DLMF:6.16.E2 (Q2311): Difference between revisions

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Property / Symbols defined
 
Property / Symbols defined: partial sum (locally) / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: sine integral / rank
 
Normal rank
Property / Symbols used: sine integral / qualifier
 
test:

Si ⁡ ( z ) sine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Si}\left(\NVar{z}\right)}}

\sinint@{\NVar{z}}
Property / Symbols used: sine integral / qualifier
 
xml-id: C6.S2.E9.m2adec
Property / Symbols used
 
Property / Symbols used: Q11307 / rank
 
Normal rank
Property / Symbols used: Q11307 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11307 / qualifier
 
xml-id: C6.S1.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q11308 / rank
 
Normal rank
Property / Symbols used: Q11308 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11308 / qualifier
 
xml-id: C6.S1.XMD3.m1dec
Property / Symbols used
 
Property / Symbols used: remainder term (locally) / rank
 
Normal rank
Property / Symbols used: remainder term (locally) / qualifier
 
test:

R n ⁢ ( x ) subscript 𝑅 𝑛 𝑥 {\displaystyle{\displaystyle R_{n}(x)}}

R_{n}(x)
Property / Symbols used: remainder term (locally) / qualifier
 
xml-id: C6.S16.XMD2.m1dec

Latest revision as of 00:35, 2 January 2020

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DLMF:6.16.E2
No description defined

    Statements

    test
    S n ⁢ ( x ) = ∑ k = 0 n - 1 sin ⁡ ( ( 2 ⁢ k + 1 ) ⁢ x ) 2 ⁢ k + 1 = 1 2 ⁢ ∫ 0 x sin ⁡ ( 2 ⁢ n ⁢ t ) sin ⁡ t ⁢ d t = 1 2 ⁢ Si ⁡ ( 2 ⁢ n ⁢ x ) + R n ⁢ ( x ) , subscript 𝑆 𝑛 𝑥 superscript subscript 𝑘 0 𝑛 1 2 𝑘 1 𝑥 2 𝑘 1 1 2 superscript subscript 0 𝑥 2 𝑛 𝑡 𝑡 𝑡 1 2 sine-integral 2 𝑛 𝑥 subscript 𝑅 𝑛 𝑥 {\displaystyle{\displaystyle S_{n}(x)=\sum_{k=0}^{n-1}\frac{\sin\left((2k+1)x% \right)}{2k+1}=\frac{1}{2}\int_{0}^{x}\frac{\sin\left(2nt\right)}{\sin t}% \mathrm{d}t=\tfrac{1}{2}\mathrm{Si}\left(2nx\right)+R_{n}(x),}}
    0 references
    DLMF id
    DLMF:6.16.E2
    0 references
    Symbols used
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    sine integral
    test
    Si ⁡ ( z ) sine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Si}\left(\NVar{z}\right)}}
    xml-id
    C6.S2.E9.m2adec
    0 references
    Q11307
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C6.S1.XMD1.m1adec
    0 references
    Q11308
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C6.S1.XMD3.m1dec
    0 references
    remainder term (locally)
    test
    R n ⁢ ( x ) subscript 𝑅 𝑛 𝑥 {\displaystyle{\displaystyle R_{n}(x)}}
    xml-id
    C6.S16.XMD2.m1dec
    0 references
     
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