Euler Basel Zeta Approximation Presentation Intro To Abstract Maths Presentation

1647 words - 7 pages

Euler Approximation for


k=1
1
k2
Niklov Rother
Johns Hopkins University
December 16, 2018
Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1
1
k2
December 16, 2018 1 / 25
What will be covered
1 History and Background
Biography Of Euler
Prerequisites
2 Proof
Proof 1
Proof 2
3 Conclusion
Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1
1
k2
December 16, 2018 2 / 25
Biography
Leonhard Euler, born April 15, 1707, Basel, Switzerland—died
September 18, 1783, St. Petersburg, Russia
Swiss mathematician and physicist, one of the founders of pure
mathematics.
He made decisive and formative contributions to the subjects of
geometry, calculus, mechanics, and number theory
He also developed methods for solving problems in observational
astronomy and demonstrated useful applications of mathematics in
technology and public affairs.
Much of the notation used by mathematicians today - including
e, i , f (x),∑, and the use of a, b and c as constants and x , y and z as
unknowns - was either created, popularized or standardized by Euler.
Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1
1
k2
December 16, 2018 3 / 25
Biography
Euler’s Formula eθi = cos(θ) + isin(θ)
He produced one of the most beautiful, mathematical equations,
e ipi = −1
1735, Euler solved an intransigent mathematical and logical problem,
known as the Seven Bridges of Ko¨nigsberg Problem
The demonstration of geometrical properties such as Euler’s Line and
Euler’s Circle;
A new method for solving quartic equations;
The Prime Number Theorem, which describes the asymptotic
distribution of the prime numbers;
Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1
1
k2
December 16, 2018 4 / 25
Weierstrass Factorization Theorem
Firstly, any finite sequence {cn} in the complex plane has an
associated polynomial p(z) that has zeroes precisely at the points of
that sequence,
p(z) = ∏
n
(z − cn).
Secondly, any polynomial function p(z) in the complex plane has a
factorization
p(z) = a∏
n
(z − cn)
where a is a non-zero constant and {cn} are the zeroes of p.
It is a necessary condition for convergence of the infinite product in
question is that for each z, the factors (z − cn) must approach 1 as
n→ ∞ .
∏bi=a f (i) This is simply the multiplication of all values from f (a) to
f (b)
Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1
1
k2
December 16, 2018 5 / 25
Riemann Zeta
The Riemann zeta function ζ(s) is one of the most significant
functions in mathematics because of its relationship to the
distribution of the prime numbers.
The zeta function is defined for any complex number s with real part
greater than 1 by the following formula:
ζ(s) =


n=1
1
ns
This is where the infinite summation we are dealing with, come from.
ζ(2) =


n=1
1
n2
Niklov Rother (Johns Hopkins University) Euler Approximation for ∑∞k=1
1
k2
December 16, 2018 6 / 25
Sine Function
f (x) = sin(x) is a very well known trigonometric function...

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