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In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formula-long regarded as the gold standard for mathematical beauty-and shows why it still lies at the heart of complex number theory. In some ways a sequel to Nahin's An Imaginary Tale, this book examines the many applications of complex numbers alongside intriguing stories from the history of mathematics. Dr. Euler's Fabulous Formula is accessible to any reader familiar with calculus and differential equations, and promises to inspire mathematicians for years to come.

Numbers, Complex. --- Euler's numbers. --- Mathematics --- Math --- Science --- Complex numbers --- Imaginary quantities --- Quantities, Imaginary --- Algebra, Universal --- Quaternions --- Vector analysis --- Numbers, Euler's --- Numerical functions --- History. --- Euler, Leonhard,

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Serves as a historical research monograph on the biographical sketch and career of Leonhard Euler and his major contributions to numerous areas in the mathematical and physical sciences. This title contains fourteen chapters describing Euler's works on number theory, algebra, geometry, trigonometry, and differential and integral calculus.

Mathematics --- Mathematicians --- History --- Euler, Leonhard, --- Geschichte 1750-2010

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Continued fractions, studied since Ancient Greece, only became a powerful tool in the eighteenth century, in the hands of the great mathematician Euler. This book tells how Euler introduced the idea of orthogonal polynomials and combined the two subjects, and how Brouncker's formula of 1655 can be derived from Euler's efforts in Special Functions and Orthogonal Polynomials. The most interesting applications of this work are discussed, including the great Markoff's Theorem on the Lagrange spectrum, Abel's Theorem on integration in finite terms, Chebyshev's Theory of Orthogonal Polynomials, and very recent advances in Orthogonal Polynomials on the unit circle. As continued fractions become more important again, in part due to their use in finding algorithms in approximation theory, this timely book revives the approach of Wallis, Brouncker and Euler and illustrates the continuing significance of their influence. A translation of Euler's famous paper 'Continued Fractions, Observation' is included as an Addendum.

Mathematical analysis --- Orthogonal polynomials --- Continued fractions --- Euler, Leonhard, --- Orthogonal polynomials. --- Continued fractions. --- Fractions, Continued --- Series --- Processes, Infinite --- Fourier analysis --- Functions, Orthogonal --- Polynomials --- Euler, Leonhard, - 1707-1783

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Polynomials play a crucial role in many areas of mathematics including algebra, analysis, number theory, and probability theory. They also appear in physics, chemistry, and economics. Especially extensively studied are certain infinite families of polynomials. Here, we only mention some examples: Bernoulli, Euler, Gegenbauer, trigonometric, and orthogonal polynomials and their generalizations. There are several approaches to these classical mathematical objects. This Special Issue presents nine high quality research papers by leading researchers in this field. I hope the reading of this work will be useful for the new generation of mathematicians and for experienced researchers as well

Shivley’s matrix polynomials --- Generating matrix functions --- Matrix recurrence relations --- summation formula --- Operational representations --- Euler polynomials --- higher degree equations --- degenerate Euler numbers and polynomials --- degenerate q-Euler numbers and polynomials --- degenerate Carlitz-type (p, q)-Euler numbers and polynomials --- 2D q-Appell polynomials --- twice-iterated 2D q-Appell polynomials --- determinant expressions --- recurrence relations --- 2D q-Bernoulli polynomials --- 2D q-Euler polynomials --- 2D q-Genocchi polynomials --- Apostol type Bernoulli --- Euler and Genocchi polynomials --- Euler numbers and polynomials --- Carlitz-type degenerate (p,q)-Euler numbers and polynomials --- Carlitz-type higher-order degenerate (p,q)-Euler numbers and polynomials --- symmetric identities --- (p, q)-cosine Bernoulli polynomials --- (p, q)-sine Bernoulli polynomials --- (p, q)-numbers --- (p, q)-trigonometric functions --- Bernstein operators --- rate of approximation --- Voronovskaja type asymptotic formula --- q-cosine Euler polynomials --- q-sine Euler polynomials --- q-trigonometric function --- q-exponential function --- multiquadric --- radial basis function --- radial polynomials --- the shape parameter --- meshless --- Kansa method

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Written by a group of international experts in their field, this book is a review of Lagrangian observation, analysis and assimilation methods in physical and biological oceanography. This multidisciplinary text presents new results on nonlinear analysis of Lagrangian dynamics, the prediction of particle trajectories, and Lagrangian stochastic models. It includes historical information, up-to-date developments, and speculation on future developments in Lagrangian-based observations, analysis, and modeling of physical and biological systems. Containing contributions from experimentalists, theoreticians, and modellers in the fields of physical oceanography, marine biology, mathematics, and meteorology, this book will be of great interest to researchers and graduate students looking for both practical applications and information on the theory of transport and dispersion in physical systems, biological modelling, and data assimilation.

Ocean currents --- Lagrange equations. --- D'Alembert equation --- Equations, Euler-Lagrange --- Equations, Lagrange --- Euler-Lagrange equations --- Lagrangian equations --- Differential equations --- Equations of motion --- Currents, Oceanic --- Ocean circulation --- Water currents --- Ocean surface topography --- Mathematical models. --- Hydrodynamique --- Océanographie