How to determine the periodicity of a function. Periodic functions Periodic function of time

Goal: summarize and systematize students’ knowledge on the topic “Periodicity of Functions”; develop skills in applying the properties of a periodic function, finding the smallest positive period of a function, constructing graphs of periodic functions; promote interest in studying mathematics; cultivate observation and accuracy.

Equipment: computer, multimedia projector, task cards, slides, clocks, tables of ornaments, elements of folk crafts

“Mathematics is what people use to control nature and themselves.”
A.N. Kolmogorov

During the classes

I. Organizational stage.

Checking students' readiness for the lesson. Report the topic and objectives of the lesson.

II. Checking homework.

We check homework using samples and discuss the most difficult points.

III. Generalization and systematization of knowledge.

1. Oral frontal work.

Theory issues.

1) Form a definition of the period of the function
2) Name the smallest positive period of the functions y=sin(x), y=cos(x)
3). What is the smallest positive period of the functions y=tg(x), y=ctg(x)
4) Using a circle, prove the correctness of the relations:

y=sin(x) = sin(x+360º)
y=cos(x) = cos(x+360º)
y=tg(x) = tg(x+18 0º)
y=ctg(x) = ctg(x+180º)

tg(x+π n)=tgx, n € Z
ctg(x+π n)=ctgx, n € Z

sin(x+2π n)=sinx, n € Z
cos(x+2π n)=cosx, n € Z

5) How to plot a periodic function?

Oral exercises.

1) Prove the following relations

a) sin(740º) = sin(20º)
b) cos(54º) = cos(-1026º)
c) sin(-1000º) = sin(80º)

2. Prove that an angle of 540º is one of the periods of the function y= cos(2x)

3. Prove that an angle of 360º is one of the periods of the function y=tg(x)

4. Transform these expressions so that the angles included in them do not exceed 90º in absolute value.

a) tg375º
b) ctg530º
c) sin1268º
d) cos(-7363º)

5. Where did you come across the words PERIOD, PERIODICITY?

Student answers: A period in music is a structure in which a more or less complete musical thought is presented. A geological period is part of an era and is divided into epochs with a period from 35 to 90 million years.

Half-life of a radioactive substance. Periodic fraction. Periodicals are printed publications that appear within strictly defined deadlines. Mendeleev's periodic system.

6. The figures show parts of the graphs of periodic functions. Determine the period of the function. Determine the period of the function.

Answer: T=2; T=2; T=4; T=8.

7. Where in your life have you encountered the construction of repeating elements?

Student answer: Elements of ornaments, folk art.

IV. Collective problem solving.

(Solving problems on slides.)

Let's consider one of the ways to study a function for periodicity.

This method avoids the difficulties associated with proving that a particular period is the smallest, and also eliminates the need to touch upon questions about arithmetic operations on periodic functions and the periodicity of a complex function. The reasoning is based only on the definition of a periodic function and on the following fact: if T is the period of the function, then nT(n?0) is its period.

Problem 1. Find the smallest positive period of the function f(x)=1+3(x+q>5)

Solution: Assume that the T-period of this function. Then f(x+T)=f(x) for all x € D(f), i.e.

1+3(x+T+0.25)=1+3(x+0.25)
(x+T+0.25)=(x+0.25)

Let's put x=-0.25 we get

(T)=0<=>T=n, n € Z

We have obtained that all periods of the function in question (if they exist) are among the integers. Let's choose the smallest positive number among these numbers. This 1 . Let's check whether it will actually be a period 1 .

f(x+1) =3(x+1+0.25)+1

Since (T+1)=(T) for any T, then f(x+1)=3((x+0.25)+1)+1=3(x+0.25)+1=f(x ), i.e. 1 – period f. Since 1 is the smallest of all positive integers, then T=1.

Problem 2. Show that the function f(x)=cos 2 (x) is periodic and find its main period.

Problem 3. Find the main period of the function

f(x)=sin(1.5x)+5cos(0.75x)

Let us assume the T-period of the function, then for any X the ratio is valid

sin1.5(x+T)+5cos0.75(x+T)=sin(1.5x)+5cos(0.75x)

If x=0, then

sin(1.5T)+5cos(0.75T)=sin0+5cos0

sin(1.5T)+5cos(0.75T)=5

If x=-T, then

sin0+5cos0=sin(-1.5T)+5cos0.75(-T)

5= – sin(1.5T)+5cos(0.75T)

sin(1.5T)+5cos(0.75T)=5

– sin(1.5T)+5cos(0.75T)=5

Adding it up, we get:

10cos(0.75T)=10

2π n, n € Z

Let us choose the smallest positive number from all the “suspicious” numbers for the period and check whether it is a period for f. This number

f(x+)=sin(1.5x+4π )+5cos(0.75x+2π )= sin(1.5x)+5cos(0.75x)=f(x)

This means that this is the main period of the function f.

Problem 4. Let’s check whether the function f(x)=sin(x) is periodic

Let T be the period of the function f. Then for any x

sin|x+Т|=sin|x|

If x=0, then sin|Т|=sin0, sin|Т|=0 Т=π n, n € Z.

Let's assume. That for some n the number π n is the period

the function under consideration π n>0. Then sin|π n+x|=sin|x|

This implies that n must be both an even and an odd number, but this is impossible. Therefore, this function is not periodic.

Task 5. Check if the function is periodic

f(x)=

Let T be the period of f, then

, hence sinT=0, Т=π n, n € Z. Let us assume that for some n the number π n is indeed the period of this function. Then the number 2π n will be the period

Since the numerators are equal, their denominators are equal, therefore

This means that the function f is not periodic.

Work in groups.

Tasks for group 1.

Tasks for group 2.

Check if the function f is periodic and find its fundamental period (if it exists).

f(x)=cos(2x)+2sin(2x)

Tasks for group 3.

At the end of their work, the groups present their solutions.

VI. Summing up the lesson.

Reflection.

The teacher gives students cards with drawings and asks them to color part of the first drawing in accordance with the extent to which they think they have mastered the methods of studying a function for periodicity, and in part of the second drawing - in accordance with their contribution to the work in the lesson.

VII. Homework

1). Check if the function f is periodic and find its fundamental period (if it exists)

b). f(x)=x 2 -2x+4

c). f(x)=2tg(3x+5)

2). The function y=f(x) has a period T=2 and f(x)=x 2 +2x for x € [-2; 0]. Find the value of the expression -2f(-3)-4f(3.5)

Literature/

Mordkovich A.G. Algebra and beginnings of analysis with in-depth study.
Mathematics. Preparation for the Unified State Exam. Ed. Lysenko F.F., Kulabukhova S.Yu.
Sheremetyeva T.G. , Tarasova E.A. Algebra and beginning analysis for grades 10-11.

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Slide captions:

Algebra and beginnings of analysis, grade 10 (profile level) A.G. Mordkovich, P.E. Semenov Teacher Volkova S.E.

Definition 1 A function y = f (x), x ∈ X is said to have period T if for any x ∈ X the equality f (x – T) = f (x) = f (x + T) holds. If a function with period T is defined at point x, then it is also defined at points x + T, x – T. Any function has a period equal to zero at T = 0, we get f(x – 0) = f(x) = f( x + 0) .

Definition 2 A function that has a non-zero period T is called periodic. If a function y = f (x), x ∈ X has a period T, then any number that is a multiple of T (that is, a number of the form kT, k ∈ Z) is also its period.

Proof Let 2T be the period of the function. Then f(x) = f(x + T) = f((x + T) +T) = f(x +2T), f(x) = f(x - T) = f((x - T) -T) = f(x - 2T). Similarly, it is proved that f(x) = f(x + 3 T) = f(x - 3 T), f(x) = f(x + 4 T) = f(x - 4 T), etc. So f(x - kT) = f(x) = f(x + kT)

The smallest period among the positive periods of a periodic function is called the main period of this function.

