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Nov 8, 2021 · A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions. The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering.
- Wesley N. Mathews Jr., Mark A. Esrick, ZuYao Teoh, James K. Freericks
- 2021
During Kummer's first period of mathematics he worked on function theory. He extended Gauss's work on hypergeometric series, giving developments that are useful in the theory of differential equations. He was the first to compute the monodromy groups of these series.
The confluenthypergeometric equation, or Kummer’s equation, is given by I d2| dI2 + (1 − I) d| dI − 0| = 0, (1.1) with 0 and 1 constants. This differential equation is in the Laplace form [4, 5], where the coefficients of the different terms are at most linear functions in I, although we will not be using the Laplace method in this work.
- Wesley N. Mathews Jr., Mark A. Esrick, ZuYao Teoh, James K. Freericks
- 2021
Topics treated include the wave equation in the hands of d’Alembert and Euler; Fourier’s solutions to the heat equation and the contribution of Kovalevskaya; the work of Euler, Gauss, Kummer, Riemann, and Poincaré on the hypergeometric equation; Green’s functions, the Dirichlet principle, and Schwarz’s solution of the Dirichlet problem; minimal ...
- Jeremy Gray
Kummer's (confluent hypergeometric) function M(a, b, z), introduced by Kummer ( 1837 ), is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name.
May 10, 2024 · Dirichlet found an error, and Kummer continued his search and developed the concept of ideal numbers. Using this concept, he proved the insolubility of the Fermat relation for all but a small group of primes, and he thus laid the foundation for an eventual complete proof of Fermat’s last theorem.
Feb 8, 2010 · Feb 8, 2010. 1 Kummer Theory of Fields. Kummer theory is concerned with classifying the abelian extensions of exponent n of a eld K, assuming that K contains the nth roots of unity. It's a generalization of the correspondence between quadratic extensions of Q and non-square squarefree integers.
