DLMF:14.5.E26 (Q4738): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11601 / rank
 
Normal rank
Property / Symbols used: Q11601 / qualifier
 
Defining formula:

E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}

\compellintEk@{\NVar{k}}
Property / Symbols used: Q11601 / qualifier
 
xml-id: C19.S2.E8.m2acdec
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.m2afdec
Property / Symbols used
 
Property / Symbols used: hyperbolic secant function / rank
 
Normal rank
Property / Symbols used: hyperbolic secant function / qualifier
 
Defining formula:

sech ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}

\sech@@{\NVar{z}}
Property / Symbols used: hyperbolic secant function / qualifier
 
xml-id: C4.S28.E6.m2adec
Property / Symbols used
 
Property / Symbols used: Q11597 / rank
 
Normal rank
Property / Symbols used: Q11597 / qualifier
 
Defining formula:

𝑸 ν ⁡ ( z ) = 𝑸 ν 0 ⁡ ( z ) shorthand-associated-Legendre-black-Q 𝜈 𝑧 associated-Legendre-black-Q 0 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=% \boldsymbol{Q}^{0}_{\nu}\left(z\right)}}

\assLegendreOlverQ[]{\NVar{\nu}}@{\NVar{z}}=\assLegendreOlverQ[0]{\nu}@{z}
Property / Symbols used: Q11597 / qualifier
 
xml-id: C14.S2.SS2.p2.m6abdec
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.m1fdec

Latest revision as of 01:25, 2 January 2020

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DLMF:14.5.E26
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    Statements

    Defining formula
    𝑸 1 2 ⁡ ( cosh ⁡ ξ ) = 2 ⁢ π - 1 / 2 ⁢ cosh ⁡ ξ ⁢ sech ⁡ ( 1 2 ⁢ ξ ) ⁢ K ⁡ ( sech ⁡ ( 1 2 ⁢ ξ ) ) - 4 ⁢ π - 1 / 2 ⁢ cosh ⁡ ( 1 2 ⁢ ξ ) ⁢ E ⁡ ( sech ⁡ ( 1 2 ⁢ ξ ) ) , shorthand-associated-Legendre-black-Q 1 2 𝜉 2 superscript 𝜋 1 2 𝜉 1 2 𝜉 complete-elliptic-integral-first-kind-K 1 2 𝜉 4 superscript 𝜋 1 2 1 2 𝜉 complete-elliptic-integral-second-kind-E 1 2 𝜉 {\displaystyle{\displaystyle\boldsymbol{Q}_{\frac{1}{2}}\left(\cosh\xi\right)=% 2\pi^{-1/2}\cosh\xi\operatorname{sech}\left(\tfrac{1}{2}\xi\right)K\left(% \operatorname{sech}\left(\tfrac{1}{2}\xi\right)\right)-4\pi^{-1/2}\cosh\left(% \tfrac{1}{2}\xi\right)E\left(\operatorname{sech}\left(\tfrac{1}{2}\xi\right)% \right),}}
    0 references
    DLMF id
    DLMF:14.5.E26
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aodec
    0 references
    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1aedec
    0 references
    Q11601
    Defining formula
    E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m2acdec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2afdec
    0 references
    hyperbolic secant function
    Defining formula
    sech ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
    xml-id
    C4.S28.E6.m2adec
    0 references
    Q11597
    Defining formula
    𝑸 ν ⁡ ( z ) = 𝑸 ν 0 ⁡ ( z ) shorthand-associated-Legendre-black-Q 𝜈 𝑧 associated-Legendre-black-Q 0 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=% \boldsymbol{Q}^{0}_{\nu}\left(z\right)}}
    xml-id
    C14.S2.SS2.p2.m6abdec
    0 references
    Q11599
    Defining formula
    ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S5.XMD1.m1fdec
    0 references
     
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