Editing Harry Bateman (section)

==Scientific contributions==
In 1907, Harry Bateman was lecturing at the [[University of Liverpool]] together with another senior wrangler, [[Ebenezer Cunningham]]. In 1908, together they came up with the idea of a [[conformal group of spacetime]] (now usually denoted as {{math|C(1,3)}})<ref name="Kosyakov_2007" /> which involved an extension of the [[method of images]].<ref name="Warwick_2003" />

[[File:DecayChain241Pu-eng.svg|thumb|Quantity calculation with the Bateman function for 241Pu]]
In nuclear physics, the [[Bateman equation]] is a mathematical model describing abundances and activities in a decay chain as a function of time, based on the decay rates and initial abundances. The model was formulated by Ernest Rutherford in 1905 and the analytical solution was provided by Harry Bateman in 1910.<ref>Bateman, H. (1910, June). The solution of a system of differential equations occurring in the theory of radioactive transformations. In Proc. Cambridge Philos. Soc (Vol. 15, No. pt V, pp. 423–427) [https://archive.org/details/cbarchive_122715_solutionofasystemofdifferentia1843]</ref>

For his part, in 1910 Bateman published ''[[:s:The Transformation of the Electrodynamical Equations|The Transformation of the Electrodynamical Equations]]''.<ref name="Bateman_1910" /> He showed that the [[Jacobian matrix and determinant|Jacobian]] [[Matrix (mathematics)|matrix]] of a [[spacetime]] [[diffeomorphism]] which preserves the [[Maxwell equations]] is proportional to an [[orthogonal matrix]], hence [[conformal transformation|conformal]]. The transformation [[Lie group|group]] of such transformations has 15 parameters and extends both the [[Poincaré group]] and the [[Lorentz group]]. Bateman called the elements of this group [[spherical wave transformation]]s.<ref name="Bateman_1909" />

In evaluating this paper, one of his students, [[Clifford Truesdell]], wrote:
{{blockquote|The importance of Bateman's paper lies not in its specific details but in its general approach. Bateman, perhaps influenced by Hilbert's point of view in mathematical physics as a whole, was the first to see that the basic ideas of electromagnetism were equivalent to statements regarding integrals of [[differential form]]s, statements for which Grassmann's calculus of extension on differentiable manifolds, Poincaré's theories of Stokesian transformations and integral invariants, and Lie's theory of continuous groups could be fruitfully applied.<ref name="Truesdell_1984" />}}
Bateman was the first to apply [[Laplace transform]] to the integral equation in 1906. He submitted a detailed report on integral equations in 1911 to the British association for the advancement of science.<ref name="Bateman_1911" /> [[Horace Lamb]] in his 1910 paper<ref name="Lamb_1910" /> solved an integral equation 
:<math>2\int_0^\infty f(y-\zeta^2)\,d\zeta = F(y)</math>  
as a double integral, but in his footnote he says, "Mr. H. Bateman, to whom I submitted the question, has obtained a simpler solution in the form" 
:<math>f(y) = \frac 1 \pi \int_{-y}^\infty \frac{F'(-z)}{\sqrt{z+x}}dz</math>.

In 1914, Bateman published ''The Mathematical Analysis of Electrical and Optical Wave-motion''. As Murnaghan says, this book "is unique and characteristic of the man. Into less than 160 small pages is crowded a wealth of information which would take an expert year to digest."<ref name="Murnaghan_1948" />
The following year he published a [[textbook]] ''Differential Equations'', and sometime later ''Partial differential equations of mathematical physics''. Bateman is also author of ''Hydrodynamics'' and ''Numerical integration of differential equations''. Bateman studied the [[Burgers' equation]]<ref name="Bateman_1915_1" /> long before [[Jan Burgers]] started to study.

Harry Bateman wrote two significant articles on the history of applied mathematics: "The influence of tidal theory upon the development of mathematics",<ref name="Bateman_1943" /> and "Hamilton's work in dynamics and its influence on modern thought".<ref name="Bateman_1944" />

In his ''Mathematical Analysis of Electrical and Optical Wave-motion'' (p.&nbsp;131), he describes the charged-corpuscle trajectory as follows:
{{blockquote|a corpuscle has a kind of tube or thread attached to it. When the motion of the corpuscle changes a wave or kink runs along the thread; the energy radiated from the corpuscle spreads out in all directions but is concentrated round the thread so that the thread acts as a guiding wire.}}
This figure of speech is not to be confused with a [[String (physics)|string in physics]], for the universes in [[string theory]] have dimensions inflated beyond four, something not found in Bateman's work. Bateman went on to study the [[luminiferous aether]] with an article "The structure of the Aether".<ref name="Bateman_1915_2" /> His starting point is the [[bivector (complex)|bivector]] form of an [[electromagnetic field]], <math>\mathbf{E} + i \mathbf{B}</math>. He recalled [[Alfred-Marie Liénard]]'s electromagnetic fields, and then distinguished another type he calls "aethereal fields":
{{blockquote|When a large number of "aethereal fields" are superposed their singular curves indicate the structure of an "aether" which is capable of supporting a certain type of electromagnetic field.}}

Bateman received many honours for his contributions, including election to the [[American Philosophical Society]] in 1924, election to the [[Royal Society]] of London in 1928, and election to the [[United States National Academy of Sciences|National Academy of Sciences]] in 1930.<ref>{{Cite web |title=Harry Bateman |url=http://www.nasonline.org/member-directory/deceased-members/20000657.html |access-date=2023-08-23 |website=www.nasonline.org}}</ref> He was elected as vice-president of the [[American Mathematical Society]] in 1935 and was the Society's Gibbs Lecturer for 1943.<ref name="Murnaghan_1948" /><ref name="Bateman_1945" /> He was on his way to New York to receive an award from the Institute of Aeronautical Science when he died of [[coronary thrombosis]]. The ''Harry Bateman Research Instructorships'' at the California Institute of Technology is named in his honour.<ref name="caltech" />

After his death, his notes on higher transcendental functions were edited by [[Arthur Erdélyi]], [[Wilhelm Magnus]], {{ill|Fritz Oberhettinger|de}}, and [[Francesco Giacomo Tricomi|Francesco G. Tricomi]], and published in 1953.<ref name="Erdélyi_1953" />