Let a single-valued function be defined in a certain domain and let the points and belong to the domain.

Definition. If there is a finite limit of the ratio when, according to any law, it tends to zero, then:

1) this limit is called derivative of a function at a point and is indicated by the symbol

2) in this case the function is called differentiable at the point.

All rules and formulas for differentiating functions of a real variable remain in force for functions of a complex variable.

Theorem. In order for a function to be differentiable at a point , it is necessary and sufficient that:

1) real functions and were differentiable at the point *);

2) at this point the conditions were met

, (4.2)

called Cauchy-Riemann conditions(C.-R.)or d'Alembert-Euler.

If the conditions are met ( C.-R.) the derivative of a function can be found using one of the following formulas:

Let us present two definitions that are of fundamental importance in the theory of functions of a complex variable.

Definition.Function called analytical in the field, if it is differentiable at every point in this region.

Definition.Function called analytical at point, if it is analytic in some neighborhood of the point, i.e. if the function is differentiable not only at a given point, but also in its neighborhood.

From the above definitions it is clear that the concepts of analyticity and differentiability of a function in a domain coincide, but the analyticity of a function at a point and differentiability at a point are different concepts. If a function is analytic at a point, then it is certainly differentiable there, but the converse may not be true. A function can be differentiable at a point, but not be differentiable in any neighborhood of that point, in which case it will not be analytic at the point in question.

The condition for a function to be analytic in a domain is that the Cauchy–Riemann conditions are satisfied for all points in this domain.

Relationship between analytical functions and harmonic ones. Can any function of two variables serve as the real and imaginary part of some analytic function?

If the function is analytic in the domain, then the functions are harmonic, that is, they satisfy Laplace’s equation.

And .

However, if the functions are arbitrarily chosen harmonic functions, then the function , generally speaking, will not be analytical, i.e. the conditions for them will not always be met.

You can construct an analytical function from one given harmonic function (for example, ), picking up another so that the conditions are satisfied. Conditions (4.2) allow us to determine an unknown function (for example, ) by its two partial derivatives or, what is the same, by its total differential. Finding a harmonic function from its differential is the problem of integrating the total differential of a function of two variables, known from real analysis.

Geometric meaning of the module and argument of the derivative. Let the function be differentiable in the domain and . The function will map a plane point to a plane point, a curve passing through a point to a curve passing through (Fig. 4.1).

Derivative modulus is the limit of the ratio of the infinitesimal distance between the mapped points and to the infinitesimal distance between their prototypes and . Therefore, the quantity can be considered geometrically as a stretching coefficient (if ) at a point when mapping a region into a region, carried out by the function

At each point in the region in each direction the stretching coefficient will be different. For the derivative argument, we can write

where and are respectively the angles and that the vectors and form with the real axis (Fig. 4.1). Let the angles formed by tangents to the curve and at points and with the real axis. Then for , a , therefore defines the angle by which the tangent to the curve at point must be rotated to obtain the direction to the tangent to the curve at point .

If we consider two curves and , and , then the angles and (Fig. 4.1) between their tangents are, generally speaking, unequal.

Definition. A mapping of a domain to a domain having the properties of constant dilations () in any direction and conservation (or conservatism) of angles between two curves intersecting at point is called conformal(similar in small). The mapping carried out by the analytic function is conformal at all points at which .

EXERCISES

55. Show that the function is differentiable and analytic in the entire complex plane. Calculate its derivative.

Solution. Let's find and. By definition we have . Hence, .

, ,

Where , .

As can be seen, the partial derivatives are continuous throughout the plane, and the functions and are differentiable at each point of the plane. The conditions are met. Consequently, it is differentiable at every point of the plane, and therefore analytic on the entire plane. Therefore, the derivative can be found using one of the formulas (4.3):

Finally, the derivative can be found using the rules of formal differentiation: .

