All posts by math
Solutions to the book Galois Theory by Ian Stewart
Chapter 2: The Fundamental Theorem of Algebra
Chapter 3: Factorisation of Polynomials
Chapter 6: The Degree of an Extension
Chapter 7: Ruler-and-Compass Constructions
Chapter 8: The Idea Behind Galois Theory
Chapter 9: Normality and Separability
Chapter 10: Counting Principles
Chapter 11: Field Automorphisms
Chapter 12: The Galois Correspondence
Chapter 14: Solubility and Simplicity
Chapter 15: Solution by Radicals
Chapter 16: Abstract Rings and Fields
Chapter 17: Abstract Field Extensions
Chapter 18: The General Polynomial Equation
Chapter 22: Calculating Galois Groups
Chapter 23: Algebraically Closed Fields
Solutions to the book Complex Variables by M. Ya. Antimirov, A. A. Kolyshkin and Remi Vaillancourt
Chapter 1: Functions of a Complex Variable
Continuity in the complex plane
Functions of a complex variable
Chapter 2: Elementary Conformal Mappings
Basic problems and principles of conformal mappings
Linear fractional transformations
Symmetry and linear fractional transformations
Mapping by z^n and w = z^(1/n)
Exponential and logarithmic mappings
Mapping by Joukowsky’s function
Mapping by trigonometric functions
Chapter 3: Complex Integration and Cauchy’s Theorem
Cauchy’s integral formula and applications
Chapter 4: Taylor and Laurent Series
Chapter 5: Singular Points and the Residue Theorem
Singular points of analytic functions
Chapter 6: Elementary Definite Integrals
Rational functions over (-Infinity,+Infinity)
Rational functions times sine or cosine
Rational functions times exponential functions
Rational functions times a power of x
Chapter 7: Intermediate Definite Integrals
Rational functions over (0, +Infinity)
Forms containing (ln x)^p in the numerator
Forms containing In g(x) or arctan g(x)
Forms containing In in the denominator
Forms containing P(n)(e^x)/Q(m)(e^x)
Chapter 8: Advanced Definite Integrals
Rational functions times trigonometric functions
Forms containing (x^2 – 2a sinx + a^2)-1
Forms containing (h sin a.x + x cos a.x)^-1
Forms containing Bessel functions
Chapter 9: Further Applications of the Theory of Residues
Counting zeros and poles of meromorphic functions
Simple-pole expansion of meromorphic functions
Infinite product expansion of entire functions
Chapter 10: Series Summation by Residues
Series with neither even nor odd terms
Solutions to the book Introduction to Analytic Number Theory by Tom M. Apostol
Chapter 1: The Fundamental Theorem of Arithmetic
Chapter 2: Arithmetical Functions and Dirichlet Multiplication
Chapter 3: Averages of Arithmetical Functions
Chapter 4: Some Elementary Theorems on the Distribution of Prime Numbers
Chapter 6: Finite Abelian Groups and Their Characters
Chapter 7: Dirichlet’s Theorem on Primes in Arithmetic Progressions
Chapter 8: Periodic Arithmetical Functions and Gauss Sums
Chapter 9: Quadratic Residues and the Quadratic Reciprocity Law
Chapter 11: Dirichlet Series and Euler Products
Chapter 12: The Functions $\zeta(s)$ and $L(s,\chi)$
One Finite Product
In this post, I wish to find the value of some finite product. I begin by defining a function as follows and trying to find it’s roots
Let , then we have
and similarly . So we have
The sine function vanishes at integer multiples of , so it follows that
where
for all integers
, that is, for
for all
. Thus,
for
where we recall that . Since
is strictly increasing on the interval
, it follows that
moreover, since tangent is an odd function, we have for each
. In particular we have found
distinct roots of
, so as a consequence of the fundamental theorem of algebra, we can write
                                      Â
                                      Â
So we have , and
More Elementary Proof for Euler’s Sine Expansion
Our goal in this post is to provide an elementary proof for the real version of Euler’s famous Sine expansion.
I first prove that for any that is a power of
and for any
,
by using the following more familiar identity
we can easily arrive to the following identity that is valid for equal to any power of
,
                  Â
Â
By considering what happens as in the recent formula, we can obtain that for
a power of
and then I replace in
and prove
.
