DLMF:7.5.E9 (Q2355): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: change of variable (locally) / rank
 
Normal rank
Property / Symbols used: change of variable (locally) / qualifier
 
Defining formula:

ζ 𝜁 {\displaystyle{\displaystyle\zeta}}

\zeta
Property / Symbols used: change of variable (locally) / qualifier
 
xml-id: C7.S5.XMD1.m1bdec

Latest revision as of 01:40, 2 January 2020

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DLMF:7.5.E9
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    Statements

    Defining formula
    C ⁡ ( z ) ± i ⁢ S ⁡ ( z ) = 1 2 ⁢ ( 1 ± i ) ⁢ ( 1 - e ± 1 2 ⁢ π ⁢ i ⁢ z 2 ⁢ w ⁡ ( i ⁢ ζ ) ) . plus-or-minus Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 1 2 plus-or-minus 1 𝑖 1 superscript 𝑒 plus-or-minus 1 2 𝜋 𝑖 superscript 𝑧 2 error-function-w 𝑖 𝜁 {\displaystyle{\displaystyle C\left(z\right)\pm iS\left(z\right)=\tfrac{1}{2}(% 1\pm i)\left(1-e^{\pm\frac{1}{2}\pi iz^{2}}w\left(i\zeta\right)\right).}}
    0 references
    DLMF id
    DLMF:7.5.E9
    0 references
    Symbols used
    Fresnel integral
    Defining formula
    C ⁡ ( z ) Fresnel-cosine-integral 𝑧 {\displaystyle{\displaystyle C\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E7.m2addec
    0 references
    Fresnel integral
    Defining formula
    S ⁡ ( z ) Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle S\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E8.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    Faddeeva (or Faddeyeva) function
    Defining formula
    w ⁡ ( z ) error-function-w 𝑧 {\displaystyle{\displaystyle w\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E3.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
    0 references
    Q11312
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1hdec
    0 references
    change of variable (locally)
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C7.S5.XMD1.m1bdec
    0 references
     
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