1. Heaviside unit inclusion function, Dirac delta function and their main properties

Heaviside identity function

Heaviside function (unit step function, unit hop function, included unit) is a piecewise constant function equal to zero for negative values of the argument and one for positive ones. At zero, this function is not defined, but it is usually extended at this point by a certain number so that the domain of the function contains all points of the real axis. Most often, it does not matter what value the function takes at zero, so various definitions of the Heaviside function can be used, convenient for one reason or another, for example:

Another common definition:

The Heaviside function is widely used in the mathematical apparatus of control theory and signal processing theory to represent signals that pass from one state to another at a certain point in time. In mathematical statistics, this function is used to write the empirical distribution function.

The Heaviside function is the antiderivative for the Dirac delta function, H" = δ, this can also be written as:

delta function

δ -function(ordelta Function,δ -Dirac function, Dirac delta, unit impulse function) allows you to write down the spatial density of a physical quantity (mass, charge, heat source intensity, force, etc.) concentrated or applied at one point.

For example, the density of a unit point mass located at a point a Euclidean space , is written using the δ-function in the form δ( x − a). Also applicable to describe distributions of charge, mass, etc. on surfaces or lines.

The δ-function is a generalized function, which means that it is formally defined as a continuous linear functional on the space of differentiable functions.

The δ-function is not a function in the classical sense; nevertheless, it is not difficult to find sequences of ordinary classical functions that converge weakly to the δ-function.

One can distinguish between one-dimensional and multidimensional delta functions, however, the latter can be represented as a product of one-dimensional ones in an amount equal to the dimension of the space on which the multidimensional one is defined.

Properties

The antiderivative of the one-dimensional delta function is the Heaviside function:

Filtering property of the delta function:

2. Filtertreble(HPF)- an electronic or any other filter that passes high frequencies of the input signal, while suppressing signal frequencies less than the cutoff frequency. The degree of suppression depends on the specific filter type. Passive filter - An electronic filter that consists of only passive components such as capacitors and resistors. Passive filters do not require any energy source to operate. Unlike active filters, passive filters do not amplify the signal in terms of power. Almost always passive filters are linear.

The simplest electronic high-pass filter consists of a capacitor and a resistor connected in series. The capacitor passes only alternating current, and the output voltage is taken from the resistor. The product of resistance and capacitance (R×C) is the time constant for such a filter, which is inversely proportional to the cutoff frequency in hertz.

(Either way)

Convert LPF response to HPF response can be done using the change of variable: where n is the cutoff frequency of the LPF passband and

Convert circuits passiveLC-filters. Change of variables (2.31) and (2.32) in the expression for the squared frequency response |H p (j )| 2 low-pass filters, when implementing this function, leads to the conversion of the low-pass circuit into high-pass and PF circuits. The inductive resistance of the low-pass filter j n.h. L n.h. passes when converting frequencies (17.31) into resistance: i.e., into the capacitance of the high-frequency filter, where C v.ch = 1/ p 2 L n.h.

Capacitive conductance: turns into inductive conduction of the high-pass filter with inductance L high-frequency = 1/ n 2 C low-frequency.

Transfer Function Transformation of Active RC Filters. In active RC filters, in order to move from the transfer function of the prototype low-pass filter to the transfer functions of the high-pass filter and PF, the complex variable p must be replaced. From (17.31) we obtain for HPF

or (17.34) where n.h = n.h/p and v.h = v.h/p.

(Or as they wrote on the elective)

DELTA FUNCTION

Definition. delta function

(2.1)

A generalized function

Fig.1. delta function

Normalization condition

, . (2.2)

a, as shown in Figure 1, b

Function parity follows from (2.1)

. (2.2a)

, (2.2b)

as shown in Figure 1, b.

orthonormality. Lots of features

DELTA FUNCTION Properties

filter property

we get

b, we find

, . (2.5)

Orthonormal basis

In (2.5) we set

, ,

. (2.7)

Performed

, (2.8)

Proof

Simplifying the Argument

If are the roots of the function , Then

. (2.9)

Proof

In a small neighborhood, we expand in a Taylor series

and restrict ourselves to the first two terms

We use (2.8)

We compare the integrands and obtain (2.9).

