Mathematical Analysis

Mathematical analysis begins with a question that seems trivial: what is a number? We are used to working with numbers since childhood, but a precise definition of a real number required three centuries of effort from the best mathematicians—from Newton and Leibniz to Cantor and Dedekind.

Real numbers form the number line—a continuous continuum, in which between any two numbers, a third can always be found. This property, called density, distinguishes real numbers from rational ones. Rational numbers are also "infinitely many," but they do not fill the number line completely—there...

Georg Cantor, in the 1870s, created set theory—the language on which all modern mathematics is written. A set is any collection of definite and well-distinguished objects. Cantor made a striking discovery: there exist infinities of different "sizes".

There are infinitely many natural numbers {1, 2, 3, ...}—this is countable infinity. Surprisingly, there are as many rational numbers as natural: a one-to-one correspondence can be constructed between them (Cantor’s diagonal argument). However, there are more real numbers—they are uncountably man...

Formulas

The concept of the limit is the heart of mathematical analysis. Intuitively, we understand that the sequence 1/2, 1/4, 1/8, 1/16, ... “tends to zero”—its terms become arbitrarily small. But what does “arbitrarily” mean? How can we make this statement mathematically rigorous?

The answer was given by Augustin-Louis Cauchy in the first half of the 19th century, and Karl Weierstrass gave it its final form: the limit of a sequence via ε-N.

A number $a$ is called the limit of the sequence $\{a_n\}$ if for any $\varepsilon > 0$ there exists a natural number $N$ such that for all $n > N$ the inequality $|a_n - a| < \varepsilon$ holds.

Let us break down this definition word by word. “For any $\varepsilon > 0$”—we specify an arbitrarily small error. “There exists $N$”—starting from a certain index. “For all $n > N$, $|a_n - a| < \varepsilon$”—all subsequent terms lie within the $\varepsilon$-neighborhood of the number $a$.

The completeness property of the real numbers is the deepest of the axioms that distinguish the real numbers from the rational ones. It has several equivalent formulations, each offering a new perspective.

Consider the sequence 1, 1.4, 1.41, 1.414, 1.4142, ... — these are the decimal approximations of √2. Each subsequent term is more precise than the previous. In the rational numbers, this limit does not exist — √2 is irrational. But we "know" there must be a point, because the real number line is ...

1. The Supremum Axiom. If a non-empty set is bounded above, then it has a least upper bound (supremum).

2. The Nested Intervals Principle (Cantor). If [a₁, b₁] ⊇ [a₂, b₂] ⊇ ... is a system of nested intervals, then their intersection is non-empty. If the lengths of the intervals tend to zero, the intersection consists exactly of one point.

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The limit of a sequence is the limit of a "discrete" function defined on natural numbers. Now we turn to continuous functions, defined on intervals of the real line.

What does it mean that "f(x) tends to L as x tends to a"? Intuition suggests: when x is close to a, the value of f(x) is close to L. But how close? The definition via ε and δ gives a precise answer.

A number L is called the limit of a function f(x) as x→a if, for any ε > 0, there exists δ > 0 such that for all x with 0 < |x - a| < δ it holds that |f(x) - L| < ε.

Note: x = a does not enter the condition (0 < |x - a|). The limit describes the behavior of the function near the point, but not at the point itself. This allows us to speak about limits even where the function is not defined.

Intuitively, a continuous function is one that can be drawn without lifting your pen from the paper. The formal definition translates this intuition into strict language.

A function $f$ is called continuous at point $a$ if: 1. $f$ is defined at $a$ 2. The limit $\lim_{x \to a} f(x)$ exists 3. $\lim_{x \to a} f(x) = f(a)$

All three conditions are important. Violation of any of them yields a discontinuity. The function $f(x) = \sin(x)/x$ for $x \neq 0$ and $f(0) = 1$ is continuous everywhere. But if we set $f(0) = 0$, then at zero there is a first-kind discontinuity (removable).

Removable discontinuity: the limit exists, but is not equal to the function value. It is enough to redefine the function at this point.

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Theorems on the mean value connect the local properties of a function (the values of the derivative at a point) with the global ones (the change of the function on an interval). They are the workhorses of mathematical analysis, used in proofs of hundreds of other results.

If $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists a point $c \in (a, b)$ such that $f'(c) = 0$.

Geometric meaning: if the function returns to its starting point, at some moment it "turns around"—the derivative becomes zero.

