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The early mathematics of Leonhard Euler

Verfasser/in: Charles Edward Sandifer
Verlag: Washington, D.C. : Mathematical Association of America, ©2007.
Serien: MAA tercentenary Euler celebration, v. 1.; MAA spectrum.
Ausgabe/Format   Buch : EnglischAlle Ausgaben und Formate anzeigen
Datenbank:WorldCat
Zusammenfassung:
"The Early Mathematics of Leonhard Euler describes Euler's early mathematical works: the 50 mathematical articles he wrote before he left St. Petersburg in 1741 to join the Academy of Frederick the Great in Berlin. These works contain some of Euler's greatest mathematics: the Konigsburg bridge problem, his solution to the Basel problem, his first proof of the Euler-Fermat theorem. It also presents important results  Weiterlesen…
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Name: Leonhard Euler; Leonhard Euler; Leonhard Euler; Leonhard Euler
Medientyp: Internetquelle
Dokumenttyp: Buch, Internet-Ressource
Alle Autoren: Charles Edward Sandifer
ISBN: 0883855593 9780883855591
OCLC-Nummer: 82370957
Beschreibung: xix, 391 p. : port., facsims. ; 26 cm.
Inhalt: Preface --
Interlude: 1725-1727. Construction of isochronal curves in any kind of resistant ; Method of finding reciprocal algebraic trajectories --
Interlude 1728. Solution to problems of reciprocal trajectories ; A new method of reducing innumerable differential equations of the second degree to equations of the first degree --
Interlude 1729-1731. On transcendental progressions, or those for which the general term cannot be given algebraically ; On the shortest curve on a surface that joins any two given points ; On the summation of innumerably many progressions --
Interlude 1732. General methods for summing progressions ; Observations on theorems that Fermat and others have looked at about prime numbers ; An account of the solution of isoperimetric problems in the broadest sense --
Interlude 1733. Construction of differential equations which do not admit separation of variables ; Example of the solution of a differential equation without separation of variables ; On the solution of problems of Diophantus about integer numbers ; Inferences on the forms of roots of equations and of their orders ; Solution of the differential equation axn dx = dy + y²dx --
Interlude 1734. On curves of fastest descent in a resistant medium ; Observations on harmonic progressions ; On an infinity of curves of a given kind, or a method of finding equations for an infinity of curves of a given kind ; Additions to the dissertation on infinitely many curves of a given kind ; Investigation of two curves, the abscissas of which are corresponding arcs and the sum of which is algebraic --
Interlude 1735. On sums of series of reciprocals ; A universal method for finding sums which approximate convergent series ; Finding the sum of a series from a given general term ; On the solution of equations from the motion of pulling and other equations pertaining to the method of inverse tangents ; Solution of a problem requiring the rectification of an ellipse ; Solution of a problem relating to the geometry of position --
Interlude 1736. Proof of some theorems about looking at prime numbers ; Further universal methods for summing series ; A new and easy way of finding curves enjoying properties of maximum or minimum --
Interlude 1737. On the solution of equations ; An essay on continued fractions ; Various observations about infinite series ; Solution to a geometric problem about lunes formed by circles --
Interlude 1738. On rectifiable algebraic curves and algebraic reciprocal trajectories ; On various ways of closely approximating numbers for the quadrature of the circle ; On differential equations which sometimes can be integrated ; Proofs of some theorems of arithmetic ; Solution of some problems that were posed by the celebrated Daniel Bernoulli --
Interlude 1739. On products arising from infinitely many factors ; Observations on continued fractions ; Consideration of some progressions appropriate for finding the quadrature of the circle ; An easy method for computing sines and tangents of angles both natural and artificial ; Investigation of curves which produce evolutes that are similar to themselves ; Considerations about certain series --
Interlude 1740. Solution of problems in arithmetic of finding a number, which, when divided by given numbers leaves given remainders ; On the extraction of roots of irrational quantities --
Interlude 1741. Proof of the sum of this series 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/ 36 + etc ; Several analytic observations on combinations ; On the utility of higher mathematics.
Serientitel: MAA tercentenary Euler celebration, v. 1.; MAA spectrum.
Verfasserangabe: by C. Edward Sandifer.
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Abstract:

