5. Charles Henry Edwards, David E. Penny. Differential Equations and the problem of eigenvalues: Modeling and calculation using Mathematica, Maple and MATLAB = Differential Equations and Boundary Value Problems: Computing and Modeling. – 3rd ed. – M.: "Williams", 2007.
6. Elsholts L. E. Differential Equations and Calculus of Variations. – M.: Science, 1969.
SOME OPERATIONS AND SPECIAL CASES OF MATHEMATICAL ANALYSIS IN THE EXPONENTIAL SET
Aliev Ibratjon Khatamovich
2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University
Ferghana State University, Ferghana, Uzbekistan
Аннотация. Важность определения и преобразования ингенциальных чисел и настоящего множества с каждым днём становится всё более очевидном, особенно с входом данного понятия в математическую физику, но и как чисто математический объект они представляют не малый интерес, хотя при этом имеют и практическое применение. В настоящей работе, описаны методы проведения некоторых алгебраических операций с ними, в том числе с использованием формулы Эйлера и интеграллами.
Ключевые слова: ингенциальные числа, математический анализ, алгебраические операции, формула Эйлера, интегрирование, производные.
Annotation. The importance of defining and converting exponential numbers and a real set is becoming more and more obvious every day, especially with the entry of this concept into mathematical physics, but as a purely mathematical object they are of no small interest, although they also have practical applications. In this paper, methods of performing some algebraic operations with them are described, including using Euler’s formula and integrals.
Keywords: inertial numbers, mathematical analysis, algebraic operations, Euler formula, integration, derivatives.
The very process of logarithmization of an exponential number of a general form can be seen in (1).
Thus, when logarithming, 2 parts of the expression itself are formed – the real one, as the natural logarithm of the coefficient of the ingential part and the logarithm of the ingential unit, which is defined in (2).
That is, in this case, the question arises to what degree it is necessary to raise the Euler number so that it gives an exponential unit. The answer is quite simple – it is a negative logarithm of zero (2) from this it follows that the logarithm of the exponential number is (3).
It is also interesting to solve the Euler equation with a tangential unit, and then with a general form of an exponential number, which was described further, taking the expressions as unknowns. And for this, we can initially proceed from Taylor expansions (4—6).
Which is easily proved, since when the unknown is zeroed, the sine in (5) is also zeroed, and the cosine in (6) is equal to one. And it already follows from this (7).
And the unknown in (7) can be all kinds of numbers, both complex, when substituting which the remarkable Euler equality follows, and exponential. And to begin with, let’s consider a special case with an exponential unit and perform the following transformations (8).
Based on this relation, we perform transformations in (9), leading to equation (10), while taking into account that this expression is identical, it is possible to differentiate both parts of the equation in (11) by performing the corresponding transformations.
Since the final equality (11) can be represented as in (12), further carrying out additional differentiation, also introducing the condition that this is an identity, and in (13) the differentiation process for the right side of equality is described in detail. And for the left part there is no need for a detailed painting.
When the differentiation is made, it is enough to make elementary transformations, we get the trigonometric form of the special case (14).
Now, when the general form for the doubly differentiated case is obtained, it is necessary to return to the primordial ones, because this is the identity, resulting in the following equalities (15—16).
And indeed, this value is close to the most potential value, so this expression can be considered the second kind of writing of the exponential unit. Now, it is possible to proceed to the solution of the Euler equation for the general form of the intentional numbers, having carried out the first substitution and the usual replacement operations at stage (17) and (18) at the beginning.
When the necessary transformations come to an end, and other actions no longer take place, it is also sufficient to differentiate both parts of equality as a valid identity (19).
Differentiating the first part of the equality, we can come to the result in (20), and for the second part, the calculations will continue throughout (21).
Then, applying (22—25), one can come to the form (26).
As a result, it is enough to equalize both results in (20) and (26), since these are two parts of the identity, and then get (27) with the necessary simplification, and already in (28) with additional simplification and differentiation as an identity.
At the same time, the differentiation of the first part of equality is obvious in (29), as well as the second in (30), after which equality and the resulting transformations can be introduced into (31).
As a result, equalities are formed that need to be integrated twice, because their derivatives were taken earlier, getting (32).
Integrating the first part, a separate result is obtained in (33) and integrating the second part in (34).
Thus, it is possible to arrive at equality (35), from where it is possible to arrive at another equality in the same equation.
The result is really quite surprising, but this is equality (35), which came out after substituting the general form of an ingential number into Euler’s formula and the solution for this case is the ingential number (36). Thus, this is the first full-fledged equation, the solution of which was an intentional number.
Although the complex numbers themselves are located on the axis of numbers, this interval can also be expressed on the tangential plane. This coordinate system has an axis starting from infinity as the ordinate, and the abscissa has all real numbers. Thus, all exponential numbers can be represented on such a rectangular coordinate system, in the case of adding complex numbers – already in space.
Used literature
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2. G. Kane. Modern elementary particle physics. Publishing house Mir. 1990.
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ABOUT RESEARCH ON THE COLLATZ HYPOTHESIS IN THE FACE OF A MATHEMATICAL PHENOMENON
Aliev Ibratjon Khatamovich
2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University
Ferghana State University, Ferghana, Uzbekistan
Аннотация. Когда об этой задаче рассказывают молодым математикам – их сразу предупреждают, что не стоит браться за её решение, ибо это кажется невозможным. Простую на вид гипотезу не смогли доказать лучшие умы человечества. Для сравнения, знаменитый математик Пол Эрдеш сказал: «Математика ещё не созрела для таких вопросов». Однако, стоит подробнее изучить данную гипотезу, что и исследуется в настоящей работе.
Ключевые слова: гипотеза Коллатца, числа-градины, ряды, алгоритм, последовательности, доказательства.
Annotation. When young mathematicians are told about this problem, they are immediately warned that it is not worth taking up its solution, because it seems impossible. A simple-looking hypothesis could not be proved by the best minds of mankind. For comparison, the famous mathematician Paul Erdos said: «Mathematics is not yet ripe for such questions.» However, it is worth studying this hypothesis in more detail, which is investigated in this paper.
Keywords: Collatz hypothesis, hailstone numbers, series, algorithm, sequences, proofs.
In short, its essence is as follows. A certain number is selected and if it is not even, it is multiplied by 3 and 1 is added, if it is even, then divided by 2.
We can give an algorithm of this series for the number 7:
7 – 22 – 11 – 34 – 17 – 52 – 26 – 13 – 40 – 20 – 10 – 5 – 16 – 8 – 4 – 2 – 1
Next, a cycle is obtained:
