Euler and Infinite Series

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Euler and Infinite Series Tom Roby
(UConn) Based on talks and books of William.
Dunham (Muhlenberg College)
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Leonhard Euler 1707 - 1783
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Timeline
1707 born in Basel 1720 studies with Johann
Bernoulli 1722 graduates from U. Basel 1727
to St. Petersburg Academy 1741 to Berlin
Academy 1766 back to St. Petersburg 1783
dies
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Eulers Mathematics
Quantity Diversity Significance
The study of Eulers works will remain the
best school for the various fields of
mathematics, and nothing will replace it.
Gauss
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Great Textbooks
Introductio in analysin infinitorum (1748)
Differential calculus (1755)
Integral calculus (1768)
Algebra (1770)
Calculus of variations (1744)
Mechanics (1736)
Optics (1769)
Letters to a German Princess (1768)
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No mathematician ever attained such a
position of leadership in all branches of
mathematics, pure and applied, as Euler did for
the best part of the 18th century.
André Weil
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Married to Katharina Gsell 13 children
Phenomenal memory
In the 1730s, he lost vision in one eye
By 1771, he was essentially blind
In 1775, he produced a paper a week!
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Selected Results
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A special case of Eulers Identity
V F E 2
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• The Basel Problem (Jakob Bernoulli, 1689)

Find the exact sum of the infinite series
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Number Theory
Def Whole numbers M and N are amicable if
each is the sum of the proper divisors of the
other.
Ex M 220 and N 284
Proper divisors of 220 1 2 4 5 10 11
20 22 44 55 110
284
Proper divisors of 284 1 2 4 71 142
220
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Brief History of Amicable Numbers
ca. 300 BCE Greeks knew 220 and 284
9th C Thabit ibn Qurras rule
1636 Fermat found 17,286 and 18,416
1638 Descartes found 9,363,584 and
9,437,056
In a 1750 paper, Euler found 58 more!
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And on it goes
Euler line of a triangle
Eulers constant, ? 0.577216
Eulers necessary condition in the calculus
of variations
36-Officer Problem
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Bridges of Königsberg
this solution bears little relationship to
mathematics, and I do not understand why to
expect a mathematician to produce it, rather
than anyone else, for the solution is based on
logic alone.
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Theory of Machines
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Gamma function
Partitions of whole numbers
Eulers product-sum formula
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Curiosities
Find four whole numbers, the sum of any two of
which is a perfect square
1 ,
3 ,
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Oops!
Euler found
18530 ,
38114 ,
45986 ,
65570
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Approximate p using the arctan series
For x 1, this yields
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Euler found p accurate to twenty places using
the arctan series and the fact that
All this calculation consumed about an hour
of work.
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Factoring polynomials into first and second
degree factors
Nicholas Bernoulli asserted that there was
no such factorization of the 4th degree polynomial
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Talent is doing easily what others find difficult
Genius is doing easily what others find impossible
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Made the mathematician Euler a hero. From the
real to complex, With our brains in great
flex He led us with zest but no fearo.
W. C. Willig
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Euler lacked only one thing to make him a
perfect genius he failed to be
incomprehensible. Frobenius
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All celebrated mathematicians now alive are
his disciples there is no one who is not
guided and sustained by the genius of
Euler. Condorcet
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Happy Birthday, Uncle Leonhard!
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Some references
1. Euler The Master of Us All, W. Dunham, MAA,
1999. 2. The Genius of Euler Reflections on His
Life and Work, W. Dunham (ed.), MAA, 2007. 3.
The Early Mathematics of Leonhard Euler, Ed
Sandifer, MAA, 2007. 4. Eulers Opera Omnia in 74
volumes! 5. The Euler Archive at Dartmouth
www.math.dartmouth.edu/euler/ 6. MacTutor Math
History Website www-gap.dcs.st-and.ac.uk/histor
y
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Integral Calculus
Evaluate
exactly
As a series,
Integrate termwise
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and use the fact that, for n even,
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(thanks to Leibniz)
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Complex Variables
Let
and integrate
Let z i y
dz i dy
Then
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Now let y sin x
dy cos x dx

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i
And thus (the most beautiful formula
in mathematics!) for