Proof: by Weierstrass's theorem, $f$ attains a maximum and a minimum on $[a, b]$. If both are at the endpoints, $f \equiv \mathrm{const}$ and $f' \equiv 0$. Otherwise, at least one internal extremum—at it, the derivative is zero.

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At the end of the 17th century, Newton and Leibniz independently developed differential calculus. Newton called the derivative "fluxion" and thought of it physically—as instantaneous velocity. Leibniz devised the convenient notation dy/dx, which we still use today. Their approaches were equivalen...

A rigorous definition of the derivative was given by Cauchy in the 19th century.

The derivative of a function f at the point $x_0$ is the limit of the ratio of the increment of the function to the increment of the argument:

$ f'(x_0) = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} $

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Finding a maximum or a minimum is one of the most practical mathematical problems. A company wants to maximize profit. An engineer wants to minimize energy consumption. A physicist seeks the configuration with minimal energy.

Fermat's Theorem: If $f$ is differentiable at the point $x_0$ and a local extremum is achieved at that point, then $f'(x_0) = 0$.

Points where $f' = 0$ or $f'$ does not exist are called critical points. Extrema can only occur at critical points. However, not every critical point is an extremum: $f(x) = x^3$, $f'(0) = 0$, but $x = 0$ is not an extremum (it is an inflection point).

First Derivative Test: If $f'$ changes sign from $+$ to $-$ as $x_0$ is passed through, then $x_0$ is a maximum. From $-$ to $+$ — a minimum. If the sign does not change — not an extremum.

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Polynomials are the simplest functions: their computation requires only addition and multiplication. Taylor's idea is to approximate an arbitrary function with a polynomial that coincides with the function at a given point as closely as possible.

"Coincide" means: the polynomial and the function match at point $a$ in value, in the first derivative, in the second, ..., up to the $n$-th derivative. This polynomial is unique and is called the Taylor polynomial.

$P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} \cdot (x-a)^k = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n$

Remainder term in Lagrange form: $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x-a)^{n+1}$ for some $c$ between $a$ and $x$.

How can one find the area under the curve $y = f(x)$ from $a$ to $b$? If $f = \text{const}$, the answer is trivial. If $f$ is a polynomial, one can divide into trapezoids. But what about an arbitrary function?

Riemann's idea (1854): divide the segment $[a, b]$ into small parts, on each part approximately replace the function with a constant, sum up the rectangles.

A partition $T$: $a = x_0 < x_1 < \ldots < x_n = b$. For each subinterval $[x_{i-1}, x_i]$ of length $\Delta x_i$ choose a point $\xi_i \in [x_{i-1}, x_i]$.

A function $f$ is Riemann integrable on $[a, b]$ if the limit of Riemann sums exists as $\lambda \to 0$, independent of partition and points $\xi_i$. This limit is the definite integral $\int_a^b f(x)dx$.

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The Newton–Leibniz theorem is the principal result of mathematical analysis. It unites two problems that might at first appear unrelated: finding the area (integral) and finding the derivative (differential).

Statement: If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ (i.e., $F' = f$), then:

This means: to compute the area under a curve, it is enough to find an antiderivative and substitute the endpoints.

First part of the theorem: The function $\Phi(x) = \int_a^x f(t) dt$ is differentiable, and $\Phi'(x) = f(x)$.

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The standard Riemann integral requires that a function is defined and bounded on a closed finite interval. But what if the domain of integration is infinite or the function is unbounded?

Type II (unbounded function): ∫ₐᵇ f(x) dx (f → ∞ as x→a+) = lim(ε→0+) ∫ₐ₊ₑᵇ f(x) dx.

If the limit exists and is finite—the integral converges. Otherwise—it diverges.

Benchmarks: ∫₁^∞ dx/xᵖ converges for p > 1, diverges for p ≤ 1. ∫₀¹ dx/xᵖ converges for p < 1, diverges for p ≥ 1.

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The sum 1 + 1/2 + 1/4 + 1/8 + ... — can an infinite sum be finite? Intuition says "no," but mathematics shows: yes, if the terms decrease fast enough.

A numerical series is the formal notation Σₙ₌₁^∞ aₙ. Its partial sum is Sₙ = a₁ + a₂ + ... + aₙ. The series converges if the sequence {Sₙ} has a finite limit S. Then S = Σₙ₌₁^∞ aₙ.

Necessary condition for convergence: If the series converges, then aₙ → 0. The converse is not true: the series Σ 1/n (the harmonic series) diverges, although 1/n → 0.

This is a fundamental series: annuity rates, business valuation problems (the series of discounted cash flows) are expressed through it.