Features of the graph of a periodic function If T is the main period of the function y = f(x), then it is enough to: construct a branch of the graph on one of the intervals of length T, carry out a parallel translation of this branch along the x axis by ±T, ±2T, ±3T, etc. . Usually a gap is chosen with ends at points

Properties of periodic functions 1. If f(x) is a periodic function with period T, then the function g(x) = A f(kx + b), where k > 0, is also periodic with period T 1 = T/k. 2. Let the function f 1 (x) and f 2 (x) be defined on the entire numerical axis and be periodic with periods T 1 > 0 and T 2 >0. Then, for T 1 /T 2 ∈ Q, the function f(x) = f(x) + f 2 (x) is a periodic function with period T equal to the least common multiple of the numbers T 1 and T 2.

Examples 1. The periodic function y = f(x) is defined for all real numbers. Its period is 3 and f(0) =4. Find the value of the expression 2f(3) – f(-3). Solution. Т = 3, f(3) =f(0+3) = 4, f(-3) = f(0–3) =4, f(0) = 4. Substituting the obtained values into the expression 2f(3) - f(-3) , we get 8 - 4 =4 . Answer: 4.

Examples 2. The periodic function y = f(x) is defined for all real numbers. Its period is 5, and f(-1) = 1. Find f(-12) if 2f(3) – 5f(9) = 9. Solution T = 5 F(-1) = 1 f(9) = f(-1 +2T) = 1⇨ 5f(9) = 5 2f(3) = 9 + 5f(9) = 14 ⇨f(3)= 7 F(-12) = f(3 – 3T) = f (3) = 7 Answer:7.

Used literature A.G. Mordkovich, P.V. Semenov. Algebra and beginnings of analysis (profile level), grade 10 A.G. Mordkovich, P.V. Semenov. Algebra and beginning of analysis (profile level), 10th grade. Methodological manual for teachers

On the topic: methodological developments, presentations and notes

Periodic law and periodic system D.I. Mendeleev.

A comprehensive lesson on this topic is conducted in the form of a game, using elements of technology from pedagogical workshops....

Extracurricular event "Periodic law and periodic system of chemical elements of D.I. Mendeleev"

An extracurricular activity reveals the history of the creation of the periodic law and the periodic system by D.I. Mendeleev. The information is presented in poetic form, which facilitates quick memorization...

Appendix to the extracurricular activity "Periodic law and the periodic system of chemical elements of D.I. Mendeleev"

The discovery of the law was preceded by long and intense scientific work by D.I. Mendeleev for 15 years, and its further deepening was given another 25 years....

Equipment: computer, multimedia projector, task cards, slides, clocks, tables of ornaments, elements of folk crafts

“Mathematics is what people use to control nature and themselves.”
A.N. Kolmogorov

During the classes

I. Organizational stage.

Checking students' readiness for the lesson. Report the topic and objectives of the lesson.

II. Checking homework.

We check homework using samples and discuss the most difficult points.

III. Generalization and systematization of knowledge.

1. Oral frontal work.

Theory issues.

y=sin(x) = sin(x+360º)
y=cos(x) = cos(x+360º)
y=tg(x) = tg(x+18 0º)
y=ctg(x) = ctg(x+180º)

tg(x+π n)=tgx, n € Z
ctg(x+π n)=ctgx, n € Z

sin(x+2π n)=sinx, n € Z
cos(x+2π n)=cosx, n € Z

5) How to plot a periodic function?

Oral exercises.

1) Prove the following relations

a) sin(740º) = sin(20º)
b) cos(54º) = cos(-1026º)
c) sin(-1000º) = sin(80º)

2. Prove that an angle of 540º is one of the periods of the function y= cos(2x)

3. Prove that an angle of 360º is one of the periods of the function y=tg(x)

4. Transform these expressions so that the angles included in them do not exceed 90º in absolute value.

a) tg375º
b) ctg530º
c) sin1268º
d) cos(-7363º)

5. Where did you come across the words PERIOD, PERIODICITY?

Half-life of a radioactive substance. Periodic fraction. Periodicals are printed publications that appear within strictly defined deadlines. Mendeleev's periodic system.

