DLMF:7.5.E1 (Q2347): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
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Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2adec
Property / Symbols used
 
Property / Symbols used: error function / rank
 
Normal rank
Property / Symbols used: error function / qualifier
 
Defining formula:

erf z error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erf}\NVar{z}}}

\erf@@{\NVar{z}}
Property / Symbols used: error function / qualifier
 
xml-id: C7.S2.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Faddeeva (or Faddeyeva) function / rank
 
Normal rank
Property / Symbols used: Faddeeva (or Faddeyeva) function / qualifier
 
Defining formula:

w ( z ) error-function-w 𝑧 {\displaystyle{\displaystyle w\left(\NVar{z}\right)}}

\Faddeevaw@{\NVar{z}}
Property / Symbols used: Faddeeva (or Faddeyeva) function / qualifier
 
xml-id: C7.S2.E3.m2adec
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.m2adec
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
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.m1dec

Latest revision as of 01:38, 2 January 2020

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DLMF:7.5.E1
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    Statements

    F ( z ) = 1 2 i π ( e - z 2 - w ( z ) ) = - 1 2 i π e - z 2 erf ( i z ) . Dawsons-integral 𝑧 1 2 𝑖 𝜋 superscript 𝑒 superscript 𝑧 2 error-function-w 𝑧 1 2 𝑖 𝜋 superscript 𝑒 superscript 𝑧 2 error-function 𝑖 𝑧 {\displaystyle{\displaystyle F\left(z\right)=\tfrac{1}{2}i\sqrt{\pi}\left(e^{-% z^{2}}-w\left(z\right)\right)=-\tfrac{1}{2}i\sqrt{\pi}e^{-z^{2}}\operatorname{% erf}\left(iz\right).}}
    0 references
    DLMF:7.5.E1
    0 references
    F ( z ) Dawsons-integral 𝑧 {\displaystyle{\displaystyle F\left(\NVar{z}\right)}}
    C7.S2.E5.m2adec
    0 references
    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2adec
    0 references
    erf z error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erf}\NVar{z}}}
    C7.S2.E1.m2adec
    0 references
    w ( z ) error-function-w 𝑧 {\displaystyle{\displaystyle w\left(\NVar{z}\right)}}
    C7.S2.E3.m2adec
    0 references
    e {\displaystyle{\displaystyle\mathrm{e}}}
    C4.S2.E11.m2adec
    0 references
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    C1.S9.E1.m2adec
    0 references
    z 𝑧 {\displaystyle{\displaystyle z}}
    C7.S1.XMD2.m1dec
    0 references