DLMF:14.17.E15 (Q4896): Difference between revisions

From DRMF
Jump to navigation Jump to search
Changed an Item: Add constraint
Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
Defining formula:

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.m2aedec

Revision as of 13:25, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:14.17.E15
No description defined

    Statements

    - 1 1 𝖯 ν ( x ) 𝖰 ν ( x ) d x = - sin ( 2 ν π ) ψ ( ν + 1 ) π ( 2 ν + 1 ) , superscript subscript 1 1 shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 shorthand-Ferrers-Legendre-Q-first-kind 𝜈 𝑥 𝑥 2 𝜈 𝜋 diffop digamma 1 𝜈 1 𝜋 2 𝜈 1 {\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}_{\nu}\left(x\right)\mathsf% {Q}_{\nu}\left(x\right)\mathrm{d}x=-\frac{\sin\left(2\nu\pi\right)\psi'\left(% \nu+1\right)}{\pi(2\nu+1)},}}
    0 references
    DLMF:14.17.E15
    0 references
    ν > 0 𝜈 0 {\displaystyle{\displaystyle\Re\nu>0}}
    0 references
    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2aedec
    0 references
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    C1.S4.SS4.m1andec
    0 references
    ψ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    C5.S2.E2.m2aedec
    0 references