DLMF:27.10.E12 (Q8069): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: complex conjugate / rank
 
Normal rank
Property / Symbols used: complex conjugate / qualifier
 
Defining formula:

z ¯ 𝑧 {\displaystyle{\displaystyle\overline{\NVar{z}}}}

\conj{\NVar{z}}
Property / Symbols used: complex conjugate / qualifier
 
xml-id: C1.S9.E11.m2aadec
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.m2aedec
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.m2aedec
Property / Symbols used
 
Property / Symbols used: Q12238 / rank
 
Normal rank
Property / Symbols used: Q12238 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12238 / qualifier
 
xml-id: C27.S1.XMD2.m1jdec
Property / Symbols used
 
Property / Symbols used: Q12239 / rank
 
Normal rank
Property / Symbols used: Q12239 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12239 / qualifier
 
xml-id: C27.S1.XMD3.m1fdec
Property / Symbols used
 
Property / Symbols used: Q12234 / rank
 
Normal rank
Property / Symbols used: Q12234 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12234 / qualifier
 
xml-id: C27.S1.XMD4.m1idec
Property / Symbols used
 
Property / Symbols used: Q12259 / rank
 
Normal rank
Property / Symbols used: Q12259 / qualifier
 
Defining formula:

χ Dirichlet-character {\displaystyle{\displaystyle\chi}}

\Dirichletchar[]
Property / Symbols used: Q12259 / qualifier
 
xml-id: C27.S10.XMD5.m1ddec

Latest revision as of 14:24, 2 January 2020

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DLMF:27.10.E12
No description defined

    Statements

    Defining formula
    χ ⁡ ( n ) = G ⁡ ( 1 , χ ) k ⁢ ∑ m = 1 k χ ¯ ⁢ ( m ) ⁢ e - 2 ⁢ π ⁢ i ⁢ m ⁢ n / k . Dirichlet-character 𝑛 𝑘 Gauss-sum 1 Dirichlet-character 𝑘 superscript subscript 𝑚 1 𝑘 Dirichlet-character 𝑚 superscript 𝑒 2 imaginary-unit 𝑚 𝑛 𝑘 {\displaystyle{\displaystyle\chi\left(n\right)=\frac{G\left(1,\chi\right)}{k}% \sum_{m=1}^{k}\overline{\chi}(m)e^{-2\pi\mathrm{i}mn/k}.}}
    0 references
    DLMF id
    DLMF:27.10.E12
    0 references
    Symbols used
    Q12252
    Defining formula
    χ ⁡ ( n ) Dirichlet-character 𝑛 𝑘 {\displaystyle{\displaystyle\chi\left(\NVar{n}\right)}}
    xml-id
    C27.S8.m1addec
    0 references
    Gauss sum
    Defining formula
    G ⁡ ( n , χ ) Gauss-sum 𝑛 Dirichlet-character {\displaystyle{\displaystyle G\left(\NVar{n},\NVar{\chi}\right)}}
    xml-id
    C27.S10.E9.m2acdec
    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
    complex conjugate
    Defining formula
    z ¯ 𝑧 {\displaystyle{\displaystyle\overline{\NVar{z}}}}
    xml-id
    C1.S9.E11.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q12238
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C27.S1.XMD2.m1jdec
    0 references
    Q12239
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C27.S1.XMD3.m1fdec
    0 references
    Q12234
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C27.S1.XMD4.m1idec
    0 references
    Q12259
    Defining formula
    χ Dirichlet-character {\displaystyle{\displaystyle\chi}}
    xml-id
    C27.S10.XMD5.m1ddec
    0 references
     
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