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DLMF:7.5.E5 - MaRDI portal

DLMF:7.5.E5 (Q3049)

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DLMF:7.5.E5
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    Statements

    DLMF defining formula
    e - 1 2 ⁢ π ⁢ i ⁢ z 2 ⁢ ℱ ⁡ ( z ) = g ⁡ ( z ) + i ⁢ f ⁡ ( z ) . superscript 𝑒 1 2 𝜋 𝑖 superscript 𝑧 2 Fresnel-integral 𝑧 Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 {\displaystyle{\displaystyle e^{-\frac{1}{2}\pi iz^{2}}\mathcal{F}\left(z% \right)=\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right).}}
    0 references
    Symbols used
    auxiliary function for Fresnel integrals
    DLMF 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
    auxiliary function for Fresnel integrals
    DLMF defining formula
    g ⁡ ( z ) Fresnel-auxilliary-function-g 𝑧 {\displaystyle{\displaystyle\mathrm{g}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E11.m2abdec
    0 references
    Fresnel integral
    DLMF defining formula
    ℱ ⁡ ( z ) Fresnel-integral 𝑧 {\displaystyle{\displaystyle\mathcal{F}\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E6.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1ddec
    0 references

    Identifiers

     
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