Convolution

From the definition of convolution (1.22)

at we get

We believe , and find

. (2.35a)

and (2.35a) give

. (2.35b)

we get

. (2.36a)

and (2.36a) give

. (2.36b)

. (2.37a)

we get

. (2.37b)

comb function

(2.53)

Models an unlimited crystal lattice, antenna and other periodic structures.

The Fourier transform transforms the comb function into a comb function.

(2.8)

we get

. (2.54)

Properties

The function is even

periodical

period . The filtering property of delta functions gives

. (2.55)

Fourier transform

For a periodic function with a period L The Fourier image is expressed in terms of the Fourier coefficients

, (1.47)

, (1.49)

For a comb function with a period, we get

where the filtering property of the delta function is taken into account. From (1.47) we find the Fourier transform

. (2.56)

The Fourier transform of the comb function is the comb function.

From (2.56), by the Fourier theorem on the scaling of the argument, we obtain

. (2.59)

Increasing the period of the comb function ()reduces the period and increases the amplitude of its spectrum .

Fourier series

We use

For , we get

DELTA FUNCTION

Definition. delta function

models a point disturbance and is defined as

(2.1)

The function is zero at all points except where its argument is zero and where the function is infinite, as shown in Fig. 1, A. Specifying the values at the points of the argument is ambiguous due to its going to infinity, so the delta function is generalized function , and requires an additional definition in the form of a normalization.

Fig.1. delta function

Normalization condition

, . (2.2)

The area under the function graph is equal to one in any interval containing a point a, as shown in Figure 1, b. Therefore, the delta function models a point perturbation of a single value.

Function parity follows from (2.1)

. (2.2a)

From symmetry about a point, we get

, (2.2b)

as shown in Figure 1, b.

orthonormality. Lots of features

forms an orthonormal infinite-dimensional basis.

The delta function was applied in optics by Kirchhoff in 1882, and in electromagnetic theory by Heaviside in the 1990s.

Gustav Kirchhoff (1824–1887) Oliver Heaviside (1850–1925)

Oliver Heaviside is a self-taught scientist who first used vectors in physics, developed vector analysis, introduced the concept of an operator, and developed operational calculus, an operator method for solving differential equations. He introduced the inclusion function, later named after him, used the point impulse function - the delta function. He applied complex numbers in the theory of electrical circuits. For the first time, he wrote Maxwell's equations in the form of 4 equalities instead of 20 equations, as Maxwell had. Introduced terms: conductivity, impedance, inductance, electret . He developed the theory of telegraph communication over long distances, predicted the presence of an ionosphere near the Earth - the Kennelly-Heaviside layer.

The mathematical theory of generalized functions was developed by Sergei L'vovich Sobolev in 1936. He was one of the founders of the Novosibirsk Academgorodok. The Institute of Mathematics of the SB RAS is named after him.

Sergei Lvovich Sobolev (1908–1989)

DELTA FUNCTION Properties

filter property

For a smooth function without discontinuities, from (2.1)

we get

Assuming , and using the delta function in the form of a limit at , shown in Fig. 1, b, we find

Integration gives the filter property in integral form

, . (2.5)

Orthonormal basis

In (2.5) we set

, ,

and we obtain the orthonormality condition for a basis with a continuous spectrum

. (2.7)

Argument scaling

Performed

, (2.8)

Proof

We integrate the product of the delta function with a smooth function over the interval , where :

where a variable substitution is made and the filtering property is used. Comparison of the initial and final expressions gives (2.8).

Simplifying the Argument

If are the roots of the function , Then

. (2.9)

Proof

The function is non-zero only near the points , at these points it is infinite.

To find the weight with which infinity enters, we integrate the product with a smooth function over the interval . The contributions are non-zero only in neighborhoods of points

. , (2.10) .. (2.35a)

Fourier's Argument Shift Theorem

and (2.35a) give

. (2.35b)

From (1.1) and integral representation (2.24)

we get

. (2.36a)

Fourier's theorem on the phase shift of a function

and (2.36a) give

. (2.36b)

From (2.35a) and the Fourier differentiation theorem

. (2.37a)

From (2.36a) and the Fourier theorem on multiplication by an argument

we get

. (2.37b)

Federal Agency for Education

State educational institution of higher professional education
Vyatka State University for the Humanities

Faculty of Mathematics

Department of Mathematical Analysis and Methods of Teaching Mathematics

Final qualifying work

Dirac function

Completed by a 5th year student

Faculty of Mathematics Prokasheva E.V.

