DLMF:7.11.E6 (Q2408): Difference between revisions

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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.m1edec

Latest revision as of 01:49, 2 January 2020

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DLMF:7.11.E6
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    Statements

    Defining formula
    C ⁡ ( z ) + i ⁢ S ⁡ ( z ) = z ⁢ M ⁡ ( 1 2 , 3 2 , 1 2 ⁢ π ⁢ i ⁢ z 2 ) = z ⁢ e π ⁢ i ⁢ z 2 / 2 ⁢ M ⁡ ( 1 , 3 2 , - 1 2 ⁢ π ⁢ i ⁢ z 2 ) . Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 𝑧 Kummer-confluent-hypergeometric-M 1 2 3 2 1 2 𝜋 𝑖 superscript 𝑧 2 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑧 2 2 Kummer-confluent-hypergeometric-M 1 3 2 1 2 𝜋 𝑖 superscript 𝑧 2 {\displaystyle{\displaystyle C\left(z\right)+iS\left(z\right)=zM\left(\tfrac{1% }{2},\tfrac{3}{2},\tfrac{1}{2}\pi iz^{2}\right)=ze^{\pi iz^{2}/2}M\left(1,% \tfrac{3}{2},-\tfrac{1}{2}\pi iz^{2}\right).}}
    0 references
    DLMF id
    DLMF:7.11.E6
    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.m2adec
    0 references
    Fresnel integral
    Defining formula
    S ⁡ ( z ) Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle S\left(\NVar{z}\right)}}
    xml-id
    C7.S2.E8.m2adec
    0 references
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
    Defining formula
    M ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E2.m2aadec
    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
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11312
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C7.S1.XMD2.m1edec
    0 references
     
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