Editing Carl Neumann (section)

== Work ==

=== Electrodynamics ===
Neuman's work on electrodynamics was focused on formalizing mathematically the theories of electrodynamics. However for a long time, Neumann's supported [[Weber electrodynamics]] over [[Maxwell's equations]].<ref name=":0" />

Neumann's research on electrodynamics started in the 1860s.<ref name=":0" /> He published three first majors works on electrodynamics in 1868 and 1873 and 1874.<ref name=":0" /> His work was stimulated by the work of his father and [[Wilhelm Eduard Weber]].<ref name=":0" /> He rederived [[Ampère's force law]] and [[Ampère's circuital law]] from his own formalism.<ref name=":0" /> He also derived Weber law in terms of [[Retarded potential|retarded potentials]], avoiding problems with [[action at a distance]].<ref name=":3" /> 

[[Hermann von Helmholtz]] criticized [[Weber electrodynamics]], including Neumann's work, for violating of the [[conservation of energy]] in the presence of velocity-dependent forces. This criticism started a debate between Neumann and Helmholtz.<ref name=":0" /> Neumann attempted to modify Weber's law by introducing an electric potential that was inversely proportional to the distance at long distances and different at short distances in analogy with the theory of [[capillary action]] and the [[luminiferous aether]].<ref name=":0" /> Helmholtz theory based on [[James Clerk Maxwell]]'s theory did not need these assumptions, but Helmholtz found himself unable to convince his peers at the time over one theory or the other.<ref name=":0" /> Due to the lack of experiments to settle the matter, Neumann's temporarily abandoned electrodynamics in the 1880s.<ref name=":0" />

In 1893, he returned to his electrodynamics research.<ref name=":0" /> He analyzed the mathematical similarity between [[fluid dynamics]] and electrodynamics, relating several common theorems.<ref name=":0" /> He also proposed that electrodynamics and thermodynamics could not be explained in terms of purely mechanical theories.<ref name=":0" /> Neumann remained critical of the works of Helmholtz and [[Heinrich Hertz]] on Maxwell's electrodynamics, but appreciated their [[action principles]].<ref name=":0" /> 

In 1901-1904, Neumann's finally discussed Maxwell's theory and praised the extension given by Hertz relating electrodynamics to the theory of [[thermal conduction]]. However Neumann worked on possible transformations of Maxwell's equations and was worried of the equations not being invariant for different reference frames.<ref name=":0" /> He also argued that for Newtonian mechanics to make sense there should exist an imovable object in the universe called the body Alpha, from which all speeds can be measure relative to it.<ref name=":2">{{Cite journal |last=Wilson |first=William |date=1950 |title=THE BODY ALPHA: An Essay on the Meaning of Relativity |url=https://www.jstor.org/stable/43413971 |journal=Science Progress (1933- ) |volume=38 |issue=152 |pages=622–636 |jstor=43413971 |issn=0036-8504}}</ref> The problems of reference frames was solved in 1905 by [[Albert Einstein]]'s [[special relativity]].<ref name=":3">{{Cite journal |last=Disalle |first=Robert |date=1993 |title=Carl Gottfried Neumann |url=https://www.cambridge.org/core/journals/science-in-context/article/abs/carl-gottfried-neumann/773C3C6FA625F5783DE422A1A679580F |journal=Science in Context |language=en |volume=6 |issue=1 |pages=345–353 |doi=10.1017/S0269889700001411 |issn=1474-0664}}</ref><ref name=":2" />

=== 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" />