# Sergei Yakovenko's blog: on Math and Teaching

## Integration and antiderivation

Again, no new notes were prepared this time.

## Calculus on complex manifolds

If $V$ is a complex vector space, then it is naturally also a real vector space (if you allow multiplication by complex numbers, then that by real numbers is automatically allowed). However, forgetting how to multiply by the imaginary unit results in the fact that the dimension $\dim_{\mathbb R}V$ of the space over the real numbers is two times higher. If we regret our decision to forget the complex multiplication, we still can restore it by introducing the $\mathbb R$-linear operator $J\colon V\to V$ such that $J^2=-E$, where $E$ is the identity operator.

An even-dimensional real vector space with such an operator is called an almost complex space, and it obviously can be made into a complex vector space (over $\mathbb C$). However, if we consider an even-dimensional manifold $M$ with the family of operators as above, it is somewhat less than a complex analytic manifold (a topological space equipped with an atlas of charts with biholomophic transition functions). For details, follow the lecture notes that will be available later.

## Symplectic manifolds

In parallel with the Riemannian manifolds equipped with a positive definite (symmetric) scalar product on each tangent space, it is interesting to consider manifolds equipped with an antisymmetric scalar product on each tangent space, i.e., with a differential 2-form $\omega\in\Omega^2(M)$. This form is called a symplectic structure, if $\mathrm d\omega=0$ and an additional nondegeneracy condition is met.

It turns out that this structure naturally arises on the cotangent bundle $M=T^*N$ of an arbitrary smooth manifold $N$. Moreover, this structure is intimately related with the mechanics of frictionless systems: the Hamiltonian differential equations can be naturally described by vector fields $X$ which satisfy the Hamiltonian condition $\mathrm i_X\omega=\mathrm d H$, where $H$ is a function (Hamiltonian, or full energy) on the symplectic manifold. Thus each Hamiltonian vector field is “encoded” by a single function, rather than by a tuple of functions. The commutator of Hamiltonian vector fields is again Hamiltonian: this is the invariant definition of the Poisson bracket.

There are two instant ramifications from this point. One can discuss integrability of the Hamiltonian vector fields. Another, less physically motivated direction is to study the symplectic geometry, first locally, then globally. It is a surprising twisted counterpart of the Riemannian geometry, which has no intrinsic curvature but nevertheless is very rich globally.

The lecture notes will be available later.

## Integral: antiderivation and area

If $f:[a,b]\to\mathbb R$ is a function, can we find another differential function $F:[a,b]\to\mathbb R$ such that the derivative of $F$ is $f$? If yes, then how many such functions exist? If we have a complete record of the car speedometer, can we trace the route? compute the end point of the travel?

“Uniqueness” of solution is easy: if there are two solutions $F_1(x),F_2(x)$ such that $F'_{1,2}=f$, then the derivative of their difference $F=F_1-F_2$ is identically zero. By the finite difference lemma, for any point $x\in[a,b]$ there exists an intermediate point $z\in[a,x]$ such that $F(x)-F(a)=F'(z)=0$, thus $F$ is a constant.

Clearly, if $F(x)$ is a solution, then $F(x)+c,c\in\mathbb R$, is also a solution. In particular, we can choose this constant so that $F(a)$ takes any specified value.

Example. If $f(x)\equiv\lambda\in\mathbb R$, then $F(x)=F(a)+\lambda (b-a)$, as one can check by the direct derivation.

Example. If $f$ can be found in the right  hand side of any table of derivatives (eventually, with a constant coefficient), then $F$ can be found in the same table. E.g., if $f(x)=\mu x^{\mu-1},~\mu\ne 0$, then $F(x)=x^\mu,~\mu\ne 0$. Denoting $\mu-1=\nu$ and dividing both sides by $\mu=\nu+1$, we conclude that if $f(x)=x^\nu,~\nu\ne -1$, then $F(x)=\frac1{\nu+1}\,x^{\nu+1}$. The case of $\nu=-1$ occurs on another line in the table: if $f(x)=\frac1x,~x>0$, then $F(x)=\ln x$.

