Analysis
The development of calculus was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Understanding the infinite was naturally the major focus of Euler's research. While some of Euler's proofs may not have been acceptable under modern standards of rigor, his ideas were responsible for many great advances. First of all, Euler introduced the concept of a function, and introduced the use of the exponential function and logarithms in analytic proofs
Euler frequently used the logarithm function as a tool in analysis problems, and discovered new ways by which they could be used. He discovered ways to express various logarithmic functions in terms of power series, and successfully defined logarithms for complex and negative numbers, thus greatly expanding the scope where logarithms could be applied in mathematics. Most researchers in the field long held the view that for any positive real since by using the additivity property of logarithms . In a 1747 letter to Jean Le Rond d'Alembert, Euler defined the natural logarithm of -1 as a pure imaginary.
Euler is well known in analysis for his frequent use and development of power series: that is, the expression of functions as sums of infinitely many terms, such as
Notably, Euler discovered the power series expansions for e and the inverse tangent function
- .
His daring (and, by modern standards, technically incorrect) use of power series enabled him to solve the famous Basel problem in 1735:
In addition, Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of complex analysis. Euler invented the calculus of variations including its most well-known result, the Euler-Lagrange equation.
Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem.
Read more about this topic: Contributions Of Leonhard Euler To Mathematics
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