DLMF:4.37.E20 (Q1939): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2addec
Property / Symbols used
 
Property / Symbols used: Q11246 / rank
 
Normal rank
Property / Symbols used: Q11246 / qualifier
 
test:

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.m9addec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2aedec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

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.m2addec
Property / Symbols used
 
Property / Symbols used: Q11222 / rank
 
Normal rank
Property / Symbols used: Q11222 / qualifier
 
test:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q11222 / qualifier
 
xml-id: C4.S1.XMD7.m1bdec

Latest revision as of 18:16, 1 January 2020

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DLMF:4.37.E20
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    Statements

    arccosh ( i y ) = ± 1 2 π i + ln ( ( y 2 + 1 ) 1 / 2 ± y ) , hyperbolic-inverse-cosine imaginary-unit 𝑦 plus-or-minus 1 2 𝜋 imaginary-unit plus-or-minus superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arccosh}\left(\mathrm{i}y\right)=\pm% \tfrac{1}{2}\pi\mathrm{i}+\ln\left((y^{2}+1)^{1/2}\pm y\right),}}
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    y 0 greater-than-or-less-than 𝑦 0 {\displaystyle{\displaystyle y\gtrless 0}}
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    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2addec
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    arccosh z hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\NVar{z}}}
    C4.S37.SS2.p1.m9addec
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    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    C1.S9.E1.m2aedec
    0 references
    ln z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    C4.S2.E2.m2addec
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    y 𝑦 {\displaystyle{\displaystyle y}}
    C4.S1.XMD7.m1bdec
    0 references