DLMF:2.5.E10 (Q786): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
test:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Mellin transform / rank
 
Normal rank
Property / Symbols used: Mellin transform / qualifier
 
test:

ℳ ⁡ ( f ) ⁡ ( s ) Mellin-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}

\Mellintrans@{\NVar{f}}@{\NVar{s}}
Property / Symbols used: Mellin transform / qualifier
 
xml-id: C1.S14.E32.m2agdec
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: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1abdec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10999 / rank
 
Normal rank
Property / Symbols used: Q10999 / qualifier
 
test:

h ⁢ ( t ) = J ν 2 ⁡ ( t ) ℎ 𝑡 Bessel-J 𝜈 2 𝑡 {\displaystyle{\displaystyle h(t)={J_{\nu}^{2}}\left(t\right)}}

h(t)=\BesselJ{\nu}^{2}@{t}
Property / Symbols used: Q10999 / qualifier
 
xml-id: C2.S5.XMD10.m1dec
Property / Symbols used
 
Property / Symbols used: Q10998 / rank
 
Normal rank
Property / Symbols used: Q10998 / qualifier
 
test:

f ⁢ ( t ) = 1 / ( 1 + t ) 𝑓 𝑡 1 1 𝑡 {\displaystyle{\displaystyle f(t)=1/(1+t)}}

f(t)=1/(1+t)
Property / Symbols used: Q10998 / qualifier
 
xml-id: C2.S5.XMD9.m1adec

Latest revision as of 16:03, 1 January 2020

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Language Label Description Also known as
English
DLMF:2.5.E10
No description defined

    Statements

    test
    ℳ ⁡ h ⁡ ( z ) = 2 z - 1 ⁢ Γ ⁡ ( ν + 1 2 ⁢ z ) Γ 2 ⁡ ( 1 - 1 2 ⁢ z ) ⁢ Γ ⁡ ( 1 + ν - 1 2 ⁢ z ) ⁢ Γ ⁡ ( z ) ⁢ π sin ⁡ ( π ⁢ z ) , Mellin-transform ℎ 𝑧 superscript 2 𝑧 1 Euler-Gamma 𝜈 1 2 𝑧 Euler-Gamma 2 1 1 2 𝑧 Euler-Gamma 1 𝜈 1 2 𝑧 Euler-Gamma 𝑧 𝜋 𝜋 𝑧 {\displaystyle{\displaystyle\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z% \right)=\frac{2^{z-1}\Gamma\left(\nu+\frac{1}{2}z\right)}{{\Gamma^{2}}\left(1-% \frac{1}{2}z\right)\Gamma\left(1+\nu-\frac{1}{2}z\right)\Gamma\left(z\right)}% \frac{\pi}{\sin\left(\pi z\right)},}}
    0 references
    DLMF id
    DLMF:2.5.E10
    0 references
    constraint
    - 2 ⁢ ν < ℜ ⁡ z < 1 2 𝜈 𝑧 1 {\displaystyle{\displaystyle-2\nu<\Re z<1}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Mellin transform
    test
    ℳ ⁡ ( f ) ⁡ ( s ) Mellin-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}
    xml-id
    C1.S14.E32.m2agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q10999
    test
    h ⁢ ( t ) = J ν 2 ⁡ ( t ) ℎ 𝑡 Bessel-J 𝜈 2 𝑡 {\displaystyle{\displaystyle h(t)={J_{\nu}^{2}}\left(t\right)}}
    xml-id
    C2.S5.XMD10.m1dec
    0 references
    Q10998
    test
    f ⁢ ( t ) = 1 / ( 1 + t ) 𝑓 𝑡 1 1 𝑡 {\displaystyle{\displaystyle f(t)=1/(1+t)}}
    xml-id
    C2.S5.XMD9.m1adec
    0 references
     
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