
    Defining formula
    V ⁡ ( a , z ) parabolic-V 𝑎 𝑧 {\displaystyle{\displaystyle V\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m2adec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1gdec
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