________________________________/signature/

Scientific adviser:

Onchukova L.V.

signature/

Reviewer:

Senior Lecturer, Department of Mathematical Analysis and MMM Faleleeva S.A.

________________________________/ signature/

Approved for defense in the state attestation commission

"___" __________2005 department M.V. Krutikhina

Introduction ................................................ ................................................. ........ 3

Chapter 1. Definition of the Dirac function ......................................... ............. 4

1.1. Basic concepts ................................................................ ................................ 4

1.2. Problems leading to the definition of the Dirac delta function………...10

1.2.1. The problem of momentum ………………………………………………….10

1.2.2. The problem of the density of a material point……………………........ 11

1.3. Mathematical definition of the delta function………………………..16

Chapter 2. Application of the Dirac function……………………………………………19

2.1. Discontinuous functions and their derivatives…………………………………….19

2.2. Finding derivatives of discontinuous functions………………………...21

Conclusion…………………………………………………………………………25

Introduction

The development of science requires for its theoretical substantiation more and more "high mathematics", one of the achievements of which are generalized functions, in particular the Dirac function. At present, the theory of generalized functions is relevant in physics and mathematics, as it has a number of remarkable properties that expand the capabilities of classical mathematical analysis, expands the range of problems under consideration, and also leads to significant simplifications in calculations, automating elementary operations.

Objectives of this work:

1) study the concept of the Dirac function;

2) consider the physical and mathematical approaches to its definition;

3) show the application to finding derivatives of discontinuous functions.

Tasks of the work: to show the possibilities of using the delta function in mathematics and physics.

The paper presents various methods for determining and introducing the Dirac delta function, its application in solving problems.

Chapter 1

Definition of the Dirac function

1.1. Basic concepts.

In various questions of mathematical analysis, the term "function" has to be understood with varying degrees of generality. Sometimes continuous but not differentiable functions are considered, in other questions one has to assume that we are talking about functions that are differentiable one or more times, and so on. However, in a number of cases the classical concept of a function, even interpreted in the broadest sense, i.e. as an arbitrary rule that assigns to each value x from the domain of this function a certain number y=f(x), it turns out to be insufficient.

Here is an important example: when applying the apparatus of mathematical analysis to certain problems, we have to face a situation where certain operations of analysis turn out to be impossible; for example, a function that has no derivative (at some points or even everywhere) cannot be differentiated if the derivative is understood as an elementary function. Difficulties of this type could be avoided by limiting the consideration of analytic functions alone. However, such a narrowing of the stock of admissible functions is highly undesirable in many cases. The need for further expansion of the concept of function has become especially acute.

In 1930, to solve the problems of theoretical physics, the largest English theoretical physicist P. Dirac, one of the founders of quantum mechanics, lacked the apparatus of classical mathematics, and he introduced a new object called the “delta function”, which went far beyond the classical definition of a function .

P. Dirac in the book “Principles of Quantum Mechanics” defined the delta function δ(x) as follows:

In addition, the condition is set:

You can visually represent the graph of a function similar to δ(x), as shown in Figure 1. The more

narrower to make a strip between the left and right branches, the higher this strip must be in order for the area of the strip (i.e., the integral) to retain its given value of 1. As the strip narrows, we approach the condition δ(x) = 0 at x ≠ 0 , the function approaches the delta function.

This idea is generally accepted in physics.

It should be emphasized that δ(x) is not a function in the usual sense, since this definition implies incompatible conditions from the point of view of the classical definition of a function and an integral:

And .

In classical analysis, there is no function that has the properties prescribed by Dirac. Only a few years later, in the works of S.L. Sobolev and L. Schwartz, the delta function received its mathematical design, but not as an ordinary, but as a generalized function.

Before proceeding to the consideration of the Dirac function, we introduce the main definitions and theorems that we will need:

Definition 1. An image of a function f(t) or L - an image of a given function f(t) is a function of a complex variable p defined by the equality:

, Where M And A are some positive constants.

Definition 2. Function f(t) defined like this:

, is called Heaviside identity function and is denoted by . The graph of this function is shown in Fig. 2

Let's find L– image of the Heaviside function:

. (1)

Let the function f(t) for t<0 тождественно равна нулю (рис.3). Тогда функция f(t-t 0) будет тождественно равна нулю при t

To find the image δ(x) using an auxiliary function, consider the delay theorem:

Theorem 1.IfF(p) there is a function imagef(t), that is, the image of the functionf(t- t 0 ), that is, ifL{ f(t)}= F(p), That

Proof.

