DLMF:20.11.E2 (Q6843): Difference between revisions

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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.m2aadec
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.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11951 / rank
 
Normal rank
Property / Symbols used: Q11951 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11951 / qualifier
 
xml-id: C20.S1.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q11947 / rank
 
Normal rank
Property / Symbols used: Q11947 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11947 / qualifier
 
xml-id: C20.S1.XMD2.m1adec
Property / Symbols used
 
Property / Symbols used: Gauss sum (locally) / rank
 
Normal rank
Property / Symbols used: Gauss sum (locally) / qualifier
 
Defining formula:

G ⁢ ( m , n ) 𝐺 𝑚 𝑛 {\displaystyle{\displaystyle G(m,n)}}

G(m,n)
Property / Symbols used: Gauss sum (locally) / qualifier
 
xml-id: C20.S11.XMD1.m1adec

Latest revision as of 14:43, 2 January 2020

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DLMF:20.11.E2
No description defined

    Statements

    Defining formula
    1 n ⁢ G ⁢ ( m , n ) = 1 n ⁢ ∑ k = 0 n - 1 e - π ⁢ i ⁢ k 2 ⁢ m / n = e - π ⁢ i / 4 m ⁢ ∑ j = 0 m - 1 e π ⁢ i ⁢ j 2 ⁢ n / m = e - π ⁢ i / 4 m ⁢ G ⁢ ( - n , m ) . 1 𝑛 𝐺 𝑚 𝑛 1 𝑛 superscript subscript 𝑘 0 𝑛 1 superscript 𝑒 𝜋 𝑖 superscript 𝑘 2 𝑚 𝑛 superscript 𝑒 𝜋 𝑖 4 𝑚 superscript subscript 𝑗 0 𝑚 1 superscript 𝑒 𝜋 𝑖 superscript 𝑗 2 𝑛 𝑚 superscript 𝑒 𝜋 𝑖 4 𝑚 𝐺 𝑛 𝑚 {\displaystyle{\displaystyle\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum% \limits_{k=0}^{n-1}e^{-\pi ik^{2}m/n}=\frac{e^{-\pi i/4}}{\sqrt{m}}\sum\limits% _{j=0}^{m-1}e^{\pi ij^{2}n/m}=\frac{e^{-\pi i/4}}{\sqrt{m}}G(-n,m).}}
    0 references
    DLMF id
    DLMF:20.11.E2
    0 references
    comment
    Warning: Falling back to standard tex;
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q11951
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C20.S1.XMD1.m1adec
    0 references
    Q11947
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C20.S1.XMD2.m1adec
    0 references
    Gauss sum (locally)
    Defining formula
    G ⁢ ( m , n ) 𝐺 𝑚 𝑛 {\displaystyle{\displaystyle G(m,n)}}
    xml-id
    C20.S11.XMD1.m1adec
    0 references
     
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