DLMF:10.65.E3 (Q3835)

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DLMF:10.65.E3
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    ker n ⁡ x = 1 2 ⁢ ( 1 2 ⁢ x ) - n ⁢ ∑ k = 0 n - 1 ( n - k - 1 ) ! k ! ⁢ cos ⁡ ( 3 4 ⁢ n ⁢ π + 1 2 ⁢ k ⁢ π ) ⁢ ( 1 4 ⁢ x 2 ) k - ln ⁡ ( 1 2 ⁢ x ) ⁢ ber n ⁡ x + 1 4 ⁢ π ⁢ bei n ⁡ x + 1 2 ⁢ ( 1 2 ⁢ x ) n ⁢ ∑ k = 0 ∞ ψ ⁡ ( k + 1 ) + ψ ⁡ ( n + k + 1 ) k ! ⁢ ( n + k ) ! ⁢ cos ⁡ ( 3 4 ⁢ n ⁢ π + 1 2 ⁢ k ⁢ π ) ⁢ ( 1 4 ⁢ x 2 ) k , Kelvin-ker 𝑛 𝑥 1 2 superscript 1 2 𝑥 𝑛 superscript subscript 𝑘 0 𝑛 1 𝑛 𝑘 1 𝑘 3 4 𝑛 𝜋 1 2 𝑘 𝜋 superscript 1 4 superscript 𝑥 2 𝑘 1 2 𝑥 Kelvin-ber 𝑛 𝑥 1 4 𝜋 Kelvin-bei 𝑛 𝑥 1 2 superscript 1 2 𝑥 𝑛 superscript subscript 𝑘 0 digamma 𝑘 1 digamma 𝑛 𝑘 1 𝑘 𝑛 𝑘 3 4 𝑛 𝜋 1 2 𝑘 𝜋 superscript 1 4 superscript 𝑥 2 𝑘 {\displaystyle{\displaystyle\operatorname{ker}_{n}x=\tfrac{1}{2}(\tfrac{1}{2}x% )^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{% 2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-\ln\left(\tfrac{1}{2}x\right)% \operatorname{ber}_{n}x+\tfrac{1}{4}\pi\operatorname{bei}_{n}x+\tfrac{1}{2}(% \tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\psi\left(k+1\right)+\psi\left(n+k+% 1\right)}{k!(n+k)!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1% }{4}x^{2})^{k},}}
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    DLMF:10.65.E3
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