6. The figures show parts of the graphs of periodic functions. Determine the period of the function. Determine the period of the function.

Answer: T=2; T=2; T=4; T=8.

7. Where in your life have you encountered the construction of repeating elements?

Student answer: Elements of ornaments, folk art.

IV. Collective problem solving.

(Solving problems on slides.)

Let's consider one of the ways to study a function for periodicity.

Problem 1. Find the smallest positive period of the function f(x)=1+3(x+q>5)

Solution: Assume that the T-period of this function. Then f(x+T)=f(x) for all x € D(f), i.e.

1+3(x+T+0.25)=1+3(x+0.25)
(x+T+0.25)=(x+0.25)

Let's put x=-0.25 we get

(T)=0<=>T=n, n € Z

f(x+1) =3(x+1+0.25)+1

Since (T+1)=(T) for any T, then f(x+1)=3((x+0.25)+1)+1=3(x+0.25)+1=f(x ), i.e. 1 – period f. Since 1 is the smallest of all positive integers, then T=1.

Problem 2. Show that the function f(x)=cos 2 (x) is periodic and find its main period.

Problem 3. Find the main period of the function

f(x)=sin(1.5x)+5cos(0.75x)

Let us assume the T-period of the function, then for any X the ratio is valid

sin1.5(x+T)+5cos0.75(x+T)=sin(1.5x)+5cos(0.75x)

If x=0, then

sin(1.5T)+5cos(0.75T)=sin0+5cos0

sin(1.5T)+5cos(0.75T)=5

If x=-T, then

sin0+5cos0=sin(-1.5T)+5cos0.75(-T)

5= – sin(1.5T)+5cos(0.75T)

sin(1.5T)+5cos(0.75T)=5

– sin(1.5T)+5cos(0.75T)=5

Adding it up, we get:

10cos(0.75T)=10

2π n, n € Z

Let us choose the smallest positive number from all the “suspicious” numbers for the period and check whether it is a period for f. This number

f(x+)=sin(1.5x+4π )+5cos(0.75x+2π )= sin(1.5x)+5cos(0.75x)=f(x)

This means that this is the main period of the function f.

Problem 4. Let’s check whether the function f(x)=sin(x) is periodic

Let T be the period of the function f. Then for any x

sin|x+Т|=sin|x|

If x=0, then sin|Т|=sin0, sin|Т|=0 Т=π n, n € Z.

Let's assume. That for some n the number π n is the period

the function under consideration π n>0. Then sin|π n+x|=sin|x|

This implies that n must be both an even and an odd number, but this is impossible. Therefore, this function is not periodic.

Task 5. Check if the function is periodic

f(x)=

Let T be the period of f, then

, hence sinT=0, Т=π n, n € Z. Let us assume that for some n the number π n is indeed the period of this function. Then the number 2π n will be the period

Since the numerators are equal, their denominators are equal, therefore

This means that the function f is not periodic.

Work in groups.

Tasks for group 1.

Tasks for group 2.

Check if the function f is periodic and find its fundamental period (if it exists).

f(x)=cos(2x)+2sin(2x)

Tasks for group 3.

At the end of their work, the groups present their solutions.

VI. Summing up the lesson.

Reflection.

VII. Homework

1). Check if the function f is periodic and find its fundamental period (if it exists)

b). f(x)=x 2 -2x+4

c). f(x)=2tg(3x+5)

2). The function y=f(x) has a period T=2 and f(x)=x 2 +2x for x € [-2; 0]. Find the value of the expression -2f(-3)-4f(3.5)

Literature/

Mordkovich A.G. Algebra and beginnings of analysis with in-depth study.
Mathematics. Preparation for the Unified State Exam. Ed. Lysenko F.F., Kulabukhova S.Yu.
Sheremetyeva T.G. , Tarasova E.A. Algebra and beginning analysis for grades 10-11.

