Comprehensive Guide to Mathematical Analysis II: Integral Calculus, Series, and Differential Equations

Fundamental Definitions of Intervallic Subdivisions and Step Functions

In the study of integral calculus, we begin by examining the segment $[a,b]$ as a closed and bounded interval in $\mathbb{R}$ where $a < b$. A subdivision of this interval, denoted as $\sigma = \{x_0, x_1, \dots, x_n\}$, is a finite suite of elements such that $a = x_0 < x_1 < \dots < x_n = b$. The "pas" or mesh of this subdivision, denoted by $\delta(\sigma)$, is defined as the maximum length of the sub-intervals: $\delta(\sigma) = \max_{0\leq i \leq n-1} |x_{i+1} - x_i|$. A uniform subdivision of order $n \in \mathbb{N}^*$ specifically divides the interval into $n$ sub-intervals of equal length, where $\delta(\sigma_n) = \frac{b-a}{n}$ and each point is defined by the formula $x_i = a + i\frac{b-a}{n}$ for $i = 0, \dots, n$. A subdivision $\sigma$ is considered finer than another subdivision $\sigma'$ if $\sigma \subset \sigma'$, which implies that $\delta(\sigma) \leq \delta(\sigma')$.

A function $f: I \rightarrow \mathbb{R}$ is defined as a step function (fonction en escalier) if there exists a subdivision $\sigma = (x_i)_{0\leq i \leq n}$ adapted to $f$, such that $f$ remains constant on every open partial interval $]x_i, x_{i+1}[$. The choice of an adapted subdivision is not unique; if $\sigma$ is adapted to $f$, any finer subdivision $\sigma'$ is also adapted. For two step functions $f$ and $g$ with respective subdivisions $\sigma$ and $\sigma'$, the union $\sigma \cup \sigma'$ serves as a subdivision adapted to both. The set of all step functions on $[a,b]$, denoted $\mathcal{E}([a,b])$, forms a vector subspace of the set of all functions $\mathcal{F}([a,b])$, meaning that for any $\alpha, \beta \in \mathbb{R}$ and $f, g \in \mathcal{E}(I)$, the combination $\alpha f + \beta g \in \mathcal{E}(I)$.

The Integral of Step Functions and Its Properties

For a step function $f: [a,b] \rightarrow \mathbb{R}$ and an adapted subdivision $\sigma = (x_i)_{0\leq i \leq n}$, where $f(x) = c_i$ on the interval $]x_i, x_{i+1}[$, the integral is defined by the expression $I(\sigma, f) = \sum_{i=0}^{n-1} c_i(x_{i+1} - x_i)$. Crucially, this value does not depend on the choice of the adapted subdivision $\sigma$. The integral of $f$ on $[a,b]$ is denoted as $\int_a^b f(x)dx$. Geometrically, this value corresponds to the algebraic area between the x-axis, the graph of the function, and the lines $x = a$ and $x = b$.

The integral of step functions follows several fundamental properties. Linearity dictates that $\int_a^b (\alpha f(x) + \beta g(x))dx = \alpha \int_a^b f(x)dx + \beta \int_a^b g(x)dx$. Positivity ensures that if $f \geq 0$ on $[a,b]$, then $\int_a^b f(x)dx \geq 0$. Monotonicity implies that if $f \geq g$, then $\int_a^b f(x)dx \geq \int_a^b g(x)dx$. The triangle inequality relates the absolute value: $| \int_a^b f(x)dx | \leq \int_a^b |f(x)|dx$. If $|f(x)| \leq \lambda$ for all $x \in [a,b]$, then $| \int_a^b f(x)dx | \leq \lambda(b-a)$. Additionally, Chasles' relation states that for any $c \in ]a,b[$, $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$. Finally, if two step functions $f$ and $g$ are equal except for at a finite number of points, their integrals are identical.