What to do if $f$ cannot be so easily found? A good idea is to try approximating $f$ by some simple functions and see what happens.

Example. Let $[a,b]=[0,N]$ and assume that the function $f$ takes the same constant value $\lambda_i$ on the (semi)interval $[i,i+1)$. Then one can easily check that the following function,

$F(x)=\lambda_0+\lambda_1+\cdots+\lambda_{i-1}+\lambda_i(x-i),\qquad x\in [i,i+1),\quad i=0,\dots,N-1,\qquad(1)$

is satisfying the following conditions:

1. continuous on the entire segment $[0,N]$,
2. differentiable everywhere except the entire points $1,2,\dots,N-1$,
3. the derivative of $F$ coincides with $f$,
4. the graph of $F$ is a broken line (קו שבור).

Instead of the cumbersome (מורכות) formula (1), one can use the equivalent verbal description:

$F(x)=$area under the graph of $f$ between the vertical segments $a$ and $x.\qquad (2)$

Indeed, if $h$ is small enough, then both $x$ and $x+h$ belong to the same interval $(i,i+1)$ and hence $F(x+h)-F(x)=\lambda_i h$, thus the derivative $F'(x)=\lambda_i=f(x)$.

Clearly, an arbitrary interval $[a,b]$ can be subdivided into $N$ equal subintervals: the formula (1) will be changed, yet its meaning (2) would obviously remain the same.

This observation suggests that  the formula (2) gives the answer also in the general case, provided that the expression “the area under the graph” makes sense. Clearly, this is the case when $f(x)$ is piecewise constant  (“the step function”, פונקציית מדרגה, as above) and even when $f$ is piecewise linear (קו שבור). Note that we allow only for step functions having only finitely many intervals of continuity, to avoid summing infinitely many areas of rectangles!

To define the area in the general case, we appeal to the following intuitively clear observation: if $X\subseteq Y$ are two nested subsets of the plane, then the areas of the sets, if it is well defined, should satisfy the inequality $s(X)\leq s(Y)$. For polygons this is an elementary theorem.

Let $f:[a,b]\to\mathbb R$ be a bounded function on a finite interval. For any step function $g_i(x):[a,b]\to\mathbb R$ which is everywhere less or equal to $f$, $g_-(x)\le f(x)$, the area $s(g_-)$ under the graph of $g_-$ is called a lower sum for $f$ on $[a,b]$. In a similar way, an upper sum for $f$ is the area under the graph of any step function $g_+(x)$ such that $f(x)\le g_+(x)$ on $[a,b]$. Clearly, for any two such functions $g_-,g_+$, the inequality $s(g_-)\le s(g_+)$ holds, thus any lower sum is less or equal to any upper sum.

Definition. A function  $f:[a,b]\to\mathbb R$ is called integrable on the segment $[a,b]$, if there exists a single number $s=s(f)$ which separates the lower and the upper sums, so that $s_-\le s \le s_+$ for any pair of lower/upper sums $s_\pm=s(g_\pm)$. This number is called the integral (האינטגרל המסוים) of the function $f(x)$ on the interval $[a,b]$ and denoted by $\displaystyle \int_a^b f(x)\,dx$.♦

An equivalent definition of integrability requires that for any positive $\varepsilon>0$ there exist an upper and a lower sum which differ by no more than $\varepsilon$:

$\forall\varepsilon>0~~\exists g_-,g_+\text{ step functions }:g_-(x)\le f(x)\le g_+(x),~~s(g_+)-s(g_-)<\varepsilon.$

Exercise. Prove that the two definitions are indeed equivalent.

Exercise. Prove that the function $f(x)=px+q,~p,q\in\mathbb R$, is integrable (in the above sense) on any interval and its integral is equal to the (geometrically calculated) area of the corresponding trapeze (טרפז).

Proposition. Any monotone (bounded) function is integrable.

