DLMF:5.11.E9 (Q2131): 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: 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.m2addec
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.m2abdec
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.m2adec
Property / Symbols used
 
Property / Symbols used: Q11281 / rank
 
Normal rank
Property / Symbols used: Q11281 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11281 / qualifier
 
xml-id: C5.S1.XMD5.m1dec
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.m1dec

Latest revision as of 19:41, 1 January 2020

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English
DLMF:5.11.E9
No description defined

    Statements

    | Γ ( x + i y ) | 2 π | y | x - ( 1 / 2 ) e - π | y | / 2 , asymptotically-equals Euler-Gamma 𝑥 𝑖 𝑦 2 𝜋 superscript 𝑦 𝑥 1 2 superscript 𝑒 𝜋 𝑦 2 {\displaystyle{\displaystyle|\Gamma\left(x+iy\right)|\sim\sqrt{2\pi}|y|^{x-(1/% 2)}e^{-\pi|y|/2},}}
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    DLMF:5.11.E9
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    Γ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    C5.S2.E1.m2addec
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    asymptotically-equals {\displaystyle{\displaystyle\sim}}
    C2.S1.E1.m2aadec
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    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2addec
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    e {\displaystyle{\displaystyle\mathrm{e}}}
    C4.S2.E11.m2abdec
    0 references
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    C1.S9.E1.m2adec
    0 references
    x 𝑥 {\displaystyle{\displaystyle x}}
    C5.S1.XMD5.m1dec
    0 references
    y 𝑦 {\displaystyle{\displaystyle y}}
    C5.S1.XMD6.m1dec
    0 references