DLMF:5.4.E4 (Q2040): Difference between revisions

From DRMF
Jump to navigation Jump to search
Changed an Item: Add constraint
Changed an Item: Add constraint
 
(3 intermediate revisions by the same user not shown)
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
 
Defining formula:

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.m2abdec
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.m2adec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

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

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11278 / rank
 
Normal rank
Property / Symbols used: Q11278 / qualifier
 
Defining formula:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q11278 / qualifier
 
xml-id: C5.S1.XMD6.m1adec

Latest revision as of 19:29, 1 January 2020

No description defined
Language Label Description Also known as
English
DLMF:5.4.E4
No description defined

    Statements

    Γ ( 1 2 + i y ) Γ ( 1 2 - i y ) = | Γ ( 1 2 + i y ) | 2 = π cosh ( π y ) , Euler-Gamma 1 2 imaginary-unit 𝑦 Euler-Gamma 1 2 imaginary-unit 𝑦 superscript Euler-Gamma 1 2 imaginary-unit 𝑦 2 𝜋 𝜋 𝑦 {\displaystyle{\displaystyle\Gamma\left(\tfrac{1}{2}+\mathrm{i}y\right)\Gamma% \left(\tfrac{1}{2}-\mathrm{i}y\right)=\left|\Gamma\left(\tfrac{1}{2}+\mathrm{i% }y\right)\right|^{2}=\frac{\pi}{\cosh\left(\pi y\right)},}}
    0 references
    DLMF:5.4.E4
    0 references
    Γ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    C5.S2.E1.m2acdec
    0 references
    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2abdec
    0 references
    cosh z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    C4.S28.E2.m2adec
    0 references
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    C1.S9.E1.m2aadec
    0 references
    y 𝑦 {\displaystyle{\displaystyle y}}
    C5.S1.XMD6.m1adec
    0 references