Category: Study Notes

Integral EquationsMathStudy Notes

Solving Integral Equations (2) – Square Integrable Functions, Norms, Trial Method

Square Integrable function or quadratically integrable function $\mathfrak{L}_2$ function

A function $y(x)$ is said to be square integrable or $\mathfrak{L}_2$ function on the interval $(a,b)$ if $$\displaystyle {\int_a^b} {|y(x)|}^2 dx <\infty$$ or $$\displaystyle {\int_a^b} y(x) \bar{y}(x) dx <\infty$$.

For further reading, I suggest this Wikipedia page.

$y(x)$ is then also called ‘regular function’.

The kernel $K(x,t)$ , a function of two variables is an $\mathfrak{L_2}$ – function if atleast one of the following is true:

  • $\int_{x=a}^b \int_{t=a}^b |K(x,t)|^2 dx dt <\infty$
  • $\int_{t=a}^b |K(x,t)|^2 dx <\infty$
  • $\int_{x=a}^b |K(x,t)|^2 dt <\infty$

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integral equation
Integral EquationsMathStudy Notes

Solving Integral Equations – (1) Definitions and Types

If you have finished your course in Calculus and Differential Equations, you should head to your next milestone: the Integral Equations. This marathon series (planned to be of 6 or 8 parts) is dedicated to interactive learning of integral equations for the beginners —starting with just definitions and demos —and the pros— taking it to the heights of problem solving. Comments and feedback are invited.

Also read:

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What is an Integral Equation?

An integral equation is an equation in which an unknown function appears under one or more integration signs. Any integral calculus statement like — $ y= \int_a^b \phi(x) dx$can be considered as an integral equation. If you noticed I have used two types of integration limits in above integral equations –their significance will be discussed later in the article.
A general type of integral equation, $ g(x) y(x) = f(x) + \lambda \int_a^\Box K(x, t) y(t) dt$ is called linear integral equation as only linear operations are performed in the equation. The one, which is not linear, is obviously called ‘Non-linear integral equation’. In this article, when you read ‘integral equation’ understand it as ‘linear integral equation’.

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Examination Strategies : Tactics & Tips

Study for Exams

Every student or graduate knows how hard the first experience of passing exams is. Preliminary preparation starves the nervous system and the physical condition of the human body, however, the exam itself is always a stressful situation, which requires a candidate a great manifestation of mental and physical abilities.

Therefore, just the knowledge of a subject is not enough for the exam. The examinee needs to have high self-organization, ability to keep his emotions and absolute confidence in his abilities.

Most people, who successfully passed a few dozen of examinations, advise to adhere to the following tactics:

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MathNumber TheoryResearchStudy Notes

Complete Elementary Analysis of Nested radicals

This is a continuation of the series of summer projects sponsored by department of science and technology, government of India. In this project work, I have worked to collect and expand what Ramanujan did with Nested Radicals and summarized all important facts into the one article. In the article, there are formulas, formulas and only formulas — I think this is exactly what Ramanujan is known for.

This article not only deals with Ramanujan’s initial work on Nested Radicals but also develops few new models and adds more information to it by catching latest research in a very elementary way. The project was initiated first at, about two years ago and done rigorously in recent days.

Download a PDF copy of the article (here) and let me know what you think about it.


Algebra and TopologyMathReal AnalysisScienceStudy Notes

Real Sequences

Sequence of real numbers

A sequence of real numbers (or a real sequence) is defined as a function $ f: \mathbb{N} \to \mathbb{R}$ , where $ \mathbb{N}$ is the set of natural numbers and $ \mathbb{R}$ is the set of real numbers. Thus, $ f(n)=r_n, \ n \in \mathbb{N}, \ r_n \in \mathbb{R}$ is a function which produces a sequence of real numbers $ r_n$ . It’s customary to write a sequence as form of functions in brackets, e.g.; $ \langle f(n) \rangle$ , $ \{ f(n) \}$ . We can alternatively represent a sequence as the function with natural numbers as subscripts, e.g., $ \langle f_n \rangle$ , $ \{ f_n \}$ . This alternate method is a better representation of a sequence as it distinguishes ‘a sequence’ from ‘a function’. We shall use $ \langle f_n \rangle$ notation and when writen $ \langle f_n \rangle$ , we mean $ \langle f_1, f_2, f_3, \ldots, f_n, \ldots \rangle$ a sequence with infinitely many terms. Since all of $ \{ f_1, f_2, f_3, \ldots, f_n, \ldots \}$ are real numbers, this kind of sequence is called a sequence of real numbers.

Examples of Sequences

  1. Like $ f(x)=\dfrac{1}{x} \forall x \in \mathbb{R}$ is a real-valued-function, $ f(n)=\dfrac{1}{n} \forall n \in \mathbb{N}$ is a real sequence.

Putting consecutive values of $ n \in \mathbb{N}$ in $ f(n)=\dfrac{1}{n}$ we obtain a real-sequence

n=1 f(1)=1

n=2 f(2)=1/2

n=3 f(3)=1/3

n=n f(n)=1/n

This real-sequence can be represented by

$ \langle \dfrac{1}{n} \rangle := \langle 1, \dfrac{1}{2}, \dfrac{1}{3}, \ldots, \dfrac{1}{n}, \ldots$ .

