DLMF:18.17.E22 (Q5763): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(2 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: Q11743 / rank
 
Normal rank
Property / Symbols used: Q11743 / qualifier
 
Defining formula:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q11743 / qualifier
 
xml-id: C18.S1.XMD1.m1qdec
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1udec
Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1sdec

Latest revision as of 15:49, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:18.17.E22
No description defined

    Statements

    Defining formula
    1 2 ⁢ π ⁢ ∫ - ∞ ∞ e - 1 4 ⁢ x 2 ⁢ 𝐻𝑒 n ⁡ ( x ) ⁢ e 1 2 ⁢ i ⁢ x ⁢ y ⁢ d x = i n ⁢ e - 1 4 ⁢ y 2 ⁢ 𝐻𝑒 n ⁡ ( y ) , 1 2 𝜋 superscript subscript superscript 𝑒 1 4 superscript 𝑥 2 Hermite-polynomial-He 𝑛 𝑥 superscript 𝑒 1 2 imaginary-unit 𝑥 𝑦 𝑥 imaginary-unit 𝑛 superscript 𝑒 1 4 superscript 𝑦 2 Hermite-polynomial-He 𝑛 𝑦 {\displaystyle{\displaystyle\frac{1}{2\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-% \frac{1}{4}x^{2}}\mathit{He}_{n}\left(x\right)e^{\frac{1}{2}\mathrm{i}xy}% \mathrm{d}x={\mathrm{i}^{n}}e^{-\frac{1}{4}y^{2}}\mathit{He}_{n}\left(y\right)% ,}}
    0 references
    DLMF id
    DLMF:18.17.E22
    0 references
    Symbols used
    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.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1audec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3audec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1qdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1udec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1sdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5763&oldid=61257"