Proof. Consider the partition of $[a,b]$ into $N$ equal parts by the points $a=x_0 and denote $\lambda_i=f(x_i)$ the values of $f$ at these points. Then the two step functions, $g_-(x)=\lambda_{i},~~x\in [x_i,x_{i+1})$ and $g_+(x)=\lambda_{i+1},~~x\in [x_i,x_{i+1})$ squeeze $f$ between them. The lower sum is $s_-=\frac1N(\lambda_0+\lambda_1+\cdots+\lambda_{N_1})$ and $s_+=\frac1N(\lambda_1+\cdots+\lambda_N)$. Their difference is less or equal to $(\lambda_N-\lambda_0)/N$ and it  becomes less than any positive number $\varepsilon$ when $N$ is large enough.

Theorem. Any continuous function (on the closed interval) is integrable.

Proof. The idea is to construct two step functions with the common set of jump points $a=x_0 such that on each interval $[x_i,x_{i+1})$ these functions squeeze $f$ between themselves and differ by less than a given $latex\varepsilon>0$. The points $x_i$ should be close enough to each other so that the continuity of $f$ implies the required proximity of the corresponding values.

More precisely, for any point $c\in[a,b]$ there exists a small interval latex $U_c\subseteq[a,b]$ containing $c$ such that on this interval the function is squeezed between $f(c)-\frac12\varepsilon$ and $f(c)+\frac12\varepsilon$.  The union of all these small intervals covers $[a,b]$ which is compact. Hence there exists a finite covering of $[a,b]$ by these intervals. The endpoints of these intervals subdivide $[a,b]$ into finitely many intervals $U_i$ in such a way that on each interval the function $f$ is squeezed between two constants $\lambda_i\le f(x) \le \lambda_+$ such that $\lambda_+-\lambda_-<\varepsilon$. The corresponding upper and lower sum differ by no more than $\sum (\lambda_+-\lambda_-)(x_{i+1}-x_i)\le \varepsilon \sum (x_{i+1}-x_i)\le \varepsilon (b-a)$. This difference can be as small as required if $\varepsilon$ is chosen small enough. ♦

Clearly, continuity is sufficient but not necessary assumption for integrability: the step functions are by definition integrable, though they are discontinuous. The accurate description of all integrable functions goes beyond the scope of these notes, yet there are certainly many non-integrable functions.

Example. The Dirichlet function is non-integrable on $[0,1]$. Indeed, any upper sum must be at least 1, and any lower sum at most 0, hence the gap between these values cannot be bridged (complete all details of this proof!).

Problems.

1. Prove that any function on $[0,1]$ which differs from identical zero at only finitely many points, is integrable and its integral is zero, no matter what are the values at these points.
2. Prove that a function that differs from identical zero at countably many points $x_1,x_2,\dots,x_n,\dots$, is integrable and the integral is zero if $\lim_{k\to\infty}f(x_k)=0$. Can one drop the limit assumption?
3. Prove that the Riemann function equal to $f(x)=1/q$ at the rational points $x=p/q$ (assuming $p,q$ mutually prime) and zero at irrational points, is integrable on $[0,1]$ and its integral is zero.

Problem. Is the function $f(x)=\sin \frac1x$ integrable on $[0,1]$? (Do not try to compute the integral :-))

Sin(1/x) for x near the origin

## The Newton-Leibniz formula

Quite obviously, if $f:[a,b]\to\mathbb R^1$ is integrable on $[a,b]$, then it is also integrable on any sub-interval $[a,c]$, $a\le c \le b$. The formula (2) relating the area (definite integral) with the antiderivative $F$ of an integrable function $f$ will take the form

$\displaystyle F(c)=F(a)+\int_a^c f(x)\,dx \iff \int_a^c f(x)\,dx=F(c)-F(a).$

This allows to express the area under the graph of a function in terms of its antiderivative (if the latter exists and is known).

However, integrability is a weaker condition than existence of the antiderivative: if $f$ is a step function with the partition points $x_1,\dots,x_N$, then the corresponding area function $F(z)=\int_0^z f(x)\,dx$ is continuous but differentiable only everywhere except these points.

