All sciences. №3, 2023. International Scientific Journal. Sultonali Mukaramovich Abduraxmonov
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).