Sergei Yakovenko's blog: on Math and Teaching

Tuesday, December 29, 2015

Lecture 10, Dec 29, 2015

Elementary transcendental functions as solutions to simple differential equations

The way how logarithmic, exponential and trigonometric functions are usually introduced, is not very satisfactory and appears artificial. For instance, the mere definition of the non-integer power x^a, a\notin\mathbb Z, is problematic. For a=1/n,\ n\in\mathbb N, one can define the value as the root \sqrt[n]x, but the choice of branch/sign and the possibility of defining it for negative x is speculative. For instance, the functions x^{\frac12} and x^{\frac 24} may turn out to be different, depending on whether the latter is defined as \sqrt[4]{x^2} (makes sense for negative x) or as (\sqrt[4]x)^2 which makes sense only for positive x. But even if we agree that the domain of x^a should be restricted to positive arguments only, still there is a big question why for two close values a=\frac12 and a=\frac{499}{1000} the values, say, \sqrt 2 and \sqrt[1000]{2^{499}} should also be close…

The right way to introduce these functions is by looking at the differential equations which they satisfy.

A differential equation (of the first order) is a relation, usually rational, involving the unknown function y(x), its derivative y'(x) and some known rational functions of the independent variable x. If the relation involves higher derivatives, we say about higher order differential equations. One can also consider systems of differential equations, involving several relations between several unknown functions and their derivatives.

Example. Any relation of the form P(x, y)=0 implicitly defines y as a function of x and can be considered as a trivial equation of order zero.

Example. The equation y'=f(x) with a known function f is a very simple differential equation. If f is integrable (say, continuous), then its solution is given by the integral with variable upper limit, \displaystyle y(x)=\int_p^x f(t)\,\mathrm dt for any meaningful choice of the lower limit p. Any two solutions differ by a constant.

Example. The equation y'=a(x)y with a known function a(x). Even the case where a(x)=a is a constant, there is no, say, polynomial solution to this equation (why?), except for the trivial one y(x)\equiv0. This equation is linear: together with any two functions y_1(x),y_2(x) and any constant \lambda, the functions \lambda y_1(x) and y_1(x)\pm y_2(x) are also solutions.

Example. The equation y'=y^2 has a family of solutions \displaystyle y(x)=-\frac1{x-c} for any choice of the constant c\in\mathbb R (check it!). However, any such solution “explodes” at the point x=c, while the equation itself has no special “misbehavior” at this point (in fact, the equation does not depend on x at all).

Logarithm

The transcendental function y(x)=\ln x satisfies the differential equation y'=x^{-1}: this is the only case of the equation y'=x^n,\ n\in\mathbb Z, which has no rational solution. In fact, all properties of the logarithm follow from the fact that it satisfies the above equation and the constant of integration is chosen so that y(1)=0. In other words, we show that the function defined as the integral \displaystyle \ell(x)=\int_1^x \frac1t\,\mathrm dt possesses all what we want. We show that:

  1. \ell(x) is defined for all x>0, is monotone growing from -\infty to +\infty as x varies from 0 to +\infty.
  2. \ell(x) is infinitely differentiable, concave.
  3. \ell transforms the operation of multiplication (of positive numbers) into the addition: \ell(\lambda x)=\ell(\lambda)+\ell(x) for any x,\lambda>0.

Exponent

The above listed properties of the logarithm ensure that there is an inverse function, denoted provisionally by E(x), which is inverse to \ell:\ \ell(E(x))=x. This function is defined for all real x\in\mathbb R, takes positive values and transforms the addition to the multiplication: E(\lambda+x)=E(\lambda)\cdot E(x). Denoting the the value E(1) by e, we conclude that E(n)=e^n for all n\in\mathbb Z, and E(x)=e^x for all rational values x=\frac pq. Thus the function E(x), defined as the inverse to \ell, gives interpolation of the exponent for all real arguments. A simple calculation shows that E(x) satisfies the differential equation y'=y with the initial condition y(0)=1.

Computation

Consider the integral operator \Phi which sends any (continuous) function f:\mathbb R\to\mathbb R to the function g=\Phi(f) defined by the formula \displaystyle g(x)=f(0)+\int_0^x f(t)\,\mathrm dt. Applying this operator to the function E(x) and using the differential equation, we see that E is a “fixed point” of the transformation \Phi: \Phi(E)+E. This suggests using the following approach to compute the function E: choose a function f_0 and build the sequence of functions f_n=\Phi(f_{n-1}), n=1,2,3,4,\dots. If there exists a limit f_*=\lim f_{n+1}=\lim \Phi(f_n)=\Phi(f_*), then this limit is a fixed point for \Phi.

Note that the action of $\Phi$ can be very easily calculated on the monomials: \displaystyle \Phi\biggl(\frac{x^k}{k!}\biggr)=\frac{x^{k+1}}{(k+1)!} (check it!). Therefore if we start with f_0(x)=1, we obtain the functions $\latex f_n=1+x+\frac12 x^2+\cdots+\frac1{n!}x^n$. This sequence converges to the sum of the infinite series \displaystyle\sum_{n=0}^\infty\frac1{n!}x^n which represents the solution E(x) on the entire real line (check that). This series can be used for a fast approximate calculation of the number e=E(1)=\sum_0^\infty \frac1{n!}.

Differential equations in the complex domain

The function E(ix)=e^{ix} satisfies the differential equation y'=\mathrm iy. The corresponding “motion on the complex plane”, x\mapsto e^{\mathrm ix}, is rotation along the (unit) circle with the unit (absolute) speed, hence the real and imaginary parts of e^{\mathrm ix} are cosine and sine respectively. In fact, the “right” definition of them is exactly like that,

\displaystyle \cos x=\textrm{Re}\,e^{\mathrm ix},\quad \sin x=\textrm{Im}\,e^{\mathrm ix} \iff e^{\mathrm ix}=\cos x+\mathrm i\sin x,\qquad x\in\mathbb R.

Thus, the Euler formula “cis” in fact is the definition of sine and cosine. Of course, it can be “proved” by substituting the imaginary value into the Taylor series for the exponent, collecting the real and imaginary parts and comparing them with the Taylor series for the sine and cosine.

In fact, both sine and cosine are in turn solutions of the real differential equations: derivating the equation y'=\mathrm iy, one concludes that y''=\mathrm i^2y=-y. It can be used to calculate the Taylor coefficients for sine and cosine.

For more details see the lecture notes.


Not completely covered in the class: solution of linear equations with constant coefficients and resonances.

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Wednesday, February 1, 2012

Last lecture: January 31, 2012

Integral: antiderivative and area

The last lecture (only partially exposed in the class) deals with the two seemingly unrelated problem: how to antidifferentiate functions (i.e., how to find a function when its derivative is known) and how to compute areas, in particular, under the graph of a given nonlinear function.

The answers turn out to be closely related by the famous Newton-Leibniz formula, which expresses the undergraph area through the antiderivative (primitive) of the function.

We discuss some tricks which allow to read the table of the derivatives from right to left (how to invert the Leibniz rule?) and find out that not all anterivatives can be “explicitly computed”. This “non-computability”, however, has its bright side: among “non-computable” antiderivatives we find functions which possess very special and useful properties, like the primitive of the power x^{-1}=\frac1x, which transforms multiplication into addition.

The lecture notes are available here.

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