Introduction of Nielshenrik Abel and his Theories in Mathematics (Essay Sample)

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NielsHenrik Abel
NielsHenrik Abel was a Norwegian mathematician who pioneered the development of various branches in modern mathematics. In 1815, Nielsenrolled in a cathedral school in Oslo, a place his poor father had moved to prior toNiels’ birth. Niels’ talent in mathematics was recognized in 1817 at the cathedral school, by a new mathematics teacher who introduced Niels to classics in mathematical literature. Moreover, the teacher, Bernt Michael Holmboeproposed originalproblems to Niels to solve. Abel considered the 17th century mathematical works of Sir Isaac Newton, 18th century Leonhard Eular, Joseph-Louis Lagrange and Carl Friedrich Gaus to prepare him for his research (Mahanti, para1).
In 1823, first of Abel’s papers were published. The papers dealt with functional equations and integrals, which made him the first person to form and resolve an integral equation. In 1824, Nielspublished his proof on the impossibility of solving the general algebraic equation in the fifth degree.The winter of 1825 – 1826 Abel spent with Norwegian friends in Germany, where he met August Leopold Crelle. Crelle was a civil engineer, as well as a self-taughtfanatic of mathematics. Crelle became Abel’s close friend, and with Abel’s encouragement, Crellefound a journal for pure and practical mathematics commonly referred to as Crelle’sJournal. In 1826, the first volume of the journal contained Abel’s papersincluding complex versions of his work in the quintic equation. Moreover, the journal contained papers that dealt with equation theory, theoretical mechanics and calculus. Later volumes of the journal presented Neils theory of elliptic functions that are complex functions, which generalize the common trigonometric functions. In Paris, Abel completed a major paper regarding the theory of integrals in algebraic functions. His result referred to as Abel’s theorem was the basis for a later ontheory of Abelianfunctions and integrals, whichgeneralized elliptic function theory to several variable functions. In 1828, Abel wrote many papers principally on equation theory, as well as elliptic functions. Among the theories,there were theories on polynomial equations worked with Abelian groups. Abel rapidly developed elliptic functions theory in competition with a German Carl Gustav Jacobi. Before this time, Abel’s reputationspread to all mathematical regions. During the fall of 1828, Abel was seriously ill, and the condition worsened when he travelled to visit his fiancée in Froland, where he died (Abel, Olav, and Ragni 3).
In his lifetime, Abel committed himself to various topics in mathematics (Ore 9). Hechose subjects in pure mathematics, which are classified intosolution of algebraic equations, transcendental functions particularly elliptic integrals and functions, as well as Abelian integrals. In addition, Abel dealt with functional equations, integral transformations and theory of series treated ina rigorous way. Abel in the study of functional equations, he published a paper with a title that characterizes composition law that is associative and commutative. The law is written as f(x, y) = f(y, x), f(z, f(x, y))=f(x, f(y, z))=f(y, f(z, x)) or f(z, r)= f(x, v) = f(y,s)(Abel, Olav, and Ragni23). Abel’s generalmethod for equation found that it was easy to determine π through functional equations. This was applicable toan equation of the form πα =F(x, y, φx, φ1x…., fy,f1y, ….)where α is a function of x and y and π,f,φare unknown functions (Abel, Olav, and Ragni33).
In the second paper, Abel studied the first part of the integral equation πα =x=x=αds(α-x)n
Where π is a given function and s an unknown function of x and n<1.When n= 12,s is interpreted as the length of a curve found when a massive fall from a height α takes time equal to πα. Therefore, total duration of fall along a curve is proportional to the integral ofx=0x=αdsα-x
The equation Abel derived is probably the first case of an integral equation in the history of mathematics. Prior to his equation, Euler had introduced a general idea for solving a differential equation through a definite integral. Moreover, Abel develops (cosφ)ncosnφand xn(x+α)nin power series in respect to n and compares the coefficients of the powers of n. through this Abel gets some definite integrals. In posthumous paper, Abel gives integral formulae for finite sums that extend from 1 to α-1. Moreover, he studies their continuation to non integral values of Î&plusm...