By the definition of an image, we have

Dirac delta function

The delta function (5-function) was introduced by the English physicist P. A. M. Dirac "out of necessity" when he created the mathematical apparatus of quantum mechanics. Mathematicians "did not recognize" it for some time, after which they created the theory of generalized functions, of which the δ-function is a special case.

According to the (naive) definition, the δ-function is equal to zero everywhere except at one point, but the area covered by this function is equal to one:

These conflicting

the requirements cannot be met by a "regular" type function.

Zeldovich Ya.B. Higher mathematics for beginner physicists and technicians. -M.: Nauka, 1982.

In fact, like a differential δх is not a number (equal to zero), and the phrase "infinitely small value" is difficult to understand qualitatively, to understand correctly δх not as a number, but as a limit (process), it is also correct to understand the δ-function as a limit (process). On fig. 3.7.1 and 3.7.2 show several functions (depending on a parameter), the limit of which is the δ-function. There are infinitely many such functions - everyone can choose their own.

The δ-function has many useful properties, being, in particular, the continuum analogue of the Kronecker symbol δkk

compare with

Another surprising relation indicates how you can differentiate by integrating:

Where 8 - derivative 8- functions.

Rice. 3.7.1 - Two successive approximations to δ-

Dirac functions. Depicted function

Rice. 3.7.2 - Two functions that are in the limit A ->∞ give δ-functions:

Finally, note that the interval of the δ-function:

Where in(x)- Heaviside function,

step, with break at point x= 0 .

Phase transitions

In order to talk about phase transitions, it is necessary to define what phases are. The concept of phases occurs in many phenomena, therefore, instead of giving a general definition (the more general it is, the more abstract and non-visible, as it should be), we will give a few examples.

First, an example of their physics. For the usual, most common liquid in our life - water, three phases are known: liquid, solid (ice) and gaseous (steam). Each of them is characterized by its own parameter values. It is essential that when external conditions change, one phase (ice) passes into another (liquid). Another favorite object of theorists is a ferromagnet (iron, nickel and many other pure metals and alloys). At low temperatures (for nickel below T= 3600 WITH) a nickel sample is a ferromagnet; when the external magnetic field is removed, it remains magnetized, i.e. can be used as a permanent magnet. At temperatures above Ts this property is lost, when the external magnetic field is turned off, it goes into a paramagnetic state and is not a permanent magnet. When the temperature changes, a transition occurs - a phase transition - from one phase to another.

Let us give one more geometric example from the theory of percolation. Randomly cutting out connections from the grid, in the end, when the concentration of the remaining connections - R becomes less than some value rs, it will no longer be possible to pass along the lattice "from one end to the other". Thus, the grid from the state of percolation - the phase of "leakage", will go into the state of the phase of "no leakage".

From these examples, it is clear that for each of the considered systems there is a so-called order parameter that determines which of the phases the system is in. In ferromagnetism, the order parameter is the magnetization in a zero external field; in the theory of percolation, it is the network connectivity, or, for example, its conductivity or the density of an infinite cluster.

Phase transitions are of various kinds. Phase transitions of the first kind are such a transition when several phases can simultaneously exist in the system. For example, at a temperature of 0° C ice floats in water. If the system is in thermodynamic equilibrium (no heat supply and removal), then the ice does not melt and does not grow. For phase transitions of the second kind, the existence of several phases simultaneously is impossible. A piece of nickel is either in a paramagnetic state or in a ferromagnetic state. A mesh with randomly cut connections is either connected or not.

Decisive in the creation of the theory of phase transitions of the second kind, the beginning of which was laid by L.D. Landau, there was an introduction of the order parameter (we will denote it G]) as a distinguishing feature of the phase of the system. In one of the phases, for example, paramagnetic, r] = 0, and in the other, ferromagnetic, G ^ 0. For magnetic phenomena, the order parameter ] is the magnetization of the system.

To describe phase transitions, a certain function of the parameters that determine the state of the system is introduced - G(n, T,...). In physical systems, this is the Gibbs energy. In each phenomenon (percolation, a network of "small worlds", etc.), this function will be determined "independently". The main property of this function, the first assumption of L.D. Landau - in a state of equilibrium, this function takes a minimum value:

In physical systems one speaks of thermodynamic equilibrium, in the theory of complex chains one can speak of stability. Note that the minimality condition is determined by varying the order parameter.

