# 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**

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Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 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 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 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 myÂ previous posts
**

**By this identity, I can write**

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 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 theseÂ sums. A****about 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:**

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Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

**Hence,**

**Step 2:**

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 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 theÂ Flajolet-Vardi theorem that you can find it’s proof in my second post:**

**Flajolet-Vardi Theorem:**

**If** **and** **converges thenâ€Ž,**

**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**

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# 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 Â Â

Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â

Â Â Â Â Â Â Â Â Â Â Â

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# 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 (1625â€“1686), in his 1650 book Novae quadraturae arithmeticae, seu de additione fractionum, posed the following question: Whatâ€™s the value of the sum
**

** Hereâ€™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 â€” the question itself still awaiting solution â€” 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 Â Â Â Â Â Â Â Â Â Â Â Â Â

**Now** **, and if** ** then** **and** **. **

**It follows that** **and so** **. Hence**

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 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** **.**