56. Find out whether the function is analytical:

Solution. a) Since, then, from where . As can be seen, the first condition (4.2) is not satisfied for any and . Consequently, the function is not differentiable at any point in the plane, and therefore not analytic.

b) We have . Function And are differentiable at every point of the plane, because their partial derivatives are continuous throughout the plane. But the conditions are not satisfied at any point in the plane, except for the point where all partial derivatives are equal to zero. Consequently, the function is differentiable only at one point, but is not analytic there, since by definition differentiability is required in a neighborhood of this point.

Thus, the function is not analytic for any value. From the above example it is clear that the analyticity of a function at a point is a stronger requirement than its differentiability at this point.

57. Is there an analytical function for which ?

Solution. Let's check if the function is harmonic. For this purpose we find

And . From the last relation it follows that it cannot be a real, as well as an imaginary part of an analytical function.

58. Find, if possible, an analytic function from its real part .

Solution. First let's check whether the function is harmonic. We find , , , And . A function that is harmonic on the entire plane is associated with the Cauchy-Riemann conditions, . From these conditions we obtain, . From the first equation of the system we find it by integrating over , assuming constant.

where is an arbitrary function to be determined. Let's find it from here and equate it to the expression previously found: . We obtain a differential equation to determine the function , where

So, . Then, i.e. at this point there is a rotation through an angle and forming an angle with each other, are respectively displayed in rays and forming an angle with each other . Therefore, at a point, the conformity of the mapping is violated due to the fact that the property of angle conservatism is violated: the angles are not preserved, but are tripled.

The main task of the theory of conformal mappings is to construct a conformal mapping of a given domain onto some given domain of the plane of the variable w.

A continuous mapping of a region of 2-dimensional Euclidean space into 2-dimensional Euclidean space is called conformal at a point if it has the properties of constant extensions and conservation of angles at this point. The property of constancy of dilations at a point during mapping is that the ratio of the distance between the images and points u to the distance between and tends to a certain limit when it tends to in an arbitrary way; the number is called the stretching coefficient at a point under the mapping in question. The property of conservation (conservatism) of angles at a point during mapping is that any pair of continuous curves located at and intersecting at a point at an angle b (i.e., having tangents at a point forming an angle b between themselves), under the mapping in question goes into a pair of continuous curves intersecting at a point at the same angle b. A continuous mapping of a domain is called conformal if it is conformal at every point of the domain.

By definition, a conformal mapping of a domain must be continuous and conformal only at internal points, and if one speaks of a conformal mapping of a closed domain, then, as a rule, they mean a continuous mapping of a closed domain, conformal at its internal points.

Conformal mappings of a region of 2-dimensional Euclidean space into 2-dimensional Euclidean space can be conveniently considered as a mapping of a region of the plane of a complex variable into the plane of a complex variable; accordingly, the mapping is a complex-valued function of a complex variable. Moreover, if at a point the mapping preserves angles, then curvilinear angles with a vertex with this mapping either retain their absolute value and sign, or retain their absolute value, changing the sign to the opposite. In the first case we say that the mapping at a point is a conformal mapping of the first kind, in the second - a conformal mapping of the second kind. If a function defines a conformal mapping of the second kind at a point, then the complex conjugate function w= defines a conformal mapping of the first kind at a point, and vice versa. Therefore, only conformal mappings of the first kind are studied, and it is they that are usually meant when talking about a conformal mapping, without specifying their kind. If the mapping is conformal at a point, then at there is a finite limit of the relation, i.e., there is a derivative. The opposite is also true. Thus, if there is, then every infinitesimal vector with origin at a point is transformed when displayed using a linear function, i.e. stretches by a factor, rotates by an angle arg and in parallel shifts by a vector.

In the theory of flat conformal mappings and its applications, the fundamental question is the possibility of univalently and conformally mapping one given domain onto another, and in practical applications the question of the possibility of doing this using relatively simple functions. The first problem, for the case of simply connected domains whose boundaries are not empty and do not degenerate into points, is solved in the positive sense by Riemann’s conformal mapping theorem. The second problem for some areas of a special type is solved by using elementary functions of a complex variable.

