DLMF:5.7.E5 (Q2081): Difference between revisions

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Property / constraint
 

z ≠ 0 , ± 1 𝑧 0 plus-or-minus 1 {\displaystyle{\displaystyle z\neq 0,\pm 1}}

z\neq 0,\pm 1
Property / constraint: z ≠ 0 , ± 1 𝑧 0 plus-or-minus 1 {\displaystyle{\displaystyle z\neq 0,\pm 1}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Euler’s constant / rank
 
Normal rank
Property / Symbols used: Euler’s constant / qualifier
 
test:

γ {\displaystyle{\displaystyle\gamma}}

\EulerConstant
Property / Symbols used: Euler’s constant / qualifier
 
xml-id: C5.S2.E3.m2acdec
Property / Symbols used
 
Property / Symbols used: Riemann zeta function / rank
 
Normal rank
Property / Symbols used: Riemann zeta function / qualifier
 
test:

ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}

\Riemannzeta@{\NVar{s}}
Property / Symbols used: Riemann zeta function / qualifier
 
xml-id: C25.S2.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2adec
Property / Symbols used
 
Property / Symbols used: cotangent function / rank
 
Normal rank
Property / Symbols used: cotangent function / qualifier
 
test:

cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}

\cot@@{\NVar{z}}
Property / Symbols used: cotangent function / qualifier
 
xml-id: C4.S14.E7.m2adec
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
test:

ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

\digamma@{\NVar{z}}
Property / Symbols used: psi (or digamma) function / qualifier
 
xml-id: C5.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11279 / rank
 
Normal rank
Property / Symbols used: Q11279 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11279 / qualifier
 
xml-id: C5.S1.XMD4.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11273 / rank
 
Normal rank
Property / Symbols used: Q11273 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11273 / qualifier
 
xml-id: C5.S1.XMD7.m1cdec

Latest revision as of 18:34, 1 January 2020

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DLMF:5.7.E5
No description defined

    Statements

    test
    ψ ⁡ ( 1 + z ) = 1 2 ⁢ z - π 2 ⁢ cot ⁡ ( π ⁢ z ) + 1 z 2 - 1 + 1 - γ - ∑ k = 1 ∞ ( ζ ⁡ ( 2 ⁢ k + 1 ) - 1 ) ⁢ z 2 ⁢ k , digamma 1 𝑧 1 2 𝑧 𝜋 2 𝜋 𝑧 1 superscript 𝑧 2 1 1 superscript subscript 𝑘 1 Riemann-zeta 2 𝑘 1 1 superscript 𝑧 2 𝑘 {\displaystyle{\displaystyle\psi\left(1+z\right)=\frac{1}{2z}-\frac{\pi}{2}% \cot\left(\pi z\right)+\frac{1}{z^{2}-1}+1-\gamma-\sum_{k=1}^{\infty}(\zeta% \left(2k+1\right)-1)z^{2k},}}
    0 references
    DLMF id
    DLMF:5.7.E5
    0 references
    constraint
    z ≠ 0 , ± 1 𝑧 0 plus-or-minus 1 {\displaystyle{\displaystyle z\neq 0,\pm 1}}
    0 references
    Symbols used
    Euler’s constant
    test
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2acdec
    0 references
    Riemann zeta function
    test
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    cotangent function
    test
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
    0 references
    Q11279
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1ddec
    0 references
    Q11273
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C5.S1.XMD7.m1cdec
    0 references
     
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