Darboux Sums and the Riemann Integral Definition

To extend integration to broader classes of functions, we consider a bounded function $f: [a,b] \rightarrow \mathbb{R}$. For a uniform subdivision $\sigma_n$ with pas $\frac{b-a}{n}$, we define $m_{ni} = \inf_{x \in ]x_i, x_{i+1}[} f(x)$ and $M_{ni} = \sup_{x \in ]x_i, x_{i+1}[} f(x)$. These generate two step functions, $f_n(x) = m_{ni}$ and $F_n(x) = M_{ni}$ on $]x_i, x_{i+1}[$. The lower and upper Darboux sums are given by $s_n = \int_a^b f_n(x)dx = \frac{b-a}{n} \sum_{i=0}^{n-1} m_{ni}$ and $S_n = \int_a^b F_n(x)dx = \frac{b-a}{n} \sum_{i=0}^{n-1} M_{ni}$, respectively. Since $m_{ni} \leq M_{ni}$, it is always true that $s_n \leq S_n$.

A bounded function $f$ is Riemann-integrable if $\lim_{n \to \infty} (S_n - s_n) = 0$. The common limit of $s_n$ and $S_n$ is the Riemann integral of $f$ on the segment $[a,b]$, denoted $\int_a^b f(t)dt$. For example, the function $f(x) = 2x$ on $[0,1]$ is Riemann-integrable with an integral of 1. Conversely, the Dirichlet function, defined as $f(x) = 1$ if $x \in \mathbb{Q}$ and $0$ if $x \notin \mathbb{Q}$, is not Riemann-integrable because on any interval, $m_{ni} = 0$ and $M_{ni} = 1$, leading to $s_n = 0$ and $S_n = 1$ for all $n \in \mathbb{N}^*$.

Families of Riemann-Integrable Functions and Sums of Riemann

Several specific classes of functions are guaranteed to be Riemann-integrable. These include monotone functions (whether increasing or decreasing) and continuous functions. For these functions, the integral can be calculated using Riemann sums: $\int_a^b f(x)dx = \lim_{n \to \infty} \frac{b-a}{n} \sum_{i=0}^{n-1} f(a + i\frac{b-a}{n}) = \lim_{n \to \infty} \frac{b-a}{n} \sum_{i=1}^{n} f(a + i\frac{b-a}{n})$. The Mean Value Theorem for integrals states that for a continuous function $f \in \mathcal{C}([a,b])$, there exists a point $c \in [a,b]$ such that $\frac{1}{b-a} \int_a^b f(x)dx = f(c)$.

Riemann-integrable functions share the same properties as step function integrals regarding linearity, positivity, growth, absolute value, and Chasles' relation. By convention, we define $\int_a^b f(x)dx = -\int_b^a f(x)dx$ and $\int_a^a f(x)dx = 0$, which allows the Chasles relation $\int_u^v f(x)dx = \int_u^w f(x)dx + \int_w^v f(x)dx$ to hold for any $u, v, w \in [a,b]$. Like step functions, changing a Riemann-integrable function at a finite number of points does not change the value of its integral.

Primitives and the Fundamental Theorem of Analysis

A function $F: I \rightarrow \mathbb{R}$ is a primitive of $f$ if $F$ is derivable and $F' = f$ on $I$. Primitives are unique up to an additive constant. The Fundamental Theorem of Analysis establishes that derivation and integration are reciprocal operations. Specifically, if $f$ is Riemann-integrable and has a primitive $F$, then $\int_a^b f(x)dx = F(b) - F(a)$. Furthermore, if $f$ is continuous on $[a,b]$ and we define $F(x) = \int_{x_0}^x f(t)dt$, then $F$ is the unique primitive of $f$ that vanishes at $x_0$.

Indefinite integrals, denoted $\int f(x)dx$, represent any primitive of $f$. Integration techniques include integration by parts: $\int_a^b f(x)g'(x)dx = [f(x)g(x)]_a^b - \int_a^b f'(x)g(x)dx$, provided $f$ and $g$ are of class $\mathcal{C}^1$. Another key method is the change of variable theorem: if $\phi: [\alpha, \beta] \rightarrow [a,b]$ is of class $\mathcal{C}^1$, then $\int_{\alpha}^{\beta} f(\phi(x))\phi'(x)dx = \int_{\phi(\alpha)}^{\phi(\beta)} f(u)du$. If $\phi$ is also a bijection, $\int_a^b f(x)dx = \int_{\phi^{-1}(a)}^{\phi^{-1}(b)} f(\phi(u))\phi'(u)du$.