Basic principles of the theory of conformal mappings on the mapping of one region to another

Riemann's theorem. Let be a simply connected region of the extended complex plane whose boundary contains at least two points. Then:

• 1) there is an analytic function that conformally maps to the unit circle
• 2) this function can be selected so that the conditions are met

where the given points are the given real number. In this case, the function is determined uniquely by conditions (1).

Two simply connected regions, each with at least two boundary points, can be mapped conformally from one to the other. An important theoretical position characterizing the behavior of a conformal mapping near the boundary of a domain is the following boundary correspondence principle.

Theorem 1. Let and be simply connected domains bounded by simple piecewise smooth contours and, and let the function univalently and conformally map a domain onto a domain. Then:

• 1) the function has a continuous extension to the boundary of the region, i.e. it can be further defined at the points of the contour that the result is a function that is continuous in closure;
• 2) the function, which is further defined at the boundary, maps the contour one-to-one onto the contour, and in such a way that a positive bypass of the contour will correspond to a positive bypass of the contour.

Theorem 2. Let the function be analytic in a simply connected domain bounded by a piecewise smooth contour and continuous in the closure of this domain. If a function carries out a one-to-one mapping of a contour onto some simple piecewise smooth contour, then it maps the region conformally and univalently onto the region bounded by the contour, and a traversal of the contour in the positive direction corresponds to a traversal of the contour also in the positive direction.

To prove the theorem it is enough to show that

• 1) for each point there is only one such that, i.e. the function has only one zero in its scope;
• 2) for each point there is no point such that i.e. the function does not take any value

Let's prove the first statement. According to the conditions of the theorem, the function does not vanish on the contour, because when the point falls on the contour, but lies in and cannot belong. This means, according to the argument principle, the number of zeros of the function in the region is equal to

Since the point lies in the area limited by the contour, then where the plus sign corresponds to the positive direction of traversing the contour. A negative value in this case is impossible, since it indicates the presence in the region of the poles of the function and, by condition, is analytic in Therefore, the equation in the region has only one solution.

Let's consider the second statement. If the point is located on the outside of the contour, then the equation has no solutions in the region. And this means that any internal point of the region, under a conformal and univalent mapping, goes to the internal point of the region. Q.E.D.

Remark 1. Theorems 1 and 2 are also true for regions and the extended complex plane bounded by simple piecewise smooth contours and.

Theorem 3 (domain conservation principle) If a function is analytic in a domain and is not constant, then the image of the domain is also a domain.

To prove the theorem, it is necessary to show that the set is linearly connected and open. Since the mapping, due to analyticity, is a continuous mapping, then the image of any linearly connected set under this mapping is a linearly connected set. Therefore, it is a linearly connected set.

Let us now prove that the open set, i.e. any point enters together with some of its neighborhood. Let one of the inverse images of a point. If, then, according to the inverse function theorem, in a certain neighborhood of a point a function is defined that is the inverse function of k. Consequently, all points of this neighborhood are images under the mapping and it belongs entirely to. If, then we come to the same conclusion based on the theorem (On the inverse function).

Theorem 4 (maximum modulus principle). If a function is analytic in a domain, and its modulus reaches a local maximum at some point, then it is constant in.

We will carry out the proof by contradiction. Let be. For a point, we choose an arbitrary neighborhood that entirely belongs to the region, and assume that it is not constant in the neighborhood under consideration. According to the principle of area conservation, the image of a circle when displayed is an area. This means that all points in a certain neighborhood of a point are images of points on a circle. In this neighborhood we choose a point for which (if, then we can take

and if, then any point in the indicated neighborhood can be taken as a point). For this point we have > Since the neighborhood of the point can be chosen to have an arbitrarily small radius, we conclude that the point is not a local maximum point of the function.

