DLMF:12.7.E3 (Q4126): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2adec

Revision as of 15:11, 2 January 2020

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DLMF:12.7.E3
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    Statements

    V ( n + 1 2 , z ) = 2 / π e 1 4 z 2 ( - i ) n 𝐻𝑒 n ( i z ) = 2 / π e 1 4 z 2 ( - i ) n 2 - 1 2 n H n ( i z / 2 ) , parabolic-V 𝑛 1 2 𝑧 2 𝜋 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑖 𝑛 Hermite-polynomial-He 𝑛 𝑖 𝑧 2 𝜋 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑖 𝑛 superscript 2 1 2 𝑛 Hermite-polynomial-H 𝑛 𝑖 𝑧 2 {\displaystyle{\displaystyle V\left(n+\tfrac{1}{2},z\right)=\sqrt{2/\pi}e^{% \frac{1}{4}z^{2}}(-i)^{n}\mathit{He}_{n}\left(iz\right)=\sqrt{2/\pi}e^{\frac{1% }{4}z^{2}}(-i)^{n}2^{-\frac{1}{2}n}H_{n}\left(iz/\sqrt{2}\right),}}
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    DLMF:12.7.E3
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    n = 0 , 1 , 2 , 𝑛 0 1 2 {\displaystyle{\displaystyle n=0,1,2,\dots}}
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    H n ( x ) Hermite-polynomial-H 𝑛 𝑥 {\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\right)}}
    C18.S3.T1.t1.r13.m2aadec
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    𝐻𝑒 n ( x ) Hermite-polynomial-He 𝑛 𝑥 {\displaystyle{\displaystyle\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)}}
    C18.S3.T1.t1.r14.m2aadec
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    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2adec
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    e {\displaystyle{\displaystyle\mathrm{e}}}
    C4.S2.E11.m2abdec
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    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    C1.S9.E1.m2adec
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