Advanced Integration of Complex Functions

Rational fractions $\frac{P(x)}{Q(x)}$ are integrated by decomposing them into a polynomial and simple elements of two types: $\frac{\gamma}{(x-x_0)^k}$ or $\frac{\alpha x + \beta}{(ax^2 + bx + c)^k}$ with $b^2 - 4ac < 0$. Specific results include $\int \frac{dx}{x-x_0} = \ln|x-x_0| + C$ and $\int \frac{dx}{(x-x_0)^k} = \frac{(x-x_0)^{-k+1}}{-k+1} + C$ for $k \geq 2$. For quadratic denominators, transformations into $\ln$ or $\arctan$ forms are required.

Trigonometric functions $\int F(\cos(x), \sin(x))dx$ can be solved using the change of variable $t = \tan(\frac{x}{2})$, where $\cos(x) = \frac{1-t^2}{1+t^2}$, $\sin(x) = \frac{2t}{1+t^2}$, and $dx = \frac{2dt}{1+t^2}$. Alternatively, Bioche's rules are used: if $\omega(x) = f(x)dx$, then $t = \cos(x)$ is used if $\omega(-x) = \omega(x)$, $t = \sin(x)$ if $\omega(\pi-x) = \omega(x)$, and $t = \tan(x)$ if $\omega(\pi+x) = \omega(x)$. For rational functions involving exponentials, $\int F(e^x)dx$, the change of variable $t = e^x$ allows substitution via $dx = \frac{dt}{t}$.

Nature and Properties of Generalized Integrals

A function $f$ is locally integrable on $I$ (denoted $R_{loc}(I)$) if it is Riemann-integrable on every segment of $I$. A generalized (or improper) integral $\int_a^b f(t)dt$ is defined for intervals that are not necessarily closed or for functions that are not bounded. If $F(x) = \int_a^x f(t)dt$, the integral is convergent if $\lim_{k \to b} F(x)$ is a finite limit $L$. Otherwise, it is divergent. Examples include $\int_0^{+\infty} \frac{dt}{1+t^2} = \frac{\pi}{2}$ (convergent) and $\int_e^{+\infty} \frac{dt}{t} = +\infty$ (divergent).

If the integral has two "uncertain points," it is doublement généralisée. For $f \in R_{loc}(]a,b[)$, the integral $\int_a^b f(t)dt$ converges if there exists a $c \in ]a,b[$ such that both $\int_a^c f(t)dt$ and $\int_c^b f(t)dt$ converge. Their sum provides the final value. An integral is "falsely generalized" if $f$ has a finite limit at the uncertain endpoint, in which case it is always convergent.

Convergence Criteria for Positive and Variable Functions

For positive functions ($f \geq 0$), $\int_a^b f(t)dt$ converges if and only if the integral function $F(x)$ is bounded. The comparison theorem states that if $0 \leq f \leq g$, the convergence of $\int g$ implies the convergence of $\int f$, and the divergence of $\int f$ implies the divergence of $\int g$. The equivalents theorem allows replacing $f$ with an equivalent function $g$ ($f \sim_b g$) to determine nature. Standard scales include the Riemann Scale: $\int_a^{+\infty} \frac{dx}{x^{\alpha}}$ converges if and only if $\alpha > 1$, whereas $\int_0^a \frac{dx}{x^{\alpha}}$ and $\int_b^a \frac{dx}{(x-b)^{\alpha}}$ converge if and only if $\alpha < 1$. The Bertrand scale $\int_e^{+\infty} \frac{dx}{x^{\alpha} \ln^{\beta}(x)}$ converges if $\alpha > 1$ or if $\alpha = 1$ and $\beta > 1$.

For functions with variable signs, an integral is absolutely convergent if $\int_a^b |f(x)|dx$ converges. Absolute convergence implies convergence. If an integral converges but is not absolutely convergent, it is semi-convergente. The Cauchy criterion provides a theoretical basis for convergence without computing the limit. The Abel Rule for integrals suggests that if $f$ is decreasing towards zero and the integral of $g$ is bounded, then $\int f(t)g(t)dt$ converges.