So, if a function is not constant in a neighborhood of a point, then it does not have a maximum at the point. If it reaches a maximum at some point in the region, then the function is constant in some neighborhood of the point, i.e. at. According to the theorem on the uniqueness of an analytic function, analytic functions and coincide in the domain. In other words, the function is constant at.

Theorem 5. If a function is analytic in a bounded domain and continuous on the closure of this domain, then the function reaches its greatest value at the boundary of the domain.

Indeed, if a function is constant in, then by virtue of continuity it is constant in and the statement of the theorem is obvious.

If it is not constant in, then, according to Theorem 4, the function cannot reach its greatest value in the region, because otherwise it would have a local maximum point. But, being continuous on a closed limited set, it reaches its greatest value on this set: this can only happen on the boundary of the region.

Theorem 6. If a function is analytic in a domain, has no zeros, and its modulus reaches a local minimum in , then it is constant in this domain.

Theorem 7 (Schwartz lemma). If a function analytic in a circle satisfies the conditions, then and, z. Moreover, equality or is possible at least at one point z 0 only when

Proof. Due to the fact that the point is the zero of the function, this function can be represented in the form where is the analytical function in, and. Consider a circle bounded by a circle. The function is analytic in and continuous in. Therefore, according to Theorem 5, it reaches its greatest value at the boundary. In this case, since according to the conditions of the theorem. Therefore, everywhere in we have.

Let us assume that the inequality holds at some point. Let's choose r<1 так, что. Тогда и, следовательно, . Получили противоречие, которое показывает, что на самом деле всюду в. В частности, в.

If, then the function reaches a maximum at a point equal to one. Similarly, equality means that it reaches a maximum at a point equal to one. In both cases, according to the principle of maximum modulus, the function is constant, and. Therefore, and.

Theorem 8. Let the function be harmonic in a bounded domain and continuous in the closure of this domain. If it is not constant in, then it reaches its maximum and minimum values ​​only at the boundary of this region.

CONFORMAL MAPPING (conformal transformation), a mapping of one region (in a plane or in space) to another region, preserving the angles between curves. The simplest examples of conformal mapping are similarity transformations and rotations (orthogonal transformations).

Conformal mapping is used in cartography when it is necessary to depict part of the surface of the globe on a plane (map) while preserving the values ​​of all angles; examples of such conformal mappings are the stereographic projection and the Mercator projection (see Map projections). A special place is occupied by conformal mappings of some regions of the plane onto others; their theory has significant applications in aero- and fluid mechanics, electrostatics and elasticity theory. The solution to many important problems is easily obtained when the area for which the problem is posed has a fairly simple form (for example, a circle or half-plane). If the problem is posed for a more complex domain, then it turns out to be sufficient to conformally map the simplest domain onto the given one in order to obtain a solution to the new problem from a known solution. This is exactly the path N. E. Zhukovsky followed when creating the theory of an airplane wing.

Not all regions of the plane admit conformal mappings onto each other. For example, a circular ring bounded by concentric circles cannot be mapped conformally onto a ring with a different radius ratio. However, any two regions, each of which is bounded by only one curve (simply connected regions), can be conformally mapped onto each other (Riemann's theorem). As for areas bounded by several curves, such an area can always be conformally mapped onto an area bounded by the same number of parallel straight line segments (Hilbert’s theorem) or circles (Köbe’s theorem), but the sizes and relative positions of these line segments or circles cannot be set arbitrarily .

If we introduce complex variables z and w in the original and image planes, then the variable w, considered in the conformal mapping as a function of z, is either an analytic function or a function complex conjugate to the analytic one. Conversely, any function that is analytic in a given domain and takes different values ​​at different points of the domain (such a function is called univalent) conformally maps this domain onto some other domain. Therefore, the study of conformal mappings of plane regions is reduced to the study of univalent analytic functions.