Theorem. If $f(x)$ is continuous at a point $c\in[a,b]$, then the area function $F(z)=\int_a^z f(x)\,dx$ is differentiable at $c$ and $F'(c)=f(c)$. ♦

## Change of the independent variable in the integrals

If $f(x)$ is a function defined on the interval $x\in [a,b]$, and $z$ is a new variable which is obtained from $x$ by a monotone differentiable transformation, $z=h(x)$, then this transformation maps bijectively (1-1-way) the interval $[a,b]$ into the interval $[h(a),h(b)]$. The function $f(x)$ becomes after such change a new function $g(z)$ of the new variable: $g(z)=f(x)$ if and only if $z=h(x)$, i.e., the two functions take equal values at the two points “connected” by the transformation $h$.

The “formal” relationship between these functions is easier written “in the opposite direction”, expressing $f$ via $g$:

$f(x)=g(h(x))=(g\circ h)(x).$

The graphs of functions of $f$ and $g$ are obtained from each other by a “non-uniform stretch along the horizontal axis” which keeps the vertical direction. In particular, any step function with the partition points $x_1 will be transformed into the step function with the partition points $z_1, $z_i=h(x_i)$, with the same values. Moreover, if $f(x)$ is squeezed between two step functions, $f_-\le f\le f_+$, then its transform is squeezed between the transforms $g_\pm$ of these functions.

Thus it is sufficient to study how the change of variables affects areas under step functions, which are equal to finite sums $\sum_1^N \lambda_i(x_{i}-x_{i-1})$ and their transforms $\sum_1^N \lambda_i(z_i-z_{i-1})$. The heights $\lambda_i$ are unchanged, and the widths $z_i-z_i$ by the finite difference lemma are equal to the initial widths multiplied by the derivative $h'(c_i)$ computed at some intermediate points $c_i\in[x_{i-1},x_i]$. The result is as if instead of the step function $f(x)$ we would integrate the function $f(x)\cdot h'(x)$. Passing to limit, we conclude that

$\displaystyle \int_a^b g(h(x))\,h'(x)\,dx=\int_{h(a)}^{h(b)}g(z)\,dz.$

Change of independent variable and integral of a step function

Of course, this is equivalent to the chain rule of differentiation for the primitive functions: if $F(X)=\int_a ^X f(x)\,dx$ and $G(Z)=\int_{h(a)}^Z g(z)\,dz$, then $F(X)=G(h(X))$ and $F'(X)=G'(h(X))\cdot h'(X)$.

The formula for change of variables of integrals can be easily memorized using the existing notation: in the formula $\int g(z)\,dz$ one has to transform not just the integrand $g(z)$ by substituting $z=h(x)$, but also the differential $dz$ should be transformed using the formula $dz=h'(x)\,dx$.

# Local theory of regular singular points of linear systems

This lecture, in an exceptional way, will take place on Sunday, 16:00-18:00, in the Room 261.

1. Regular and irregular singularities: growth matters.
2. Local gauge equivalence (holomorphic, meromorphic, formal). Meromorphic classification of regular singularities.
3. Fuchsian singularities as a particular class of regular singularities (Sauvage lemma).
4. Formal classification of Fuchsian singularities (Poincaré-Dulac theorem revisited). Resonances. Levelt upper triangular normal form.
5. Coincidence of formal and holomorphic classification in the Fuchsian case.
6. Integrability of the normal form.
7. Towards global theory of Fuchsian systems on $\mathbb C P^1$: Monopoles as special classes of rational matrix functions.

Reminder: Today (actually, on Friday) was the deadline for submission of the home exam 😦

## Finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$, I. Formal theory.

1. Formal normal form for a single holomorphic self-map from $\text{Diff}(\mathbb C^1,0)$. Parabolic germs.
2. Bochner theorem on holomorphic linearization of finite groups.
3. Stratification of the subgroup of parabolic germs $\text{Diff}_1(\mathbb C^1,0)$.
4. Tits alternative for finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$: every such subgroup is either metabelian (its commutator is commutative, e.g., trivial), or non-solvable (all iterated commutators are nontrivial).
5. Centralizers and symmetries: formal classification of solvable subgroups.
6. Integrable germs and their holomorphic linearizability.

Recommended reading: Section 6 (first part) from the book (printing disabled)

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