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Semester 4: B.Sc. Mathematics
Sequences and series of functions
Sequences and Series of Functions
Introduction to Sequences of Functions
A sequence of functions is a list of functions indexed by natural numbers. Formally, a sequence of functions f_n : A -> R is an assignment of a real-valued function to each natural number n where A is a subset of the real numbers.
Pointwise Convergence
A sequence of functions f_n converges pointwise to a function f on a set A if for every point x in A, the sequence f_n(x) converges to f(x). This means for each x in A and for every ε > 0, there exists an integer N such that for all n > N, |f_n(x) - f(x)| < ε.
Uniform Convergence
Uniform convergence is a stronger form of convergence where a sequence of functions f_n converges uniformly to f on A if for every ε > 0, there exists an integer N such that for all n > N and for all x in A, |f_n(x) - f(x)| < ε. This ensures the convergence does not depend on the choice of x.
Limit Functions
The limit of a sequence of functions is defined as the function f that the sequence converges to. If a sequence of functions converges to a function f pointwise, we denote this by lim_{n->∞} f_n(x) = f(x) for every x.
Cauchy Criterion for Uniform Convergence
A sequence of functions f_n converges uniformly on a set A if and only if for every ε > 0, there exists an integer N such that for all m, n > N and for all x in A, |f_n(x) - f_m(x)| < ε. This criterion allows us to verify uniform convergence.
Series of Functions
A series of functions is the sum of a sequence of functions. If f_n is a sequence of functions, the series is denoted by Σ f_n and converges pointwise to a function f if the sequence of partial sums S_n = f_1 + f_2 + ... + f_n converges pointwise.
Uniform Convergence of Series
A series of functions Σ f_n converges uniformly to a function f on a set A if the sequence of partial sums S_n converges uniformly to f. This implies that we can interchange the limit and the sum under certain conditions.
Examples and Applications
Common examples include power series and Fourier series. Understanding the convergence behavior of these series is crucial in mathematical analysis, particularly in the context of approximating functions.
Continuity and differentiability
Continuity and Differentiability
Definition of Continuity
A function f(x) is said to be continuous at a point c if the following three conditions are satisfied: 1. f(c) is defined. 2. lim (x -> c) f(x) exists. 3. lim (x -> c) f(x) = f(c). A function is continuous on an interval if it is continuous at every point in that interval.
Types of Discontinuity
Discontinuities can be classified into three types: 1. Jump Discontinuity - where the left-hand and right-hand limits exist but are not equal. 2. Infinite Discontinuity - where the function approaches infinity at the point. 3. Removable Discontinuity - where the limit exists but does not equal the function's value at that point.
Differentiability
A function f(x) is differentiable at a point c if the derivative f'(c) exists. This means the limit lim (h -> 0) [f(c+h) - f(c)] / h must exist. A function must be continuous at a point to be differentiable there, but continuity does not imply differentiability.
Relation Between Continuity and Differentiability
While every differentiable function is continuous, the converse is not necessarily true. A function may be continuous at a point but not differentiable due to sharp corners or cusps in the graph, e.g., f(x) = |x| is continuous everywhere but not differentiable at x = 0.
Applications of Continuity and Differentiability
Continuity and differentiability are fundamental concepts in calculus and analysis, used for determining the behavior of functions, optimizing problems, and in various fields like physics and engineering. They are also crucial in the formulation of theorems such as the Mean Value Theorem.
Riemann integration
Riemann Integration
Introduction to Riemann Integration
Riemann integration is a method of defining the integral of a function based on the limit of Riemann sums. It provides a way to calculate the area under curves represented by functions.
Riemann Sums
Riemann sums approximate the area under a curve by dividing the interval into subintervals, calculating the area of rectangles formed on these intervals, and summing these areas. There are different types of Riemann sums: left sum, right sum, and midpoint sum.
Definition of Riemann Integrable Functions
A function is Riemann integrable on an interval if the limit of the Riemann sums exists as the width of the subintervals approaches zero. All continuous functions on a closed interval are Riemann integrable.