Any conformal mapping of three-dimensional regions transforms spheres and planes into spheres and planes and is reduced either to a similarity transformation, or to one inversion transformation and one similarity transformation performed sequentially (Liouville’s theorem). Therefore, conformal mappings of three-dimensional (and generally multidimensional) regions do not have such great importance and such diverse applications as conformal mappings of two-dimensional regions.

The theory of conformal mapping began with L. Euler (1777), who discovered the connection between functions of a complex variable and the problem of conformally mapping parts of a sphere onto a plane (for constructing geographic maps). The study of the general problem of conformal mapping of one surface onto another led K. Gauss (1822) to the development of the general theory of surfaces. B. Riemann (1851) formulated the conditions under which a conformal mapping of one region of the plane onto another is possible, but the approach he outlined was only substantiated at the beginning of the 20th century (A. Poincaré and C. Carathéodory). The studies of N. E. Zhukovsky and S. A. Chaplygin, who opened a wide field of applications of conformal mapping in aero- and hydromechanics, served as a powerful stimulus for the development of the theory of conformal mapping as a large branch of the theory of analytic functions.

Lit.: Goluzin G.M. Geometric theory of functions of a complex variable. 2nd ed. M., 1966; Markushevich A.I. Theory of analytic functions. 2nd ed. M., 1968. T. 2; Lavrentyev M.A., Shabat B.V. Methods of the theory of functions of a complex variable. 6th ed. M., 2002.

Here we will talk in more detail about the geometric methods of the theory of analytic and generalized analytic functions, which we will most use in applications.

§ 10. Riemann problem

This main boundary value problem of the theory of conformal mappings has already been discussed in the previous chapter. It consists in constructing a conformal mapping of one region to another.

Existence and uniqueness. Let's start with the remark that it is enough to learn how to map an arbitrary simply connected region conformally onto a circle, and then we can map any two such regions conformally onto each other.

This remark is based on two simple properties of conformal maps: 1) the inverse of a conformal map and 2) a complex map composed of two conformal maps (i.e., the map ) are again conformal maps. The properties are clear from the definition of a conformal mapping as a one-to-one analytical transformation and from the rules for differentiating inverse and complex functions.

Having these properties, it is not at all difficult to substantiate the remark made: if the functions conformally map respectively the domains onto the unit

circle then the function will display on

Riemann's problem was completed at the beginning of this century. It turned out that any simply connected region whose boundary consists of more than one point can be mapped conformally onto the unit circle. This is Riemann’s famous theorem, which he formulated back in 1851, supported by physical considerations, but did not prove (more precisely, his proof had a significant gap).

Let us deal with the question of how defined the Riemann problem is, how many solutions it has for given domains. According to the remark, to solve this question it is enough to find out in how many ways one can conformally map the unit circle onto itself. It is easy to check that for any complex and any real number the function

maps the circle conformally onto itself (indeed, with we have and, therefore, i.e. (1) transforms the unit circle into itself; in addition, it is one-to-one, since equation (1) is uniquely solvable with respect to and takes point a of the circle to its center). Mapping (1) depends on three real parameters - two coordinates of the point a, which goes to the center of the circle, and the number 0, the change of which means the rotation of the circle relative to the center.

It can be proven that formula (1) contains all conformal mappings of the unit disk onto itself. This means that the arbitrariness in solving the Riemann problem is exhausted by three real parameters:

the conformal mapping of one region to another is determined uniquely if we specify the correspondence of three pairs of boundary points (the position of a point on the boundary is specified by one parameter) or the correspondence of one pair of internal points (two parameters) and another pair of boundary points (one parameter). Such conditions that uniquely determine the mapping - they are called normalization conditions - can take different forms, but each time these conditions must determine three parameters.

Examples. Let us indicate several simple examples of conformal mappings.

1) Mapping the appearance of the circle onto itself. Function (1) can also be considered as mapping the exterior, i.e., the area onto itself; it takes a point called symmetrical to infinity with respect to the unit circle

2) The upper half-plane on the circle is also displayed by a fractional linear function:

here a is an arbitrary point of the upper half-plane; it is transferred when mapping (2) to the center of the circle; the point of the circle to which the infinite point of the plane goes (the limit of the right side of (2) with is obviously equal to ).