Repeating its values at some regular argument interval, that is, not changing its value when adding some fixed non-zero number to the argument ( period functions) over the entire domain of definition.

More formally speaking, the function is called periodic with period T ≠ 0 (\displaystyle T\neq 0), if for each point x (\displaystyle x) from its domain of definition of the point x + T (\displaystyle x+T) And x − T (\displaystyle x-T) also belong to its domain of definition, and for them the equality holds f (x) = f (x + T) = f (x − T) (\displaystyle f(x)=f(x+T)=f(x-T)).

Based on the definition, the equality is also true for a periodic function f (x) = f (x + n T) (\displaystyle f(x)=f(x+nT)), Where n (\displaystyle n)- any integer.

However, if a set of periods ( T , T > 0 , T ∈ R ) (\displaystyle $T,T>0,T\in \mathbb (R) $) there is a smallest value, then it is called main (or main) period functions.

Examples

Sin ⁡ (x + 2 π) = sin ⁡ x, cos ⁡ (x + 2 π) = cos ⁡ x, ∀ x ∈ R. (\displaystyle \sin(x+2\pi)=\sin x,\;\cos(x+2\pi)=\cos x,\quad \forall x\in \mathbb (R) .)

The Dirichlet function is periodic; its period is any non-zero rational number. It also does not have a main period.

Some features of periodic functions

And T 2 (\displaystyle T_(2))(however, this number will simply be a period). For example, the function f (x) = sin ⁡ (2 x) − sin ⁡ (3 x) (\displaystyle f(x)=\sin(2x)-\sin(3x)) the main period is 2 π (\displaystyle 2\pi ), at the function g (x) = sin ⁡ (3 x) (\displaystyle g(x)=\sin(3x)) the period is equal to 2 π / 3 (\displaystyle 2\pi /3), and their sum f (x) + g (x) = sin ⁡ (2 x) (\displaystyle f(x)+g(x)=\sin(2x)) the main period is obviously equal to π (\displaystyle \pi ).

The sum of two functions with incommensurable periods is not always a non-periodic function.

UDC 517.17+517.51

PERIOD OF THE SUM OF TWO PERIODIC FUNCTIONS

A/O. Evnin

The work completely solves the question of what the main period of a periodic function, which is the sum of two periodic functions with known main periods, can be. The case of the absence of a main period for a periodic sum of periodic functions is also studied.

We consider real-valued functions of a real variable. In the encyclopedic edition, in the article “Periodic Functions,” you can read: “The sum of periodic functions with different periods is periodic only if their periods are commensurate.” This statement is true for continuous functions1, but does not hold in the general case. A counterexample of a very general form was constructed in . In this article we find out what the main period of a periodic function, which is the sum of two periodic functions with known main periods, can be.

Preliminary information

Recall that a function / is said to be periodic if for a certain number T F O for any x from the domain of definition D(f) the numbers x + T and x - T belong to D(f) and the equalities f(x + T) = f( x) =f(x ~ T). In this case, the number Г is called the period of the function.

We will call the smallest positive period of the function (if, of course, it exists) the main period. The following fact is known.

Theorem 1. If a function has a main period To, then any period of the function has the form nTo, where n Ф 0 is an integer.

The numbers T\ and T2 are said to be commensurable if there is a number T0 that fits into both T\ and T2 an integer number of times: T\ = T2 = n2T0, n2e Z. Otherwise, the numbers T\ and T2 are called incommensurable. The commensurability (incommensurability) of periods means, therefore, that their ratio is a rational (irrational) number.

From Theorem 1 it follows that for a function that has a fundamental period, any two periods are commensurate.

A classic example of a function that does not have the smallest period is the Dirichlet function, which is equal to 1 at rational points and zero at irrational points. Any rational number other than zero is the period of the Dirichlet function, and any irrational number is not its period. As we see, here too any two periods are comparable.

Let us give an example of a non-constant periodic function that has incommensurable periods.

