Kelvin Functions - Ker(x)

Ker(x)

For integers n, Kern(x) has the (complicated) series expansion

\begin{align} \mathrm{Ker}_n(x) & = \frac{1}{2} \left(\frac{x}{2}\right)^{-n} \sum_{k=0}^{n-1} \cos\left \frac{(n-k-1)!}{k!} \left(\frac{x^2}{4}\right)^k - \ln\left(\frac{x}{2}\right) \mathrm{Ber}_n(x) + \frac{\pi}{4}\mathrm{Bei}_n(x) \\ & {} \quad + \frac{1}{2} \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \cos\left \frac{\psi(k+1) + \psi(n + k + 1)}{k! (n+k)!} \left(\frac{x^2}{4}\right)^k \end{align}

where is the Digamma function. The special case Ker, commonly denoted as just Ker, has the series expansion

and the asymptotic series

where, and


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