The second assumption of L.D. Landau - during the phase transformation n = 0. According to this assumption, the function b(n, T, ...) near the phase transition point can be expanded in a series in powers of the order parameter n:

where n = 0 in one phase (paramagnetic, if we are talking about magnetism and disconnected, if we are talking about a grid) and n ^ 0 in the other (ferromagnetic or connected).

From the condition

which gives us two solutions

For T > Tc there must be a solution n = 0, and for T< Тс solution n ^ 0. This can be satisfied if for the case T > Tc and n = 0 choose A > 0 . In this case, there is no second root. And for the case T < Ts the second solution must take place, i.e. must be carried out A< 0. This way:

A > 0 at T > Tc, A< 0 at T< Тс ,

Landau's second assumption requires that A(Tc) = 0. The simplest form of the function A(T) that satisfies these requirements is

The so-called critical index, and the function C(g], T) takes the form:

On fig. 3.8.1 shows the dependence b(n, T) for T > Tc And T< Тс .

Rice. 3.8.1 - Parameter Function Plots G(n, T) For T > Tc And T< Тс

Poston T., Stuart I. Catastrophe theory and its applications. - M.: Mir, 1980. Gilmour R. Applied catastrophe theory. - M.: Mir, 1984.

Qualitative dependence of parameters G(j], T) on the order parameter ] is shown in fig. 3.8.1 (G0 = 0). The temperature dependence of the order parameter ] is shown in Fig. . 3.8.2.

A more advanced theory takes into account that when T > Tc the order parameter ] , although very small, is not exactly zero.

Transition of the system from the state with h = 0 at T > Tc in state with h- 0 when decreasing T and reaching values T £ Tc can be understood as the loss of stability of the position h = 0 at T £ Tc. A recent mathematical theory

with the sonorous name "Theory of Catastrophes" describing from a single point of view many different phenomena. From the point of view of catastrophe theory, a phase transition of the second kind is a "assembly catastrophe".

Rice. 3.8.2 - Order parameter dependency n temperature: at T< Tc and near Tc order parameter n behaves like a power function, and when T > Tc n = 0

Definition. delta function

models a point disturbance and is defined as

(2.1)

The function is equal to zero at all points except
, where its argument is zero, and where the function is infinite, as shown in Fig. 1, A. Exercise
values at the points of the argument is ambiguous due to its going to infinity, so the delta function is generalized function , and requires an additional definition in the form of a normalization.

Fig.1. delta function

Normalization condition

,
. (2.2)

The area under the function graph is equal to one in any interval containing a point a, as shown in Figure 1, b. Therefore, the delta function models a point perturbation of a single value.

Function parity follows from (2.1)

. (2.2a)

From symmetry
relative to the point
we get

, (2.2b)

as shown in Figure 1, b.

orthonormality. Lots of features

,
,

forms an orthonormal infinite-dimensional basis.

The delta function was applied in optics by Kirchhoff in 1882, and in electromagnetic theory by Heaviside in the 1990s.

Gustav Kirchhoff (1824–1887) Oliver Heaviside (1850–1925)

Oliver Heaviside is a self-taught scientist who first used vectors in physics, developed vector analysis, introduced the concept of an operator, and developed operational calculus, an operator method for solving differential equations. He introduced the inclusion function, later named after him, used the point impulse function - the delta function. He applied complex numbers in the theory of electrical circuits. For the first time, he wrote Maxwell's equations in the form of 4 equalities instead of 20 equations, as Maxwell had. Introduced terms: conductivity, impedance, inductance, electret . He developed the theory of telegraph communication over long distances, predicted the presence of an ionosphere near the Earth - Kennelly–Heaviside layer .

Sergei Lvovich Sobolev (1908–1989)

Delta Function Properties Filter Property

For a smooth function
, which has no discontinuities, from (2.1)

we get filtering property of delta function in differential form affecting one point
:

We believe
, and use the limit for the delta function at
shown in fig. 1, b. We find

. (2.4)

We integrate (2.3) over the interval
, including the point a, we take into account the normalization (2.2) and obtain filtering property of the delta function in integral form

,
. (2.5)

Orthonormal basis

In (2.5) we set

,
,

and we obtain the orthonormality condition for the basis
with a continuous range of values

. (2.7)