Let the function /(x) be equal to 1 at points of the form /u + la/2, m, n e Z, and equal to

zero. Among the periods of this function there are 1 and l

Period of a sum of functions with commensurate periods

Theorem 2. Let fug be periodic functions with main periods mT0 and “That, where the type

Mutually prime numbers. Then the main period of their sum (if it exists) is equal to -

where k is a natural number coprime to the number mn.

Proof. Let h = / + g. Obviously, the number mnT0 is the period of h. By virtue of

of Theorem 1, the main period h has the form where k is some natural number. Presumably

Let us assume that k is not relatively prime with the number m, that is, k - dku m = dm\, where d> 1 is the most

1 A beautiful proof that the sum of any finite number of continuous functions with pairwise incommensurable periods is non-periodic is contained in the article See also.

greater common divisor of the numbers m and k. Then the period of the function k is equal to

and the function f=h-g

has a period mxnTo, which is not a multiple of its main period mTQ. A contradiction with Theorem 1 is obtained. This means that k is coprime with m. Similarly, the numbers k and n are coprime. Thus, A: is coprime with m. □

Theorem 3. Let m, n and k be pairwise coprime numbers, and T0 be a positive number. Then there exist periodic functions fug such that the main periods f, g and (f + g) are

we are respectively tT$, nTQ and -

Proof. The proof of the theorem will be constructive: we will simply construct a corresponding example. Let us first formulate the following result. Statement. Let m be relatively prime numbers. Then the functions

fx - cos- + cos--- and f2= cos- m n m

cos- have a fundamental period of 2ktp. P

Proof of the statement. Obviously, the number 2ptn is the period of both functions. You can easily check that this period is the main one for the function. Let's find its maximum points.

x = 2lM, te Z.

We have = n!. From the mutual simplicity of the type it follows that 5 is a multiple of /r, i.e. i = I e b. This means that /x(x) = 2 o x = 2mstp1,1 e 2, and the distance between neighboring points of maximum of the function /\ is equal to 2ktp, and the positive period /1 cannot be less than the number 2 spp.

For the function, we apply reasoning of a different kind (which is also suitable for the function but

less elementary). As Theorem 1 shows, the main period Г of function/2 has the form -,

where k is some natural number coprime to type. The number G will also be the period of the function

(2 ^ 2 xn g t t /2 + /2 = - -1 cos

all periods of which have the form 2pp1. So,

2nnl, i.e. t = kl. Since t and k are mutually

sty, it follows that k = 1.

Now, to prove Theorem 3, we can construct the required example. Example. Let m, n and k be pairwise relatively prime numbers and at least one of the numbers n or k is different from 1. Then pf k and by virtue of the proven statement of the function

/ (x) = cos--- + cos- t to

And g(x) = cos-cos - p to

have main periods of 2 ltk and 2 tk respectively, and their sum

k(x) = f(x) + = cos- + cos-

the main period is 2 ttp.

If n = k = 1, then a pair of functions will do

f(x)-2 cos- + COS X and g(x) - COS X. m

Their main periods, as well as the period of the function k(x) - 2, are equal to 2lm, 2/gi 2type, respectively.

how easy it is to check.

Mathematics

Let's denote T = 2lx. For arbitrary pairwise coprime numbers mn, n and k, functions f and £ are indicated such that the main periods of the functions f, g and f + g are equal to mT, nT and

The conditions of the theorem are satisfied by the functions / - n;

Period of a sum of functions with incommensurable periods

The next statement is almost obvious.

Theorem 4. Let fug be periodic functions with incommensurable main periods T) and T2, and the sum of these functions h = f + g is periodic and has a main period T. Then the number T is incommensurable with neither T] nor T2.

Proof. On the one hand, if the numbers TnT) are commensurable, then the function g = h-f has a period commensurate with Г]. On the other hand, by virtue of Theorem 1, any period of the function g is a multiple of the number T2. We obtain a contradiction with the incommensurability of the numbers T\ and T2. The incommensurability of the numbers T and T2 is proved in a similar way, d

A remarkable, and even somewhat surprising, fact is that the converse of Theorem 4 is also true. There is a widespread misconception that the sum of two periodic functions with incommensurable periods cannot be a periodic function. In fact, this is not so. Moreover, the period of the sum can be any positive number that satisfies the statement of Theorem 4.

