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

    Defining formula
    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 id
    DLMF:12.7.E3
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    constraint
    n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots}}
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    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.m2aadec
    0 references
    Q11525
    Defining formula
    𝐻𝑒 n ⁡ ( x ) Hermite-polynomial-He 𝑛 𝑥 {\displaystyle{\displaystyle\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r14.m2aadec
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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.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
     
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