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Integral and Integro-Differential Equations
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
This is the Abel type of integral equation. Niels Henrik Abel was a Norwegian mathematician and was born in 1802 and passed away at the age of 26. Abel had shown that it is impossible to solve the quintic equation (5th order algebraic equation) in radical forms. He also discovered elliptic function, which was subsequently improved by Jacobi and is now called the Jacobi elliptic function (see Chapter 1 and Appendix D). Independent of Galois, he also developed the group theory. He lived in poverty for his whole life, and passed away just two days short of receiving the good news from his friend Crelle (editor of Crelle’sjournal) that he was appointed professor at University of Berlin. The renowned French mathematician Charles Hermite (the discoverer of Hermite polynomials) remarked on Abel’s six years of works as “Abel left mathematicians enough to keep them busy for five hundred years.” Another renowned French mathematician Adrien-Marie Legendre (the discoverer of Legendre polynomials) commented, “What a head the young Norwegian has!”
Fracture Mechanics and Dynamics
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
Niels Henrik Abel was a Norwegian mathematician who was born in 1802 and passed away in 1828 at the age of 26. Abel had shown that it is impossible to solve the quintic equation in radical forms. He also discovered the elliptic function, which was subsequently improved by Jacobi and is now called the Jacobi elliptic function (see Chapter 1 and the appendix of Chau, 2018). He lived in poverty for his whole life, and passed away just two days short of receiving the good news from his friend Crelle that he was appointed as professor at University of Berlin.
Mathematica
Published in Paul W. Ross, The Handbook of Software for Engineers and Scientists, 2018
For the algebraic equation: x3− 7x 2+3x − 10==0; the exact value for the real root reads: The general quintic equation is insoluble in terms of radicals. For example, Direct numerical evaluation is still possible: N[%, 20]{{x→−1.16730397826141868426},{x→ − 0.181232444469875383902 − 1.08395410131771066843 I},{x→ − 0.181232444469875383902 + 1.08395410131771066843 I},{x→ 0.76488443360058472603 − 0.35247154603172624932 I},{x→ 0.76488443360058472603 + 0.35247154603172624932 I}}
Remainder and quotient without polynomial long division
Published in International Journal of Mathematical Education in Science and Technology, 2021
The method used in the previous example to determine the remainder avoids long division (cf. Mac Lane & Birkhoff, 1985, Theorem 21, p. 161), and it allows us to reduce the number of calculations in many cases. Furthermore, it extends the classic remainder theorem (cf. Mac Lane & Birkhoff, 1985, Corollary 1, p. 162), which allows us to calculate the remainder on division by polynomials of type x−c. This theorem, probably already known by Descartes (cf. Smith & Latham, 2007, p. 179), goes back to the works of William George Horner (1786–1837) (cf. Cajori, 1911) and Paolo Ruffini (1765–1822) (cf. Ruffini, 1812, pp. 380–381), the latter of whom is usually associated with the impossibility of algebraically solving the general quintic equation (cf. Ayoub, 1982 and references therein).