DLMF:7.12.E6 (Q2417): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1adec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: Q11312 / rank
 
Normal rank
Property / Symbols used: Q11312 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11312 / qualifier
 
xml-id: C7.S1.XMD2.m1edec
Property / Symbols used
 
Property / Symbols used: Q11314 / rank
 
Normal rank
Property / Symbols used: Q11314 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11314 / qualifier
 
xml-id: C7.S1.XMD3.m1bdec

Latest revision as of 01:50, 2 January 2020

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DLMF:7.12.E6
No description defined

    Statements

    Defining formula
    R n ( f ) ⁢ ( z ) = ( - 1 ) n π ⁢ 2 ⁢ ∫ 0 ∞ e - π ⁢ z 2 ⁢ t / 2 ⁢ t 2 ⁢ n - ( 1 / 2 ) t 2 + 1 ⁢ d t , superscript subscript 𝑅 𝑛 Fresnel-auxilliary-function-f 𝑧 superscript 1 𝑛 𝜋 2 superscript subscript 0 superscript 𝑒 𝜋 superscript 𝑧 2 𝑡 2 superscript 𝑡 2 𝑛 1 2 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle R_{n}^{(\mathrm{f})}(z)=\frac{(-1)^{n}}{\pi\sqrt{% 2}}\int_{0}^{\infty}\frac{e^{-\pi z^{2}t/2}t^{2n-(1/2)}}{t^{2}+1}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:7.12.E6
    0 references
    Symbols used
    auxiliary function for Fresnel integrals
    Defining formula
    f ⁡ ( z ) Fresnel-auxilliary-function-f 𝑧 {\displaystyle{\displaystyle\mathrm{f}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E10.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q11312
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1edec
    0 references
    Q11314
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C7.S1.XMD3.m1bdec
    0 references
     
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