Let the series Σfₙ(x) converge at each point x to a function S(x). Can we assert that S is continuous if all the fₙ are continuous? Or that the integral of the series equals the series of the integrals?

Answer: no, if the convergence is only pointwise! A stronger form is required — uniform convergence.

A series Σfₙ converges uniformly to S on a set E if for any ε > 0 there exists N such that for all n > N and all x ∈ E: |S(x) - Sₙ(x)| < ε. The same N works for all x.

Cauchy Criterion: The series converges uniformly ⟺ for any ε > 0 there exists N such that for m > n > N: |∑ₖ₌ₙ₊₁ᵐ fₖ(x)| < ε for all x ∈ E.

Jean-Baptiste Joseph Fourier in 1807 proposed a revolutionary idea: any “reasonable” function can be expanded into a series of trigonometric functions. This enabled the solution of the heat equation—a problem Fourier was working on as an engineer.

Fourier’s trigonometric series: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$, where

$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos(nx)dx$, $b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin(nx)dx$.

The functions $1$, $\cos x$, $\sin x$, $\cos 2x$, $\sin 2x$, ... are orthogonal in the space $L^2[-\pi, \pi]$:

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Real-world problems seldom depend on just one variable. Temperature depends on the coordinates x, y, z, and time t. Firm profit depends on prices, quantities, costs. The potential energy of a particle system depends on the positions of all particles. Analysis of functions of many variables is an ...

The partial derivative of $f$ with respect to $x_i$ at point $a$ is the ordinary derivative with respect to $x_i$ when the other variables are held fixed:

$ \frac{\partial f}{\partial x_i} (a) = \lim_{h \to 0} \frac{f(a + h \cdot e_i) - f(a)}{h}. $

For $f(x, y) = x^2 y + \sin(y)$: $\frac{\partial f}{\partial x} = 2 x y$ (differentiate with respect to $x$, $y$ is a parameter), $\frac{\partial f}{\partial y} = x^2 + \cos(y)$ (differentiate with respect to $y$, $x$ is a parameter).

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A one-dimensional integral finds the area under a curve. Multiple integrals generalize this to several dimensions: the double integral computes the volume under a surface, the mass of a flat plate with variable density, the probability of hitting a region for a two-dimensional distribution. Tripl...

Historically, multiple integrals arose with Leibniz and Newton in solving mechanics problems, but a rigorous justification appeared only with Cauchy and Riemann in the 19th century. The key obstacle — the region of integration can be an arbitrary shape, not a rectangle. Fubini’s theorem, which we...

Let $f(x,y)$ be defined and bounded on a closed region $D \subset \mathbb{R}^2$. Divide $D$ into small pieces with areas $\Delta A_k$, choose a point $(\xi_k, \eta_k)$ in each, and form the integral sum $\sum f(\xi_k, \eta_k) \Delta A_k$. If this limit exists as the partition becomes finer and do...

Geometric meaning: if $f(x,y) \geq 0$, then the double integral equals the volume under the surface $z = f(x,y)$ above region $D$. For $f = 1$ we get the area of the region: $S(D) = \iint_D dA$.

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Most important mathematical relationships are defined implicitly: by equations, systems of equations, equilibrium conditions. The equation of state of a gas links pressure, volume, and temperature, but we wish to know how pressure changes as temperature changes at a fixed volume. The equation of ...

Theorems on implicit and inverse mappings answer a fundamental question: when does a nonlinear system “locally” behave like a linear one? The answer is: when the corresponding linear approximation (the Jacobian matrix) is invertible. This is essentially the principle of linearization, which perme...

Theorem (Cauchy, 1831): Let $F: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}$ be continuously differentiable, $F(x_0, y_0) = 0$ and $\frac{\partial F}{\partial y}(x_0, y_0) \neq 0$. Then there exist neighborhoods $V \ni x_0$ and $W \ni y_0$ such that for each $x \in V$, the equation $F(x, y) = 0...

The condition $\frac{\partial F}{\partial y} \neq 0$ is key. Geometrically, this means: the level surface $F = 0$ “intersects” the vertical direction nontransversally. Analogy: the linear equation $ax + by = 0$ is solvable for $y$ if and only if $b \neq 0$.

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Three problems lead to curvilinear integrals. The first: find the mass of a thin wire with variable linear density ρ(x, y)—you cannot simply multiply length by density if the density changes. The second: find the work performed by a force field F(x, y) = (P, Q) when moving a particle along a curv...