In Fig. Figure 22 shows what straight lines h turn into - these are circles tangent to the unit at the point

3) The exterior of a unit circle is mapped onto the exterior of a segment by the so-called Zhukovsky function

In this case, the circles transform into ellipses with semi-axes and with foci ±1, and the rays into arcs of hyperbolas orthogonal to the ellipses (Fig. 23).

4) The stripe on a unit circle is displayed by the function

In this case, vertical straight and horizontal segments turn into “meridians” and “parallels” (Fig. 24).

5) The upper half-plane with a circular segment thrown onto the upper half-plane during normalization is displayed by the function

where a and a are the parameters of the segment (Fig. 25), and c is a real constant (note that our normalization conditions specify only two real parameters, so the third remains arbitrary).

This formula is too cumbersome for applications. For small a and a, using the first terms of the Taylor expansions, it can be replaced by the approximate formula

It can also be noted that, up to small higher orders, it gives the area from the ejected segment, therefore (6) can be rewritten in the form

6) A circle with a small hole thrown onto the circle is also displayed by a rather cumbersome recording function. An approximate formula for such a mapping, provided that the area of ​​the ejected hole is small, can be written as follows:

here is the top of the hole or (with the same accuracy) its other point.

7) The same approximate formula for mapping a strip with an ejected hole of small area c onto the strip has the form

where a is the abscissa of one of the points of the hole; hyperbolic tangent.

Flow in the channel. The ability to solve the Riemann problem determines the success of solving some hydrodynamic problems. We will illustrate this using classical examples of problems of steady flows of an ideal incompressible fluid past bodies. We will, of course, have to assume that the bodies are in the form of infinite cylinders (with arbitrary leading lines) in order to use the planar motion scheme.

Suppose we need to find a flow in a channel with walls that are perpendicular to a certain plane and intersect it along two infinite curves without common points (Fig. 26), and the flow velocities are parallel to this plane and are the same at all perpendiculars to it. The velocity field in the channel is described by a flat field in a band limited by curves

As we saw in the previous chapter, the assumption about the absence of sources and vortices in the flow leads to the conclusion about the existence of a complex potential - analytical in the function. Finding a flow means finding this function.

The flow must flow around the walls of the channel, i.e., each of the curves must be a stream line; this gives the boundary condition of the problem. We can ask

also the flow rate which, as shown in the last chapter, is equal to

where y is a line with ends, i.e., any cross section of the flow. Since we are interested in the potential up to a constant term, we can assume that on G.

In this formulation, the problem is still very uncertain. For example, for the case when it is a straight strip, its solution is any function

For any real and integer (the imaginary part vanishes at To state the problem more clearly, we will have to assume that the width of the strip remains limited at infinity, impose some smoothness conditions and consider only flows with limited speed at infinity. It can be proven that for these additional restrictions, the solution to the problem will be only a conformal mapping of the domain onto a strip with normalization . This mapping is determined up to a (real) constant term, which is not essential, i.e., the flow problem in the accepted restrictions is solved uniquely. Its solution is thus reduced to. solving the Riemann problem.

graduate work

1.1 The concept of conformal mapping and its basic properties

A one-to-one mapping that has the property of preserving angles in magnitude and direction and the property of constancy of the dilations of small neighborhoods of mapped points is called a conformal mapping.

To ensure one-to-one reflection, areas of function univalence are identified. A domain D is called a domain of univalence of the function f(z) if.

Basic properties of conformal mappings:

1) constancy of stretching. Linear at a point is the same for all curves passing through that point and is equal;

2) preservation of angles. All curves at a point rotate through the same angle, equal.