Properties of Riemann Integrals
The Riemann integral has several properties that make it useful: linearity, additivity over intervals, and behavior with respect to limits of functions.
Fundamental Theorem of Calculus
This theorem connects differentiation and integration, stating that if a function is Riemann integrable, then the integral of its derivative can be computed by evaluating the function at the boundaries of the interval.
Examples of Riemann Integration
Common examples include integrating polynomial functions, trigonometric functions, and piecewise functions. These examples illustrate the application of Riemann sums and the computation of integrals.
Limitations of Riemann Integration
Not all functions are Riemann integrable. Functions with too many discontinuities or those that do not satisfy the criteria for integrability cannot be integrated by the Riemann method.
Uniform convergence
Uniform convergence
Definition of Uniform Convergence
Uniform convergence refers to a type of convergence of a sequence of functions in which the rate of convergence is uniform across the entire domain. Specifically, a sequence of functions fn converges uniformly to a function f on a set S if for every ε greater than 0, there exists an integer N such that for all n greater than or equal to N and for all x in S, the inequality |fn(x) - f(x)| < ε holds.
Comparison with Pointwise Convergence
Pointwise convergence means that for each x in the domain, the sequence fn(x) converges to f(x) individually, but the rate of convergence can vary with x. In contrast, uniform convergence requires that the convergence be uniform across all x in the domain.
Examples of Uniform Convergence
Consider the sequence of functions fn(x) = x/n. This sequence converges uniformly to the function f(x) = 0 on the interval [0, 1] because for any ε > 0, we can choose N = ceil(1/ε), ensuring that |fn(x) - f(x)| = |x/n| < ε for all x in [0, 1] when n >= N.
Importance of Uniform Convergence
Uniform convergence is significant because it preserves properties of functions, such as continuity and integrability. If fn converges uniformly to f and each fn is continuous, then f is also continuous. Uniform convergence also allows interchange of limits and integrals or derivatives under certain conditions.
Theorems related to Uniform Convergence
Key theorems involving uniform convergence include the Weierstrass M-test for series of functions and the Arzelà-Ascoli theorem, which provides conditions under which a sequence of functions has a uniformly convergent subsequence.
Applications of Uniform Convergence
Uniform convergence is applied in various fields such as approximation theory, numerical analysis, and in establishing the validity of different calculus operations in functional analysis.
Metric spaces basics
Metric spaces basics
Definition of Metric Space
A metric space is a set equipped with a function called a metric that defines the distance between any two points in the set. Formally, a metric space is a pair (X, d), where X is a non-empty set and d: X x X -> R is a function satisfying three properties: positivity, symmetry, and the triangle inequality.
Properties of Metrics
1. Positivity: d(x, y) >= 0 for all x, y in X and d(x, y) = 0 if and only if x = y. 2. Symmetry: d(x, y) = d(y, x) for all x, y in X. 3. Triangle Inequality: d(x, z) <= d(x, y) + d(y, z) for all x, y, z in X.
Examples of Metric Spaces
Common examples include: 1. The Euclidean space R^n with the standard distance metric d(x, y) = ||x - y||. 2. Discrete metric space where d(x, y) = 1 if x != y, and d(x, y) = 0 if x = y. 3. The space of continuous functions on a closed interval with the metric defined by the maximum absolute difference.
Open and Closed Sets
In a metric space, a set is open if for every point in the set, there exists a radius such that the entire ball around that point is contained within the set. A set is closed if it contains all its limit points.
Convergence and Completeness
A sequence in a metric space is said to converge if the distance between the elements of the sequence and the limit approaches zero as the sequence progresses. A metric space is complete if every Cauchy sequence converges to a limit that is within the space.
Compactness
A metric space is compact if every open cover has a finite subcover. In Euclidean spaces, compactness is equivalent to being closed and bounded according to the Heine-Borel theorem.
Applications of Metric Spaces
Metric spaces are foundational in analysis and topology, facilitating the study of concepts such as continuity, limits, and compactness in a generalized setting.
