Editing Carl Neumann (section)

=== Mathematics ===
Neumann has a series of publications on the [[Dirichlet problem]].<ref name=":0" /> In 1861, Neumann solved the Dirichlet problem in a plane in using a logarithmic potential, a term that he coined.<ref name=":0" /> This work was extended in 1870 to solve a more general Dirichlet problem by introducing his method of the [[arithmetic mean]].<ref name=":0" /> Due to his work on the [[Dirichlet's principle|Dirichlet principle]] of potential theory, Neumann might be considered one of the initiators of the theory of [[integral equation]]s. The [[Neumann series]], which is analogous to the [[geometric series]]
:<math> \frac{1}{1-x} = 1 + x + x^2 + \cdots </math>

but for [[Matrix (mathematics)|infinite matrices]] or for [[bounded operator]]s, is named after him. The [[Neumann boundary condition]] for certain types of ordinary and [[partial differential equation]]s is named after him.<ref>{{Cite journal |last1=Cheng |first1=Alexander H.-D. |last2=Cheng |first2=Daisy T. |date=2005 |title=Heritage and early history of the boundary element method |url=https://linkinghub.elsevier.com/retrieve/pii/S0955799705000020 |journal=Engineering Analysis with Boundary Elements |language=en |volume=29 |issue=3 |pages=268–302 |doi=10.1016/j.enganabound.2004.12.001}}</ref>

In 1865, he wrote ''Vorlesungen über Riemanns Theorie der Abelschen Integrale'' on [[Abelian integral|abelian integrals]]. This book popularized [[Bernhard Riemann]]’s work on multivalued functions among mathematicians.<ref name=":0" />