DLMF:2.5.E11 (Q787): Difference between revisions

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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.m1adec
Property / Symbols used
 
Property / Symbols used: Q11000 / rank
 
Normal rank
Property / Symbols used: Q11000 / qualifier
 
test:

a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}

a_{n}
Property / Symbols used: Q11000 / qualifier
 
xml-id: C2.S5.XMD11.m1dec
Property / Symbols used
 
Property / Symbols used: Q11001 / rank
 
Normal rank
Property / Symbols used: Q11001 / qualifier
 
test:

b n subscript 𝑏 𝑛 {\displaystyle{\displaystyle b_{n}}}

b_{n}
Property / Symbols used: Q11001 / qualifier
 
xml-id: C2.S5.XMD12.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.m1bdec

Latest revision as of 16:03, 1 January 2020

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DLMF:2.5.E11
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    Statements

    test
    res z = n ⁡ [ x - z ⁢ ℳ ⁡ f ⁡ ( 1 - z ) ⁢ ℳ ⁡ h ⁡ ( z ) ] = ( a n ⁢ ln ⁡ x + b n ) ⁢ x - n , subscript residue 𝑧 𝑛 delimited-[] superscript 𝑥 𝑧 Mellin-transform 𝑓 1 𝑧 Mellin-transform ℎ 𝑧 subscript 𝑎 𝑛 𝑥 subscript 𝑏 𝑛 superscript 𝑥 𝑛 {\displaystyle{\displaystyle\Residue_{z=n}\left[x^{-z}\mathscr{M}\mskip-3.0mu % f\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z% \right)\right]=(a_{n}\ln x+b_{n})x^{-n},}}
    0 references
    DLMF id
    DLMF:2.5.E11
    0 references
    Symbols used
    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.m2ahdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q10960
    test
    res residue {\displaystyle{\displaystyle\Residue}}
    xml-id
    C1.S10.SS3.p5.m4aadec
    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.m1adec
    0 references
    Q11000
    test
    a n subscript 𝑎 𝑛 {\displaystyle{\displaystyle a_{n}}}
    xml-id
    C2.S5.XMD11.m1dec
    0 references
    Q11001
    test
    b n subscript 𝑏 𝑛 {\displaystyle{\displaystyle b_{n}}}
    xml-id
    C2.S5.XMD12.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.m1bdec
    0 references
     
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