Theorem 5. Let T\, T2 and T~ be pairwise incommensurable positive numbers. Then there exist periodic functions fug such that their sum h =/+ g is periodic, and the main periods of the function f guh are equal to Th T2 and T, respectively.

Proof. The proof will again be constructive. Our constructions will significantly depend on whether the number T is representable or not in the form of a rational combination T = aT\ + pT2 (a and P are rational numbers) of the periods T\ and T2.

I. T is not a rational combination of Tg and J2-

Let A = (mT\ + nT2 + kT\m,n, k ∈ Z) be the set of integer linear combinations of the numbers T1 T2 and T. We note immediately that if a number is representable in the form mT\ + nT2 + kT, then such a representation is unique . Indeed, if mxT\ + n\Tg + k\T- m2Tx + n2T2 + k2T9 then

(k) - k2)T - (ot2 - m\)T] + (n2 - π)Тъ and for k\ * k2 we obtain that T is rationally expressed through T] and T2. This means k\ = k2. Now, from the incommensurability of the numbers T\ and T2, the equalities m\ = m2 and u = n2 are immediately obtained.

An important fact is that the sets A and its complement A are closed under the addition of numbers from A: if x e A and y e A, then x + y e A; if x e A and y e A, then x + y e A.

Let us assume that at all points of the set A the functions / and g are equal to zero, and on the set A we define these functions as follows:

f(mTi + nT2 + kT) = nT2 + kT g(mT1 + nT2 + kT) - gnT\ - kT.

Since, as has been shown, from the number x e A the coefficients m, peak of the linear combination of the periods T1 T2 and T are uniquely restored, the indicated assignments of the functions / and g are correct.

The function h =/ + g on set A is equal to zero, and at points of set A it is equal to

h(mT\ + nT2 + kT) - mT\ + nT2.

By direct substitution it is easy to verify that the number T\ is the period of the function f, the number T2 is the period of g, and T~ is the period of h. Let us show that these periods are the main ones.

First, we note that any period of the function / belongs to the set A. Indeed,

if 0 fx in A,y e A, then ox + y e A and f(x + y) = 0 *f(x). This means that y e A is not the period of the function /

Now let x2 be unequal numbers and f(x 1) ~f(x2). From the definition of the function / we get from here that x\ - x2 = 1Т where I is some non-zero integer. Therefore, any period of the function is a multiple of T\. Thus, Tx is really the main period/

Statements regarding T2 and T are checked in the same way.

Comment. In the book on p. 172-173 another general construction for case I is given.

II. T is a rational combination of T\ and T2.

Let us present a rational combination of periods T\ and T2 in the form Г = - (кхТх + к2Т2), where кх and

k2 ™ coprime integers, k(Γ\ + k2T2 > 0, a/? and d are natural numbers. Let us introduce leZ>.

reni set B----

Let us assume that at all points of set B the functions f and g are equal to zero, and on set B we define these functions as follows:

^ mT\ + nT2 L I

^ mTx + nT2 L

Here, as usual, [x] and (x) denote the integer and fractional parts of numbers, respectively. The function k =/+ d on set B is equal to zero, and at points of set B it is equal to

fmTx +pT: l H

By direct substitution it is easy to verify that the number Tx is the period of the function /, the number T2 is the period g, and T is the period h. Let us show that these periods are the main ones.

