An analytic function is given by a locally convergent power series. Both real and complex are infinitely differentiable, but there are some properties of the second that are true. A function f defined on an open subset U, R, or C is called analytic only if it is defined locally by a convergent power series.
Definition of this concept
Complex analytic functions: R (z)=P (z) / Q (z). Here P (z)=am zm + am-1 zm-1 + ⋯ + a1 z + a0 and Q (z)=bn zn + bn-1 zn-1 + ⋯ + b1 z + b0. Moreover, P (z) and Q (z) are polynomials with complex coefficients am, am-1, …, a1, a0, bn, bn-1, …, b1, b0.
Assume that am and bn are non-zero. And also that P(z) and Q(z) have no common factors. R (z) is differentiable at any point C → SC → S, and S is a finite set inside C for which the denominator of Q (z) vanishes. The maximum of two powers from the numerator and the power of the denominator is called the power of the rational function R(z), just like the sum of two and the product. In addition, it can be verified that the space satisfies the field axioms using these operations of addition and multiplication, and it is denoted by C(X). This is an important example.
Number concept for holomorphic values
The fundamental theorem of algebra allows us to calculate the polynomials P (z) and Q (z), P (Z)=am (z − z1) p1 (z − z2) p2….(z − zr) prP(Z)=am (z − z1) p1 (z − z2) p2….(z − zr) pr and Q (Z)=bn (z − s1) q1 (z − s2) q2….(z − sr) qr. Where the exponents denote the multiplicities of the roots, and this gives us the first of two important canonical forms for a rational function:
R (Z)=a m (z − z1) p1 (z − z2) p2….(z − zr) / p r bn(z−s1)q1(z−s2)q2….(z− sr)qr. Zeros z1, …, zr of the numerator are so called in a rational function, and s1, …, sr of the denominator are considered to be its poles. The order is its multiplicity, as the root of the above values. The fields of the first system are simple.
We will say that the rational function R (z) is correct if:
m=deg P (z) ≦≦ n=degF(o) Q (z) and strictly correct if m <n. If R(z) is not strictly eigenvalue then we can divide by the denominator to get R(z)=P1(z) + R1(z) where P1(z) is a polynomial and R1(z) remainder is strictly own rational function.
Analytic with differentiability
We know that any analytic function can be real or complex and the division is infinite, which is also called smooth, or C∞. This is the case for material variables.
When considering complex functions that are analytic and derivative, the situation is very different. It's easy to provethat in an open set any structurally differentiable function is holomorphic.
Examples of this function
Consider the following examples:
1). All polynomials can be real or complex. This is because for a polynomial of degree (highest) 'n', variables greater than n in the corresponding Taylor series expansion merge immediately into 0 and hence the series will converge trivially. Also, adding each polynomial is a Maclaurin series.
2). All exponential functions are also analytic. This is because all Taylor series for them will converge for all values that can be real or complex "x" very close to "x0" as in the definition.
3). For any open set in the respective domains, trigonometric, power and logarithmic functions are also analytic.
Example: find possible values i-2i=exp ((2) log (i))
Decision. To find the possible values of this function, we first see that, log? (i)=log? 1 + i arg? [Because (i)=0 + i pi2pi2 + 2ππki, for every k that belongs to the whole set. This gives, i-2i=exp? (ππ + 4ππk), for every k that belongs to the set of integers. This example shows that the complex quantity zαα can also have different values, infinitely similar to logarithms. Even though square root functions can only have a maximum of two values, they are also a good example of multivalued functions.
Properties of holomorphic systems
The theory of analytic functions is as follows:
1). Compositions, sums or products are holomorphic.
2). For an analytic function, its inverse, if it is not equal to zero at all, is similar. Also, the inverse derivative of which must not be 0 is again holomorphic.
3). This function is continuously differentiable. In other words, we can say that it is smooth. The converse is not true, that is, all infinitely differentiable functions are not analytic. This is because, in a sense, they are sparse compared to all opposites.
Holomorphic function with multiple variables
With the help of power series, these values can be used to determine the indicated system by several indicators. Analytic functions of many variables have some of the same properties as those with one variable. However, especially for complex measures, new and interesting phenomena emerge when working in 2 or more dimensions. For example, null sets of complex holomorphic functions in more than one variable are never discrete. The real and imaginary parts satisfy the Laplace equation. That is, in order to perform the analytical assignment of the function, the following values and theories are needed. If z=x + iy, then an important condition that f(z) is holomorphic is the fulfillment of the Cauchy-Riemann equations: where ux is the first partial derivative of u with respect to x. Therefore, it satisfies the Laplace equation. As well as a similar calculation showing the result v.
