Editing Helmholtz decomposition (section)

== Extensions to fields not decaying at infinity ==

Most textbooks only deal with vector fields decaying faster than <math>|\mathbf{r}|^{-\delta}</math> with <math>\delta > 1</math> at infinity.<ref name="petrascheck2015" /><ref name="petrascheck2017"/><ref name="gregory1996" /> However, [[Otto Blumenthal]] showed in 1905 that an adapted integration kernel can be used to integrate fields decaying faster than <math>|\mathbf{r}|^{-\delta}</math> with <math>\delta > 0</math>, which is substantially less strict.
To achieve this, the kernel <math>K(\mathbf{r}, \mathbf{r}')</math> in the convolution integrals has to be replaced by <math>K'(\mathbf{r}, \mathbf{r}') = K(\mathbf{r}, \mathbf{r}') - K(0, \mathbf{r}')</math>.<ref name="blumenthal1905" />
With even more complex integration kernels, solutions can be found even for divergent functions that need not grow faster than polynomial.<ref name="trancong1993" /><ref name="petrascheck2017"/><ref name="glotzl2020" /><ref name="gurtin1962"/>

For all [[Analytic function|analytic]] vector fields that need not go to zero even at infinity, methods based on [[Integration by parts|partial integration]] and the [[Cauchy formula for repeated integration]]<ref name="cauchy1823" /> 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 function]]s.<ref name="glotzl2023" />