A portrait of Euler's early mathematics between 1725 and 1741, rich in technical detail.  Weiterlesen…

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schema:description"Preface -- Interlude: 1725-1727. Construction of isochronal curves in any kind of resistant ; Method of finding reciprocal algebraic trajectories -- Interlude 1728. Solution to problems of reciprocal trajectories ; A new method of reducing innumerable differential equations of the second degree to equations of the first degree -- Interlude 1729-1731. On transcendental progressions, or those for which the general term cannot be given algebraically ; On the shortest curve on a surface that joins any two given points ; On the summation of innumerably many progressions -- Interlude 1732. General methods for summing progressions ; Observations on theorems that Fermat and others have looked at about prime numbers ; An account of the solution of isoperimetric problems in the broadest sense -- Interlude 1733. Construction of differential equations which do not admit separation of variables ; Example of the solution of a differential equation without separation of variables ; On the solution of problems of Diophantus about integer numbers ; Inferences on the forms of roots of equations and of their orders ; Solution of the differential equation axn dx = dy + y²dx -- Interlude 1734. On curves of fastest descent in a resistant medium ; Observations on harmonic progressions ; On an infinity of curves of a given kind, or a method of finding equations for an infinity of curves of a given kind ; Additions to the dissertation on infinitely many curves of a given kind ; Investigation of two curves, the abscissas of which are corresponding arcs and the sum of which is algebraic -- Interlude 1735. On sums of series of reciprocals ; A universal method for finding sums which approximate convergent series ; Finding the sum of a series from a given general term ; On the solution of equations from the motion of pulling and other equations pertaining to the method of inverse tangents ; Solution of a problem requiring the rectification of an ellipse ; Solution of a problem relating to the geometry of position -- Interlude 1736. Proof of some theorems about looking at prime numbers ; Further universal methods for summing series ; A new and easy way of finding curves enjoying properties of maximum or minimum -- Interlude 1737. On the solution of equations ; An essay on continued fractions ; Various observations about infinite series ; Solution to a geometric problem about lunes formed by circles -- Interlude 1738. On rectifiable algebraic curves and algebraic reciprocal trajectories ; On various ways of closely approximating numbers for the quadrature of the circle ; On differential equations which sometimes can be integrated ; Proofs of some theorems of arithmetic ; Solution of some problems that were posed by the celebrated Daniel Bernoulli -- Interlude 1739. On products arising from infinitely many factors ; Observations on continued fractions ; Consideration of some progressions appropriate for finding the quadrature of the circle ; An easy method for computing sines and tangents of angles both natural and artificial ; Investigation of curves which produce evolutes that are similar to themselves ; Considerations about certain series -- Interlude 1740. Solution of problems in arithmetic of finding a number, which, when divided by given numbers leaves given remainders ; On the extraction of roots of irrational quantities -- Interlude 1741. Proof of the sum of this series 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/ 36 + etc ; Several analytic observations on combinations ; On the utility of higher mathematics."
schema:description""The Early Mathematics of Leonhard Euler describes Euler's early mathematical works: the 50 mathematical articles he wrote before he left St. Petersburg in 1741 to join the Academy of Frederick the Great in Berlin. These works contain some of Euler's greatest mathematics: the Konigsburg bridge problem, his solution to the Basel problem, his first proof of the Euler-Fermat theorem. It also presents important results that we seldom realize are due to Euler: that mixed partial derivatives are equal, our f(x) notation, and the integrating factor in differential equations. The book provides some of the way mathematics is actually done. For example, Euler found partial results towards the Euler-Fermat theorem well before he discovered a proof of the Fermat theorem itself, and the Euler-Fermat version came 30 years later, beyond the scope of this book. The book shows how results in diverse fields are related, how number theory relates to series, which, in turn relate to elliptic integrals and then to differential equations, There are dozens of such strands in this beautiful web of mathematics. At the same time, we see Euler grow in power and sophistication, from his first work on differential equations as an 18-year old student, a paper with a serious flaw in it, to the most celebrated mathematician and scientist of his times, when, at the age of 34, he was lured away like a superstar athlete might be traded today. The book is a portrait of the world's most exciting mathematics between 1725 and 1741, rich in technical detail. Woven with connections within Euler's work and with the work of other mathematicians in other times and places, laced with historical context."--Publisher's website."
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