Characteristic of fulfillment of inequalities for functions
Conversely, given the harmonic variable, it is the real part of the holomorphic (at least locally). If the trial form, then the Cauchy-Riemann equations will be satisfied. This ratio does not determine ψ, but only its increments. It follows from the Laplace equation for φ that the integrability condition for ψ is satisfied. And, therefore, ψ can be given a linear denominator. It follows from the last requirement and Stokes' theorem that the value of a line integral connecting two points does not depend on the path. The resulting pair of solutions to the Laplace equation is called the conjugate harmonic functions. This construction is only valid locally or provided that the path does not cross a singularity. For example, if r and θ are polar coordinates. However, the angle θ is unique only in the region that does not cover the origin.
The close connection between the Laplace equation and the basic analytic functions means that any solution has derivatives of all orders and can be expanded in a power series, at least within a circle that does not contain some singularities. This is in stark contrast to the solutions of the wave inequality, which usually have less regularity. There is a close relationship between power series and Fourier theory. If the function f is expanded into a power series inside a circle of radius R, this means that, with appropriately defined coefficients, the real and imaginary parts are combined. These trigonometric values can be expanded using multiple angle formulas.
Information-analytical function
These values were introduced in Release 2 of 8i and greatly simplified the ways in which summary reports and OLAP queries can be evaluated in straight, non-procedural SQL. Prior to the introduction of analytical management features, complex reports could be created in the database using complex self joins, subqueries and inline views, but these were resource intensive and very inefficient. Moreover, if the question to be answered is too complex, it can be written in PL/SQL (which by its nature is usually less efficient than a single statement in the system).
Types of magnifications
There are three types of extensions that fall under the banner of an analytic function view, although one could say that the first is to provide "holomorphic functionality" rather than being similar exponents and views.
1). Grouping extensions (rollup and cube)
2). Extensions to the GROUP BY clause allow precomputed result sets, summaries, and summaries to be supplied from Oracle itself rather than using a tool like SQLPlus.
Option 1: totals the salary for the task, and then each department, and then the entire column.
3). Method 2: Consolidates and calculates wages per job, each department and question type (similar to the total sum report in SQLPlus), then the entire capital row. This will provide counts for all columns in the GROUP BY clause.
Ways to find a function in detail
These simple examples demonstrate the power of methods specifically designed to find analytic functions. They can break down the result set into workgroups to compute, organize, and aggregate data. The above options would be significantly more complex with standard SQL and would require something like three scans of the EMP table instead of one. The OVER app has three components:
- PARTITION, with which the result set can be partitioned into groups such as departments. Without this, it is treated as one section.
- ORDER BY, which can be used to order a group of results or sections. This is optional for some holomorphic functions, but essential for those that need access to lines on each side of the current one, such as LAG and LEAD.
- RANGE or ROWS (in AKA), with which you can make row or value inclusion modes around the current column in your calculations. The RANGE windows work on values, and the ROWS windows work on records, such as the X item on each side of the current section or all previous ones in the current section.
Restore analytic functions with the OVER application. It also allows you to distinguish between PL/SQL and other similar values, indicators, variables that have the same name such as AVG, MIN and MAX.
Description of function parameters
APPLICATIONS PARTITION and ORDER BYshown in the first example above. The result set was divided into individual departments of the organization. In each grouping, the data was ordered by ename (using the default criteria (ASC and NULLS LAST). The RANGE application was not added, which means that the default value RANGE UNABUNDED PRECEDING was used. This indicates that all previous records in the current partition in the calculation for the current line.
The easiest way to understand analytic functions and windows is through examples that demonstrate each of the three components for the OVER system. This introduction demonstrates their power and relative simplicity. They provide a simple mechanism for computing result sets that prior to 8i were inefficient, impractical, and in some cases impossible in "straight SQL".
To the uninitiated, the syntax may seem cumbersome at first, but once you have one or two examples, you can actively look for opportunities to use them. In addition to their flexibility and power, they are also extremely efficient. This can be easily demonstrated with SQL_TRACE and compare the performance of analytic functions with database statements that would have been needed in the days prior to 8.1.6.
Analytical Marketing Function
Studies and researches the market itself. Relationships in this segment are not controlled and are free. In the market form of the exchange of goods, services and other important elements, there is no control between trading entities and objects of power. To get the maximumprofit and success, it is necessary to analyze its units. For example, supply and demand. Thanks to the last two criteria, the number of customers is increasing.
In fact, the analysis and systematic observation of the state of consumer needs quite often leads to positive results. At the heart of marketing research is an analytical function that involves the study of supply and demand, it also monitors the level and quality of the supplied products and services that are being implemented or appear. In turn, the market is divided into consumer, world, trade. Among other things, it helps to explore the corporate structure, which is based on direct and potential competitors.
The main danger for a novice entrepreneur or firm is considered to be entering several types of market at once. In order to improve the demand for a newcomer's goods or services, a full study of the specific type of selected division where the sale will be realized is necessary. In addition, it is important to come up with a unique product that will increase the chances of commercial success. Thus, the analytical function is an important variable not only in the narrow sense, but also in the ordinary, as it comprehensively and comprehensively studies all segments of market relations.
