DLMF:1.14.E48 (Q540)

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DLMF:1.14.E48
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    Statements

    test
    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}},}}
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    DLMF id
    DLMF:1.14.E48
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    constraint
    m = 0 , 1 , 2 , … 𝑚 0 1 2 … {\displaystyle{\displaystyle m=0,1,2,\dots}}
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    Symbols used
    Stieltjes transform
    test
    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
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    derivative of $$f$$ with respect to $$x$$
    test
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2aadec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aakdec
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    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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