  1. $ \langle {(-1)}^n \rangle$ is the sequence $ \langle -1, 1, -1, 1, \ldots, {(-1)}^n, \ldots \rangle$ .
  2. $ \langle -3n \rangle$ is the sequence $ \langle -3, -6, -9, \ldots, -3n, \ldots \rangle$
  3. A sequence can also be formed by a recurrence relation with boundary values. If $ f_n= f_{n-1}+f_{n-2} \ \text{for} n \ge 2$ and $ f_0=f_1=1$ , then we obtain the sequence $ \langle f_n \rangle$ as
    n=1 $ f_1=1$ (given)
    n=2 $ f_2=f_1 +f_0=1+1=2$ (given $ f_0=1=f_1$ )
    n=3 $ f_3=f_2+f_1=2+1=3$
    n=4 $ f_4=f_3+f_2=3+2=5$
    and so on…
    This sequence, $ \langle 1, 1,2, 3, 5, 8, 13, 21, \ldots \rangle$ is a real-sequence known as Fibonacci Sequence.

Range Set of a Sequence

The set of all ‘distinct’ elements of a sequence is called the range set of the given sequence.

For example:

  • The range set of $ \langle \dfrac{1}{n}\rangle:= \{ \dfrac{1}{n} : n \in \mathbb{N} \}$ , which is an infinite set.
  • The range set of $ \langle {(-1)}^n \rangle := \{ -1, 1 \}$ , a finite set.

Remark: The range set of a sequence may be either infinite or finite, but a sequence has always an infinite number of elements.

Sub-sequence of the Sequence

A sub-sequence of the sequence is another sequence containing some of the values of the sequence in the same order as in the original sequence. Alternatively, a sub-sequence of the sequence is another sequence which range set is a subset of the range set of the sequence.

For example:

  • <1, 3, 5, 7, …> is a sub-sequence of the sequence <1, 2, 3, 4, …>.
  • <1, 5, 13, 21, …> is a sub-sequence of the sequence <1,1,2,3,5,8,13,21, 34, …>.
  • <1,1,1,1,1,…> is a sub-sequence of the sequence <-1, 1, -1, 1, …>. Since, the sequence <1,1,1,1,…> has only one value for each term, it’s called a constant sequence.

Remark: A sub-sequence is also a sequence hence it satisfy and follow all the properties of a sequence.

Equality of two sequences

Two sequences $ \langle S_n \rangle$ and $ \langle T_n \rangle$ are said to be equal, if and only if $ S_n=T_n, \forall n \in \mathbb{N}$ .

For example: The sequences $ \langle \dfrac{n+1}{n} \rangle$ and $ \langle 1+\dfrac{1}{n} \rangle$ are equal to each other.

Remark: From the definition the sequences <-1,1,-1,1, …> and <1,-1,1,-1,…> are not equal to each other, though they look alike and has same range set.

Algebra of Sequences

Let $ \langle S_n \rangle$ and $ \langle T_n \rangle$ be two sequence, then the sequences having n-th terms $ S_n+T_n, \ S_n-T_n, \ S_n \cdot T_n, \ \dfrac{S_n}{T_n}$ (respectively) are called the SUM, DIFFERENCE, PRODUCT, QUOTIENT of $ \langle S_n \rangle$ and $ \langle T_n \rangle$ .

For example: The sequence <1, 8, 19,30, …> is the sum of sequences <0, 1, 2, 3, …> and <1, 7, 17, 27, …> obtained after adding n-th term of one sequence to corresponding n-th term of other sequence. Similarly, other operations can be carried.

If $ S_n \ne 0 \forall n$ , then the sequence $ \langle \dfrac{1}{S_n} \rangle$ is known as the reciprocal of the sequence $ \langle S_n \rangle$ .

For example: $ \langle \dfrac{1}{1}, \dfrac{-1}{2}, \dfrac{1}{3}, \ldots \rangle$ is the reciprocal of the sequence $ \langle 1, -2, 3, \ldots \rangle$ .

Remark: The sequences <-1,1,-1,1, …> and <1,-1,1,-1,…> have their reciprocals equal to the original sequence, hence these are called identity-sequences.

If $ c \in \mathbb{R}$ then the sequence with n-th term $ cS_n$ is called the scalar multiple of sequence $ \langle S_n \rangle$ . This sequence is denoted by $ \langle cS_n \rangle$ .

Bounds of a Sequence

  • A sequence $ \langle S_n \rangle$ is said to be bounded above, if there exists a real number M such that $ S_n \le M, \forall n \in \mathbb{N}$ . M is called an upper bound of the sequence $ \langle S_n \rangle$ .
  • A sequence $ \langle S_n \rangle$ is said to be bounded below, if there exists a real number m such that $ S_n \ge m, \forall n \in \mathbb{N}$ . m is called a lower bound of the sequence $ \langle S_n \rangle$ .
  • A sequence $ \langle S_n \rangle$ is said to be bounded, if it is both bounded above and bounded below. Thus, if $ \langle S_n \rangle$ is a bounded sequence, there exist two real numbers m & M such that $ m \le S_n \le M \forall n \in \mathbb{N}$ .
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