DLMF:14.5.E24 (Q4736): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
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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.m2adec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
Defining formula:

cosh z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.m2addec
Property / Symbols used
 
Property / Symbols used: Q11447 / rank
 
Normal rank
Property / Symbols used: Q11447 / qualifier
 
Defining formula:

P ν ( z ) = P ν 0 ( z ) shorthand-Legendre-P-first-kind 𝜈 𝑧 Legendre-P-first-kind 0 𝜈 𝑧 {\displaystyle{\displaystyle P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}% \left(z\right)}}

\assLegendreP[]{\NVar{\nu}}@{\NVar{z}}=\assLegendreP[0]{\nu}@{z}
Property / Symbols used: Q11447 / qualifier
 
xml-id: C14.S2.SS2.p2.m3abdec
Property / Symbols used
 
Property / Symbols used: Q11599 / rank
 
Normal rank
Property / Symbols used: Q11599 / qualifier
 
Defining formula:

ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}

\xi>0
Property / Symbols used: Q11599 / qualifier
 
xml-id: C14.S5.XMD1.m1ddec

Latest revision as of 01:25, 2 January 2020

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DLMF:14.5.E24
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    Statements

    P 1 2 ( cosh ξ ) = 2 π e ξ / 2 E ( ( 1 - e - 2 ξ ) 1 / 2 ) , shorthand-Legendre-P-first-kind 1 2 𝜉 2 𝜋 superscript 𝑒 𝜉 2 complete-elliptic-integral-second-kind-E superscript 1 superscript 𝑒 2 𝜉 1 2 {\displaystyle{\displaystyle P_{\frac{1}{2}}\left(\cosh\xi\right)=\frac{2}{\pi% }e^{\xi/2}E\left(\left(1-e^{-2\xi}\right)^{1/2}\right),}}
    0 references
    DLMF:14.5.E24
    0 references
    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2amdec
    0 references
    E ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    C19.S2.E8.m2abdec
    0 references
    e {\displaystyle{\displaystyle\mathrm{e}}}
    C4.S2.E11.m2adec
    0 references
    cosh z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    C4.S28.E2.m2addec
    0 references
    P ν ( z ) = P ν 0 ( z ) shorthand-Legendre-P-first-kind 𝜈 𝑧 Legendre-P-first-kind 0 𝜈 𝑧 {\displaystyle{\displaystyle P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}% \left(z\right)}}
    C14.S2.SS2.p2.m3abdec
    0 references
    ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}
    C14.S5.XMD1.m1ddec
    0 references