The function displays points on the z-plane (or Riemann surface). At each point z such that f(z) is analytic (i.e. uniquely determined and differentiable in some neighborhood of this point) and the mapping is conformal, i.e. the angle between two curves passing through point z transforms into an angle equal in magnitude and direction of reference between two corresponding curves in the plane.

An infinitesimal triangle near such a point z is mapped into a similar infinitesimal triangle - the plane; each side of the triangle is stretched in ratio and rotated by an angle. The distortion coefficient (local ratio of small areas) during mapping is determined by the Jacobian of the mapping

at every point z where the mapping is conformal.

Conformal mapping transforms lines into a family of orthogonal trajectories in the w-plane.

The region of the z-plane mapped onto the entire w-plane by the function f(z) is called the fundamental region of the function f(z).

The points where are called critical mapping points.

A mapping that preserves the magnitude, but not the direction, of the angle between two curves is called an isogonal or conformal mapping of the second kind.

A mapping is conformal at a point at infinity if the function conformally maps the origin to the - plane.

Two curves intersect at an angle at a point if the transformation transforms them into two curves intersecting at an angle at a point.

Similarly, maps a point conformally to a point .

Algebraic matrix groups

Let and be arithmetic linear spaces of columns of height and, respectively. Let, further, be a size matrix. Let us define a mapping by setting for any where are the columns of the matrix. Since they are tall...

Biectors in finite groups

Definition. Let --- a group and --- a class of groups. If and, then --- is a subgroup of the group. Definition. A -maximal subgroup of a group is a -subgroup of a group that is not contained in any larger -subgroup. Definition...

Vector fields

Definition of a rotor of a vector field: A rotor or vortex of a vector field is a vector with projections. Basic properties of a rotor: -- this is a vector quantity that is a differential (i.e., point) characteristic of a vector field...

External geometry of surfaces with a constant type of points

Saddle surfaces, in a certain sense, are opposite in their properties to convex surfaces. Like convex surfaces, they can be defined purely geometrically...

Chinese Remainder Theorem and its Corollaries

In this section we will look at integers, and we will denote them using Latin letters. Let's take an arbitrary fixed natural number p and consider the remainders when dividing various integers by p...

Mathematical foundations of the residual class system

Let's take an arbitrary fixed natural number p and consider the remainders when dividing various integers by p. When considering the properties of these remainders and performing operations on them, it is convenient to introduce the concept of comparison modulo...

Mathematical modeling of technical objects

A model is a physical or abstract image of a modeled object, convenient for research and allowing one to adequately display the physical properties and characteristics of the object that are of interest to the researcher...

Definite integral

1. The value of a definite integral does not depend on the designation of the integration variable: . 2. A definite integral with the same limits of integration is equal to zero: 3. If, then, by definition, we put 4...

Practical application of quadrature formulas with Chebyshev-Hermite weights

Let an even weight function be specified on the entire axis.

(1.1) Differentiating this function successively, we find (1.2) It is easy to prove by induction that the order n derivative of function (1.1) is the product of this function by some polynomial of degree n...

A spherical polygon is a part of a sphere bounded by arcs of great circles, smaller semicircles, the ends of which are the points of intersection of these great circles, taken in sequential order...

Solution of the problem of an ideal fluid flowing around a circular cylinder in quaternions

Quaternions were introduced into mathematics by William Rowan Hamilton 1]. They are a good tool for solving many problems related to three-dimensional space, and take into account its features...

Statistical Modeling

In order for an assessment to have practical value, it must have the following properties. 1. A parameter estimate is called unbiased if its mathematical expectation is equal to the estimated parameter, i.e. M= .(22.1) If equality (22...

Trigonometric functions

Cycloid

The definition of a cycloid, introduced earlier, never satisfied scientists: after all, it is based on mechanical concepts - speed, addition of movements, etc...

We will need two facts from . 1. For anyone there is a unique DF. 2. If, then the set is single-element. If, then there are continuous, one-parameter families (i.e., for and (the symbol denotes weak convergence)) and DFs such...