A smooth curve C is defined parametrically: r(t) = (x(t), y(t)), t ∈ [a, b], where x(t), y(t) are continuously differentiable and |r'(t)| = √(x'² + y'²) ≠ 0 (no “stops”). Differential arc length element: ds = |r'(t)| dt = √(x'²(t) + y'²(t)) dt.

Example: semicircle C: x = cos t, y = sin t, t ∈ [0, π]. Length: ∫₀^π |r'(t)| dt = ∫₀^π 1 dt = π. ✓

This is the integral of a scalar function f over the length of the curve. Does not depend on the orientation of the curve (the sign of ds is always positive).

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A flat plate with variable density required a double integral. A surface in space—a three-dimensional curve—requires a surface integral. Physical problems leading to them: the mass of a surface with variable density, heat flow through an unevenly heated shell, fluid flow through a membrane, elect...

A surface $S$ is defined by a mapping $r: D \subset \mathbb{R}^2 \to \mathbb{R}^3$:

Tangent vectors: $r_u = \partial r / \partial u = (\partial x/\partial u, \partial y/\partial u, \partial z/\partial u)$ and $r_v = \partial r / \partial v$—lie in the tangent plane to the surface.

Normal vector: $N = r_u \times r_v$—is perpendicular to the tangent plane. Its length $|N| = |r_u \times r_v|$ is the "stretching factor" of parametrization. Element of area: $dS = |r_u \times r_v| du dv$.

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The Newton–Leibniz theorem, Green's theorem, Stokes' theorem, and the Gauss–Ostrogradsky theorem appear to be four different theorems about four different types of integrals. In reality, they are one theorem in different dimensions: the integral of an exterior derivative over a manifold equals th...

Differential forms emerged at the end of the 19th century in works by Poincaré, Carleman, Cartan. They unify analysis, topology, and geometry and constitute the proper language for the general theory of integration on manifolds—from classical mechanics to general relativity.

A 0-form on ℝⁿ is simply a smooth function f: ℝⁿ → ℝ. Its “integral” over a point is the value of the function.

A 1-form is an expression ω = P₁ dx₁ + P₂ dx₂ + ... + Pₙ dxₙ, where Pᵢ are smooth functions and dx₁,...,dxₙ are “basis forms.” The integral of a 1-form over a curve is the line integral of the second kind.

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By the end of the 19th century, mathematicians encountered the limitations of the Riemann integral. The first problem: functions with “bad” discontinuities. The Dirichlet function $D(x) = 1$ for rational $x$ and $D(x) = 0$ for irrational numbers is not Riemann integrable: for any partition of $[0...

The second problem: passage to the limit. If $f_n \to f$ and each $f_n$ is integrable, is $f$ integrable? Is it true that $\lim \int f_n = \int f$? For Riemann, the answer is “not necessarily”—uniform convergence is required. But in probability theory, functional analysis, and the theory of diffe...

Henri Lebesgue (1875–1941) in his 1902 doctoral thesis created a theory that solves both problems. His approach: measure the “size” of sets, rather than partitioning the domain into intervals.

Let’s start simple: the length of the interval $[a,b]$ is $b - a$. The length of a finite union of non-overlapping intervals equals the sum of lengths (additivity). Can this “measurement” be extended to arbitrary sets?

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After the introduction of the Lebesgue integral, it is natural to ask: which functions “behave nicely”? Why do we need a class of functions at all, rather than just individual functions? The answer is that analysis problems rarely exist in “a single function.” We approximate functions by series, ...

Lp spaces are the proper functional spaces for analysis, probability theory, signal processing, and quantum mechanics.

Lᵖ(X, μ) is the space (of equivalence classes) of measurable functions f: X → ℝ such that ∫_X |f(x)|ᵖ dμ(x) < ∞.

Technically, L^p consists of equivalence classes: f ~ g if f = g almost everywhere (μ-a.e.). This makes ‖f − g‖_p = 0 equivalent to f = g a.e., which is necessary for the norm to be non-degenerate.

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Fourier series decompose a periodic function into frequencies 1/T, 2/T, 3/T, ... But what should we do with a non-periodic signal — for example, a finite pulse or a function on the entire real line? A step toward the non-periodic case: increase the period T → ∞. The discrete set of frequencies n/...

It is one of the most universal tools in mathematics — it is simultaneously used in number theory, quantum mechanics, partial differential equation theory, statistics, signal processing, and machine learning.

For f ∈ L¹(ℝ) (absolutely integrable function), the Fourier transform is defined by:

(The convention e^{-iωx} is used here; physics often uses e^{-2πiξx}, engineering — e^{-jωt}.)