The 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 of U, R, or C is called analytic only when it is defined locally by a convergent power series.
The 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. In addition, 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 not equal to zero. And also that P (z) and Q (z) do not have common factors. R (z) is differentiable at any point C → SC → S, and S is a finite set inside C for which the denominator Q (z) vanishes. The maximum of two degrees from the numerator and the degree of the denominator is called the degree of the rational function R (z), as well as the sum of two and the product. In addition, it can be verified that with the help of these addition and multiplication operations, the space satisfies the field axioms, and it is denoted by C (X). This is an important example.
The concept of numerical values 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 indicators indicate the multiplicities of the roots, and this gives us the first of two important canonical forms for a rational function:
R (Z) = am (z - z1) p1 (z - z2) p2 .... (z - zr) / pr bn (z − s1) q1 (z − s2) q2 .... (z − sr ) qr. The zeros z1, ..., zr of the numerator are called in the rational function, and the denominator s1, ..., sr are considered its poles. Order is its multiplicity, as the root of the above values. The fields of the first system are simple.
We say that a rational function R (z) is correct if:
m = deg P (z) ≤≤ n = degF (o) Q (z) and is strictly correct if m <n. If R (z) is not strictly proper, then we can divide by the denominator to get R (z) = P1 (z) + R1 (z), where P1 (z) is a polynomial and the remainder R1 (z) is strictly own rational function.
Differentiality analyticity
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 derivatives, the situation is very different. It is easy to prove that 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 (higher) 'n' the variables greater than n in the corresponding expansion in the Taylor series immediately merge into 0 and, therefore, the series will converge trivially. In addition, the addition of each polynomial is a Maclaurin series.
2). All exponential functions are also analytic. This is due to the fact that 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 corresponding domains, trigonometric, power, and logarithmic functions are also analytic.
Example: find out the possible values of 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 each k belonging to the whole set. This gives i-2i = exp? (ππ + 4ππk), for each k belonging to the set of integers. This example shows that the complex quantity zαα can also have different values, infinitely similar to the logarithms. Even if functions with a square root can have only at maximum two values, then they are also a good example of multi-valued functions.
Properties of holomorphic systems
The theory of analytic functions is as follows:
1). Compositions, sums or products are holomorphic.
2). For the analytic function, its inverse, if it is not non-zero at all, is similar. In addition, the inverse of which should 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 due to the fact that in a sense they are sparse compared to all the opposite.
Holomorphic function with several variables
Using power series for these values, you can determine the specified system for several indicators. The analytic functions of many variables have some of the same properties as with a single variable. However, especially for complex indicators, new and interesting phenomena appear when working in 2 or more dimensions. For example, zero 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 task of the function, the following values and theories are needed. If z = x + iy, then the 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 of v.
Characteristic of inequalities for functions
Conversely, given the harmonic variable, it is the real part of the holomorphic (at least locally). If the test form, then the Cauchy-Riemann equations will be satisfied. This relation 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 by a linear denominator. From the last requirement and Stokes theorem it follows that the value of the linear integral connecting two points does not depend on the path. The resulting pair of solutions to the Laplace equation is called conjugate harmonic functions. This construction is only valid locally or provided that the path does not intersect the 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 beginning.
The close relationship between the Laplace equation and the main analytic functions means that any solution has derivatives of all orders and can be expanded in a power series, at least inside a circle that does not contain some features. This contrasts sharply with wave inequality solutions, which usually have less regularity. There is a close connection between the power series and the Fourier theory. If we expand the function f in a power series inside a circle of radius R, this means that with suitable certain coefficients, the real and imaginary parts are combined. These trigonometric values can be extended using multiple angular formulas.
Information and analytical function
These values were introduced in Issue 2 of 8i and greatly simplified the ways in which summary reports and OLAP queries can be calculated in direct, non-procedural SQL. Prior to the introduction of analytic management functions, complex reports could be created in the database using complex independent joins, subqueries, and embedded views, but they were resource-intensive and very inefficient. Moreover, if the question that needs to be answered is too complicated, it can be written in PL / SQL (by its nature, it is usually less efficient than a single statement in the system).
Types of Magnification
There are three types of extensions that fall under the banner of the type of analytic function, although it could be said that the first thing is to provide “holomorphic functionality”, and not be similar indicators and types.
1). Grouping extensions (rollup and cube)
2). Extensions to the GROUP BY clause allow pre-computed result sets, summaries, and generalizations to be shipped from the Oracle server itself, rather than using a tool such as SQL * Plus.
Option 1: summarizes the salary for the assignment, and then each department, and then the entire column.
3). Method 2: combines and calculates the salary for the task, each department and the type of question (similar to the report on the total amount in SQL * Plus), then the entire line of capital. This will provide counts for all columns in the GROUP BY clause.
Methods for finding the 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 working groups 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. There are three components to an OVER application:
- PARTITION , with which the result set can be divided into groups, such as departments. Without it, it is considered as one section.
- ORDER BY, with which you can organize a group of results or sections. This is not necessary for some holomorphic functions, but is important and essential for those that need access to the lines on each side of the current, such as LAG and LEAD.
- RANGE or ROWS (in AKA) , with which you can make modes of including rows or values around the current column in your calculations. RANGE windows work on values, and ROWS windows work on records, such as the X item on each side of the current or all of the preceding sections.
Restore analytic functions using the OVER application. It also allows you to distinguish between PL / SQL and other similar values, measures, variables that have the same name, such as AVG, MIN and MAX.
Description of functional parameters
The PARTITION and ORDER BY applications are demonstrated in the first example above. The set of results was divided into separate departments of the organization. In each grouping, the data was ordered ename (using the default criteria (ASC and NULLS LAST). The RANGE application was not added, which means that they used the default value RANGE UNABUNDED PRECEDING. This shows that all previous records in the current section in the calculation for the current line.
The easiest way to understand analytic functions and windows are examples that demonstrate each of the three components for the OVER system. This introduction demonstrates their strength and relative simplicity. They provide a simple mechanism for calculating result sets that prior to 8i were inefficient, impractical, and in some cases impossible in “direct SQL”.
For the uninitiated, the syntax may seem cumbersome at first, but as soon as there is one or two examples, you can actively look for opportunities to use them. In addition to their flexibility and power, they are also extremely effective. This can be easily demonstrated using SQL_TRACE and compare the performance of analytic functions with the database operators that would be needed on days prior to 8.1.6.
Analytic marketing function
It studies and explores the market as such. Relations in this segment are not controlled and are free. In a market-based form of exchange of goods, services and other important elements, there is no control between the trading entities of the objects of power. To get maximum profit 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, analysis and systematic monitoring of the state of consumer needs quite often leads to positive results. The basis of marketing research is the analytical function, which implies the study of supply and demand, it also monitors the level and quality of the delivered products and services that are sold or appear. In turn, the market is divided into consumer, global, and commercial. In addition, it helps to research the corporate structure, which is based on direct and potential competitors.
The main danger for a novice entrepreneur or company is considered entry into several types of markets. To improve the demand for newbie goods or services, a full study of a certain type of the selected unit where the sale will be implemented 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 everyday, as it comprehensively and comprehensively studies all segments of market relations.