Any period of the function / belongs to the set B. Indeed, if 0 * x e B, y e B, then f(x) Ф 0, j(x + y) = 0 */(*)■ Hence, y e B _ Not function period/

So, every period of the function / has the form Тy =

Where 5i and 52 are integers. Let

x = -7] 4- -Г2, x e 5. If i = 0, then f(i) is a rational number. Now from the rationality of the number /(x + 7)) the equality -I - I - 0 follows. This means that we have the equality 52 = Xp, where X is some integer

number. The relation /(x + 7)) = /(x) takes the form

^P + I + I w +

This equality must hold for all integer types. At t-n~ 0, the right side of (1) is equal to

to zero. Since the fractional parts are non-negative, we get from this that -<0, а при

m = n = d - ] the sum of the fractional parts on the right side of equality (1) is not less than the sum of the fractional parts h-X

tey on the left. This means - >0. Thus, X = 0 and 52 = 0. Therefore, the period of the function / has the form

and equality (1) becomes

n\ | and 52 are integers. From the relations

th(0) = 0 = th(GA) =

we find that the numbers 51 and ^ must be multiples of p, i.e. for some integers Ax and A2 we have 51 = A\p, E2 = A2p. Then relation (3) can be rewritten as

From the equality A2kx = k2A\ and the mutual primeness of the numbers k\ and k2, it follows that A2 is divisible by k2. From here

for some integer t the equalities A2 = k2t and Ax ~ kxt are valid, i.e. Th ~-(kxTx + k2T2).

It is shown that any period of the function h is a multiple of the period T = - (k(Gx + k2T2)9 which, thus

zom, is the main one. □

No main period

Theorem 6. Let Tx and T2~ be arbitrary positive numbers. Then there exist periodic functions fug such that their main periods are equal to T\ and T2, respectively, and their sum h=f+g is periodic, but has no main period.

Proof. Let's consider two possible cases.

I. The periods Tx and T2 are incommensurable.

Let A = + nT2 +kT\ . As above, it is easy to show that if the number

can be represented in the form mTx + nT2 + kT, then such a representation is unique.

Let us assume that at all points of the set A the functions / and g are equal to zero, and on the set A we define these functions as follows:

/from; + nT2 + kT) = nT2 + kT, g(mTx + nT2 + kT) = mTx - kT.

It is easy to verify that the number Tx is the main period of the function /, the number T2 is the main period g, and for any rational k, the number kT is the period of the function h - f + g, which, therefore, does not have the smallest period.

II. The periods Tx and T2 are comparable.

Let Tx = mT0, T2 = nT0, where T0 > O, m and n are natural numbers. Let us introduce the set I = + into consideration.

Let us assume that at all points of set B the functions fug are equal to zero, and on set B we define these functions as follows:

/((/ + ShT0) = Shch + Jit, g((/ + 4lk)T0) - Shch - 42k.

The function h ~ / + g on set B is equal to zero, and at points of set B it is equal to

It is easy to check that the number 7j = mTQ is the main period of the function /, the number T2 ~ nT0 is the main period of g, while among the periods of the function h~ f + g there are all numbers of the form l/2kT0, where k is an arbitrary rational number. □

The constructions proving Theorem 6 are based on the incommensurability of the periods of the function h~ / + g with the periods of the functions / and g. In conclusion, let us give an example of functions fug such that all periods of the functions /, g and / + g are commensurate with each other, but / and g have basic periods, while f + g do not.

Let m be some fixed natural number, M the set of irreducible non-integer fractions whose numerators are multiples of m. Let's put

1 if heM; 1

ifhe mZ;

EcnuxeZXmZ; 2

O in other cases; 1 if xeMU

~,ifhe2 2

[Oh otherwise.

It is easy to see that the main periods of the functions fug are equal to m and 1, respectively, while the sum / + g has a period of any number of the form m/n, where n is an arbitrary natural number coprime to m.

Literature

1. Mathematical encyclopedic dictionary/Ch. ed. Yu.V. Prokhorov - M.: Sov. encyclopedia, 1988.

2. Mikaelyan L.V., Sedrakyan N.M. On the periodicity of the sum of periodic functions//Mathematical education. - 2000. - No. 2(13). - pp. 29-33.

3. Gerenshtein A.B., Evnin A.Yu. On the sum of periodic functions // Mathematics at school. -2002. - No. 1. - P. 68-72.

4. Ivlev B.M. and others. Collection of problems on algebra and principles of analysis for grades 9 and 10. - M.: Education, 1978.