Numerical Series and General Convergence Properties

A series $\sum u_n$ is defined as the suite of partial sums $S_n = \sum_{k=0}^n u_k$. The series converges if $S_n$ has a finite limit as $n \to \infty$. The sum is denoted $\sum_{n=0}^{+\infty} u_n$, and the remainder is $R_n = S - S_n$. A geometric series $\sum q^n$ converges if and only if $|q| < 1$, with a sum of $\frac{1}{1-q}$. Telescopic series $\sum (a_{n+1} - a_n)$ converge if and only if the sequence $(a_n)$ converges to a limit $l$, yielding a sum of $l - a_0$.

A necessary condition for the convergence of any series is $\lim_{n \to \infty} u_n = 0$. If this limit is non-zero or nonexistent, the series is said to "diverge grossièrement." However, this is not a sufficient condition, as seen with the harmonic series $\sum \frac{1}{n}$, which diverges despite the limit being zero. Series obey linearity: if $\sum u_n$ and $\sum v_n$ converge, then $\sum (\alpha u_n + \beta v_n)$ converges to $\alpha \sum u_n + \beta \sum v_n$.

Convergence Criteria for Series with Positive and Arbitrary Terms

For series with positive terms ($u_n \geq 0$), convergence is equivalent to partial sums being bounded. The Comparison Criterion states that if $u_n \leq v_n$, then convergence of $\sum v_n$ implies convergence of $\sum u_n$. The Equivalents Criterion holds that if $u_n \sim v_n$, the series share the same nature. Positive series can also be compared to generalized integrals of decreasing functions: $\sum f(n)$ is of the same nature as $\int_0^{+\infty} f(t)dt$. Riemann series $\sum \frac{1}{n^{\alpha}}$ converge for $\alpha > 1$, and Bertrand series $\sum \frac{1}{n^{\alpha} \ln^{\beta}(n)}$ converge if $\alpha > 1$ or ($\alpha = 1$ and $\beta > 1$).

Additional tests include D'Alembert's Ratio Test: if $\lim \frac{u_{n+1}}{u_n} = l$, the series converges if $l < 1$ and diverges if $l > 1$. Cauchy's Root Test: if $\lim ∑(u_n)^{1/n} = l$, the series converges for $l < 1$ and diverges for $l > 1$. For arbitrary terms, absolute convergence ($\sum |u_n|$) implies convergence. The Leibniz Criterion for alternating series $\sum (-1)^n a_n$ states that if $a_n$ is a positive, decreasing sequence tending to zero, the series converges, and the remainder satisfies $|R_n| \leq |u_{n+1}|$.

Equations Différentielles: First and Second Order

An ordinary differential equation (ODE) of order $n$ involves an unknown function $y(x)$ and its derivatives up to $y^{(n)}$. First-order equations include those with separable variables, which take the form $g(y)y' = f(x)$. These are solved by calculating primitives: $\int g(y)dy = \int f(x)dx$, leading to $G(y) = F(x) + C$. Linear first-order equations are defined as $a(x)y' + b(x)y = c(x)$. One first solves the homogeneous equation $a(x)y' + b(x)y = 0$, which has the solution $y_0 = K e^{-F(x)}$ where $F(x) = \int \frac{b(x)}{a(x)} dx$. A particular solution $y_p$ is then found via the Variation of Constants method, setting $y_p = K(x) e^{-F(x)}$. The general solution is $y = y_0 + y_p$.

Second-order linear ODEs with constant coefficients are of the form $ay'' + by' + cy = f(x)$. The associated characteristic equation is $ar^2 + br + c = 0$. If the discriminant $\Delta > 0$, the homogeneous solution is $y_0 = \lambda e^{r_1 x} + \mu e^{r_2 x}$; if $\Delta = 0$, then $y_0 = (\lambda + \mu x) e^{r_0 x}$; if $\Delta < 0$, the solution is $y_0 = e^{\alpha x} (\lambda \cos(\beta x) + \mu \sin(\beta x))$ where the roots are $r = \alpha \pm i\beta$. Particular solutions are found based on the form of $f(x)$. For $f(x) = e^{\alpha x} P_n(x)$, one looks for $y_p = e^{\alpha x} x^m Q_n(x)$, where $m$ depends on whether $\alpha$ is a root of the characteristic equation. For trigonometric RHS ($e^{\alpha x} P_n(x) \cos(\beta x)$), the choice relates to whether $\alpha + i\beta$ is a root.