Helmholtz decomposition
Extensions to fields not decaying at infinity

Most textbooks only deal with vector fields decaying faster than | r | − δ {\displaystyle |\mathbf {r} |^{-\delta }} with δ > 1 {\displaystyle \delta >1} at infinity. However, Otto Blumenthal showed in 1905 that an adapted integration kernel can be used to integrate fields decaying faster than | r | − δ {\displaystyle |\mathbf {r} |^{-\delta }} with δ > 0 {\displaystyle \delta >0}, which is substantially less strict.To achieve this, the kernel K (r, r ′) {\displaystyle K(\mathbf {r},\mathbf {r} ')} in the convolution integrals has to be replaced by K ′ (r, r ′) = K (r, r ′) − K (0, r ′) {\displaystyle K'(\mathbf {r},\mathbf {r} ')=K(\mathbf {r},\mathbf {r} ')-K(0,\mathbf {r} ')}. With even more complex integration kernels, solutions can be found even for divergent functions that need not grow faster than polynomial.

For all analytic vector fields that need not go to zero even at infinity, methods based on partial integration and the Cauchy formula for repeated integration can be used to compute closed-form solutions of the rotation and scalar potentials, as in the case of multivariate polynomial, sine, cosine, and exponential functions.
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