Now I choose and break up
as follows
Since for ,
, we have
and if I choose and
large enough such that
, then
which implies that
Therefore from we have
Also we have
and once again by , we get
To summarize, I have shown that
Now taking in
I arrive to
After rearrangement and then taking absolute values, we get
but since we have
I can write
Finally since , this inequality implies our goal, indeed
One Finite Sum
In this post I want to find the value of the following finite sum,
I use the following amazing identity that you can see it’s proof in my previous posts
By this identity, I can write
                  Â
                                                         Â
                                                        Â
                                                        Â
and this follows that
Hence,
Euler Sum
Sums of the form where
, sometimes are called Euler sums. There are several ways to evaluate these sums. Aabout 240 years ago, Leonhard Euler(1707-1783) in 1775 proved that for
,
In the following you can see an elementary proof of this formula in the three steps:
Step 1:
                  Â
                  Â
                  Â
                  Â
Hence,
Step 2:
                            Â
                            Â
Now by recent formula that is also an alternate definition of the Harmonic numbers, we can write:
Step 3:
Another Result By the Flajolet-Vardi Theorem
By using the Flajolet-Vardi theorem we can find the value of the another amazing convergent series. Indeed, following series
I first recall the Flajolet-Vardi theorem that you can find it’s proof in my second post:
Flajolet-Vardi Theorem:
If and
converges then‎,
This theorem shows that , because if we let
, then
and by this theorem
Therefore now we must find the value of . we use the Taylor expansion of
and the fact that the sum
is
if
divides
and
otherwise.
There are some ‘s of complex numbers. Those numbers have always non-negative real part, for the Argument we take the angle between
and
, so that it fits with the power series for
.
but also
We thus have
We have to take the limit . The
term disappears, so we get
Examples:
For we have
and also for ,
A Beautiful Convergent Series II
In this post I want to generalize the given formula at my second post as follows
To prove this , I first prove following useful identity
PROOF: (Trigonometric Method) Note that
Repeatedly applying , we arrive at the following formula:
but
and now note that
Because,
By using this recent identity we can write
                                                                       Â
                                                                       Â
                                                                       Â
                                                                       Â
The Basel Problem, Double Integral Method II
One another way that can evaluate by double integral is to write
as follows
This is true, since we have
Now integrating by parts yields
and by using the monotone convergence theorem, I can write
Hence,
          Â
          Â
          Â
          Â
          Â
          Â
          Â
          Â
          Â
The Basel Problem, Double Integral Method I
Solutions to the book Introduction to Analytic Number Theory by Tom M. Apostol
Solutions to the book Complex Variables by M. Ya. Antimirov, A. A. Kolyshkin and Remi Vaillancourt
The Italian mathematician Pietro Mengoli (1625–1686), in his 1650 book Novae quadraturae arithmeticae, seu de additione fractionum, posed the following question: What’s the value of the sum
Here’s what Mengoli said:
Having concluded with satisfaction my consideration of those arrangements of fractions,
I shall move on to those other arrangements that have the unit as numerator, and square
numbers as denominators. The work devoted to this consideration has bore some fruit — the question itself still awaiting solution — but it [the work] requires the support of a richer
mind, in order to lead to the evaluation of the precise sum of the arrangement [of fractions]
that I have set myself as a task.Â
First time, Leonhard Euler (1707-1783) in 1735 proved that above series converges to .
In page 122 of the book “Topics in Number Theory”(1956), by William J. LeVeque there is an exercise for evaluating the following integral in two ways.
First way is to write the integrand as a geometric series,
and the second way by use of a suitable change of variables () which is also published by Tom M. Apostol in 1983 in Mathematical Intelligencer.
Note that
and by the monotone convergence theorem we get
We change variables in this by putting , so that
. Hence
where is the square with vertices
and
.
Exploiting the symmetry of the square we get
             Â
Now , and if
then
and
.
It follows that and so
. Hence
                       Â
Flajolet-Vardi Theorem and Calculating Infinite Sums
In this post I want to find the value of the sum
Note that for ,
.
I use following formula that is called Gregory-Leibniz-Madhava’s series
If we define function as follows:
then
Now I use this theorem
THEOREM (Flajolet–Vardi): If and
converges then,
.
PROOF:
Because ,
Hence, by Cauchy’s double series theorem, we can switch the order of summation:
This theorem implies that
and
Constructive Dilemma in the Elementary Set Theory
My first post is about two properties in the elementary set theory. You can see an amazing proof for them from me in the following.
In proof I used the following property in logic
that is called “constructive dilemma”.
First Property:
Let , then I can write
which implies that .
Second Property:
Let , then
which implies that .