Throughout the history of mathematics and physics, there has been many examples where mathematics was discovered first prior to its application in physics. Consider $i=\sqrt {-1}$ as an example, among many others.
I am interested to know about discoveries in physics, for which a mathematical counterpart did not exist at the time and for which new mathematics had to be created. Are there such examples? If so, a few examples would be appreciated (preferably with sources).
Potential theory (Green's formulas, Green's function etc.) was discovered by George Green who was doing physics. His work was called "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism". Laplace equation was first written in a paper on Saturn rings.
Eigenvalues, eigenvectors and adjoint operators were discovered by mathematicians who were doing physics (namely celestial mechanics). Fourier transform was discovered by Fourier who was doing physics (in his studies of heat flow). Same applies to Bessel functions and theta functions which appear for the first time in Fourier's book on heat. Fourier series were discovered by Daniel Bernoulli who was doing physics (oscillations of a string).
Vector analysis was invented by physicists Herz and Gibbs. Maxwell's book on electromagneitsm is literally packed with mathematical discoveries.
Calculus of operators (a.k.a. operational calculus) was discovered by physicist/electrical engineer Oliver Heaviside, who was studying the "telegrapher's equation".
The theories of unbounded operators, operator algebras and quantum logic were developed by von Neumann to give mathematical foundations of quantum mechanics.
Lee and Yang Theorem was discovered by two physicists who studied phase transitions.
Yang-Mills equations were discovered by physicists Yang and Mills working on quantum field theory.
The list can be made almost infinite.
Paul Dirac developed the Dirac Delta for physics. Mathematicians said it made no sense. It took the development of the theory of distributions to show they were wrong.
Joseph Fourier created Fourier series to study the propagation of heat in solid bodies initially but it became an important field in pure mathematics later.
Oliver Heaviside developed Heaviside calculus to investigate electric circuits, which was popular among electrical engineers but was initially rejected by mathematicians for lacking mathematical rigor. Later, Heaviside calculus was reworked by Bromwich into what is known as Laplace transform today.
