DLMF:5.4.E19 (Q2057): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(2 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11280 / rank
 
Normal rank
Property / Symbols used: Q11280 / qualifier
 
Defining formula:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11280 / qualifier
 
xml-id: C5.S1.XMD10.m1dec
Property / Symbols used
 
Property / Symbols used: Q11279 / rank
 
Normal rank
Property / Symbols used: Q11279 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11279 / qualifier
 
xml-id: C5.S1.XMD4.m1adec

Latest revision as of 19:30, 1 January 2020

No description defined
Language Label Description Also known as
English
DLMF:5.4.E19
No description defined

    Statements

    Defining formula
    ψ ⁡ ( p q ) = - γ - ln ⁡ q - π 2 ⁢ cot ⁡ ( π ⁢ p q ) + 1 2 ⁢ ∑ k = 1 q - 1 cos ⁡ ( 2 ⁢ π ⁢ k ⁢ p q ) ⁢ ln ⁡ ( 2 - 2 ⁢ cos ⁡ ( 2 ⁢ π ⁢ k q ) ) . digamma 𝑝 𝑞 𝑞 𝜋 2 𝜋 𝑝 𝑞 1 2 superscript subscript 𝑘 1 𝑞 1 2 𝜋 𝑘 𝑝 𝑞 2 2 2 𝜋 𝑘 𝑞 {\displaystyle{\displaystyle\psi\left(\frac{p}{q}\right)=-\gamma-\ln q-\frac{% \pi}{2}\cot\left(\frac{\pi p}{q}\right)+\frac{1}{2}\sum_{k=1}^{q-1}\cos\left(% \frac{2\pi kp}{q}\right)\ln\left(2-2\cos\left(\frac{2\pi k}{q}\right)\right).}}
    0 references
    DLMF id
    DLMF:5.4.E19
    0 references
    Symbols used
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
    0 references
    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2agdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
    0 references
    Q11280
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C5.S1.XMD10.m1dec
    0 references
    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1adec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q2057&oldid=28551"