DLMF:14.15.E27 (Q4875): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11246 / rank
 
Normal rank
Property / Symbols used: Q11246 / qualifier
 
Defining formula:

arccosh ⁡ z hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\NVar{z}}}

\acosh@@{\NVar{z}}
Property / Symbols used: Q11246 / qualifier
 
xml-id: C4.S37.SS2.p1.m9adec
Property / Symbols used
 
Property / Symbols used: Q11248 / rank
 
Normal rank
Property / Symbols used: Q11248 / qualifier
 
Defining formula:

arctanh ⁡ z hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}\NVar{z}}}

\atanh@@{\NVar{z}}
Property / Symbols used: Q11248 / qualifier
 
xml-id: C4.S37.SS2.p1.m10adec
Property / Symbols used
 
Property / Symbols used: Q11589 / rank
 
Normal rank
Property / Symbols used: Q11589 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11589 / qualifier
 
xml-id: C14.S1.XMD1.m1qdec
Property / Symbols used
 
Property / Symbols used: Q11620 / rank
 
Normal rank
Property / Symbols used: Q11620 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11620 / qualifier
 
xml-id: C14.S15.XMD13.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11621 / rank
 
Normal rank
Property / Symbols used: Q11621 / qualifier
 
Defining formula:

ζ 𝜁 {\displaystyle{\displaystyle\zeta}}

\zeta
Property / Symbols used: Q11621 / qualifier
 
xml-id: C14.S15.XMD14.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11619 / rank
 
Normal rank
Property / Symbols used: Q11619 / qualifier
 
Defining formula:

α 𝛼 {\displaystyle{\displaystyle\alpha}}

\alpha
Property / Symbols used: Q11619 / qualifier
 
xml-id: C14.S15.XMD9.m1jdec

Latest revision as of 01:53, 2 January 2020

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DLMF:14.15.E27
No description defined

    Statements

    Defining formula
    1 2 ⁢ ζ ⁢ ( ζ 2 - α 2 ) 1 / 2 - 1 2 ⁢ α 2 ⁢ arccosh ⁡ ( ζ α ) = ( 1 - a 2 ) 1 / 2 ⁢ arctanh ⁡ ( 1 x ⁢ ( x 2 - a 2 1 - a 2 ) 1 / 2 ) - arccosh ⁡ ( x a ) , 1 2 𝜁 superscript superscript 𝜁 2 superscript 𝛼 2 1 2 1 2 superscript 𝛼 2 hyperbolic-inverse-cosine 𝜁 𝛼 superscript 1 superscript 𝑎 2 1 2 hyperbolic-inverse-tangent 1 𝑥 superscript superscript 𝑥 2 superscript 𝑎 2 1 superscript 𝑎 2 1 2 hyperbolic-inverse-cosine 𝑥 𝑎 {\displaystyle{\displaystyle\frac{1}{2}\zeta\left(\zeta^{2}-\alpha^{2}\right)^% {1/2}-\frac{1}{2}\alpha^{2}\operatorname{arccosh}\left(\frac{\zeta}{\alpha}% \right)=\left(1-a^{2}\right)^{1/2}\operatorname{arctanh}\left(\frac{1}{x}\left% (\frac{x^{2}-a^{2}}{1-a^{2}}\right)^{1/2}\right)-\operatorname{arccosh}\left(% \frac{x}{a}\right),}}
    0 references
    DLMF id
    DLMF:14.15.E27
    0 references
    constraint
    α ≤ ζ < ∞ 𝛼 𝜁 {\displaystyle{\displaystyle\alpha\leq\zeta<\infty}}
    0 references
    Symbols used
    Q11246
    Defining formula
    arccosh ⁡ z hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m9adec
    0 references
    Q11248
    Defining formula
    arctanh ⁡ z hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m10adec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1qdec
    0 references
    Q11620
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C14.S15.XMD13.m1cdec
    0 references
    Q11621
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C14.S15.XMD14.m1bdec
    0 references
    Q11619
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S15.XMD9.m1jdec
    0 references
     
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