DLMF:2.3.E9 (Q725)

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DLMF:2.3.E9
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    0 e - x t q ( t ) ln t d t s = 0 Γ ( s + λ μ ) a s x ( s + λ ) / μ - ( ln x ) s = 0 Γ ( s + λ μ ) a s x ( s + λ ) / μ , asymptotic-expansion superscript subscript 0 superscript 𝑒 𝑥 𝑡 𝑞 𝑡 𝑡 𝑡 superscript subscript 𝑠 0 diffop Euler-Gamma 1 𝑠 𝜆 𝜇 subscript 𝑎 𝑠 superscript 𝑥 𝑠 𝜆 𝜇 𝑥 superscript subscript 𝑠 0 Euler-Gamma 𝑠 𝜆 𝜇 subscript 𝑎 𝑠 superscript 𝑥 𝑠 𝜆 𝜇 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-xt}q(t)\ln t\mathrm{d}t\sim% \sum_{s=0}^{\infty}\Gamma'\left(\frac{s+\lambda}{\mu}\right)\frac{a_{s}}{x^{(s% +\lambda)/\mu}}-(\ln x)\sum_{s=0}^{\infty}\Gamma\left(\frac{s+\lambda}{\mu}% \right)\frac{a_{s}}{x^{(s+\lambda)/\mu}},}}
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    DLMF:2.3.E9
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