DLMF:1.14.E48 (Q540): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / rank
 
Normal rank
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
Defining formula:

d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}

\deriv{\NVar{f}}{\NVar{x}}
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
xml-id: C1.S4.E4.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aakdec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
Defining formula:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aahdec
Property / Symbols used
 
Property / Symbols used: Q10761 / rank
 
Normal rank
Property / Symbols used: Q10761 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q10761 / qualifier
 
xml-id: C1.S1.XMD6.m1dec

Latest revision as of 16:27, 1 January 2020

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DLMF:1.14.E48
No description defined

    Statements

    Defining formula
    d m d s m ⁡ 𝒮 ⁡ f ⁡ ( s ) = ( - 1 ) m ⁢ m ! ⁢ ∫ 0 ∞ f ⁢ ( t ) ⁢ d t ( s + t ) m + 1 , derivative 𝑠 𝑚 Stieltjes-transform 𝑓 𝑠 superscript 1 𝑚 𝑚 subscript superscript 0 𝑓 𝑡 𝑡 superscript 𝑠 𝑡 𝑚 1 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{m}}{{\mathrm{d}s}^{m}}\mathcal% {S}\mskip-3.0mu f\mskip 3.0mu \left(s\right)=(-1)^{m}m!\int^{\infty}_{0}\frac{% f(t)\mathrm{d}t}{(s+t)^{m+1}},}}
    0 references
    DLMF id
    DLMF:1.14.E48
    0 references
    constraint
    m = 0 , 1 , 2 , … 𝑚 0 1 2 … {\displaystyle{\displaystyle m=0,1,2,\dots}}
    0 references
    Symbols used
    Stieltjes transform
    Defining formula
    𝒮 ⁡ ( f ) ⁡ ( s ) Stieltjes-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}
    xml-id
    C1.S14.E47.m2aadec
    0 references
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aakdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aahdec
    0 references
    Q10761
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C1.S1.XMD6.m1dec
    0 references
     
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