Real Analysis

Real analysis is a branch of mathematical analysis that analyses the behaviour of real numbers, sequences and series, and real functions. Convergence, limits, continuity, smoothness, differentiability, and integrability are some of the features of real-valued sequences and functions that real analysis explores. Complex analysis, on the other hand, is concerned with the study of complex numbers and their functions. In this article, let us discuss the brief introduction about the real analysis and the concepts involved with a complete explanation.

Introduction to Real Analysis

As discussed above, real analysis is a branch of mathematics that was created to define the study of numbers and functions, as well as to analyze key concepts like limits and continuity. Calculus and its applications are based on these ideas. In a wide range of applications, real analysis has become a vital tool. Now, let us have a brief look at some of the important concepts covered under real analysis.

Real Number System

The real number system (usually referred to as the reals) is first and foremost a set of numbers {a, b, c,…} on which the operations of addition and multiplication are defined such that every pair of real numbers does have a unique sum and product, with the properties listed below.

• Commutative Law: a+b = b + a and ab = ba
• Associative Law: (a + b) + c = a + (b + c) and (ab)c = a(bc)
• Distributive Law: a (b + c) = ab + ac
• For all a, there are unique real numbers 0 and 1, such that a+0 = a and a1=a.
• There is a real number -a for each a such that a + (-a) = 0, and if a ≠ 0 there is a real number 1/a for each a such that a(1/a) = 1.

Sequence

Sequence: A sequence is defined as a function whose domain is the collection of positive integers.

(i.e) a_n = a(n), where n = 1, 2, 3, ….

Assume that P_n is the nth prime number, then

Convergence of Sequence: The sequence

Series

Infinite Series: An infinite series is defined as a pair

Here, a_n is the nth term of the series and S_n is the nth term of the partial sum of the series.

Convergence of Series: The converge of series states that, if

Also, read: Sequence and Series

Maxima and Minima

Maximum Value: The continuous function f(x) is considered to have a maximum value for x = a, if f(a) should be greater than any other values of f(x) that lies in the small neighbourhood of x = a.

Minimum Value: The continuous function f(x) is considered to have a minimum value at x = a, if f(a) should be smaller than any other values of f(x) that lie in the small neighbourhood of x = a.

Note: The tangent at maximum point or the minimum point of a curve should be parallel to x-axis.

Metric Space

A metric space <x, P> is defined as a non-empty set X of points (elements) and P: X×X→R, such that x, y, z belongs to X.

• P(x, y) ≥ 0
• P(x, y) = 0, if and only if x = y
• P(x, y) = P(y, x)
• P(x, y) ≤ P(x, y) + P(x, y)

Here, the function “P” is called a metric.

Solved Problems on Real Analysis

Example 1:

Determine the nth term of the sequence {0, 1, 0, 1, …}

Solution:

Given sequence: {0, 1, 0, 1, …}

From the given sequence, we can write as follow:

The 1st term of sequence = 0 = a₁ = (1-1)/2 = 0

The 2nd term of sequence = 1 = a₂ = [1+(-1)²]/2 = 1

The 3rd term of sequence = 0 = a₃ = [1+(-1)³]/2 = 0

The 4th term of sequence = 1 = a₄ = [1+(-1)⁴]/2 = 1

Hence, the nth term of sequence = a_n = [1+(-1)ⁿ]/2.

Example 2:

Evaluate using Green’s theorem, ∮_c (y- sinx dx + cos x dy), where c represents the triangle which is enclosed by the lines x = 0, x = π/2, xy = 2x and P = y-sinx and Q = cos x.

Solution:

From the given conditions, we can write

∂P/∂y = 1

∂Q/∂x = – sin x

By using the Green’s theorem, we can write

∮_c (y- sinx dx + cosx dy) = ∫∫_S (-1 – sin x) dx dy

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