The explanatory note for the course work is made in the volume of 36 sheets. It contains a table of the values of the gamma function for certain values of the variables and texts of programs for calculating the values of the gamma function and for plotting a graph, as well as 2 figures.

To write the course work, 7 sources were used.

Introduction

There is a special class of functions that can be represented in the form of a proper or improper integral, which depends not only on the formal variable, but also on the parameter.

Such functions are called parameter-dependent integrals. These include Euler's gamma and beta functions.

Beta functions can be represented by the Euler integral of the first kind:

The gamma function is represented by the Euler integral of the second kind:

The gamma function is one of the simplest and most significant special functions, knowledge of the properties of which is necessary for studying many other special functions, for example, cylindrical, hypergeometric and others.

Thanks to its introduction, our capabilities in calculating integrals are significantly expanded. Even in cases where the final formula does not contain functions other than elementary ones, obtaining it still often facilitates the use of the function Г, at least in intermediate calculations.

Euler integrals are well-studied non-elementary functions. The problem is considered solved if it leads to the calculation of Euler integrals.

1. Beta features I am Euler

Beta functions are determined by the Euler integral of the first kind:

=(1.1)

It represents a function of two variable parameters

and : function B. If these parameters satisfy the conditions and , then the integral (1.1) will be an improper integral depending on the parameters and , and the singular points of this integral will be the points and

Integral (1.1) converges at

.Assuming we get: = -

i.e. argument

and enter symmetrically. Taking into account the identity

by the honors integration formula we have

Where do we get it from?

For integer b = n, successively applying (1.2)

for integers

= m,= n, we have

but B(1,1) = 1, therefore:

Let us put in (1.1)

.Since the graph of the function

is symmetrical about a straight line, then

and as a result of substitution

, we get

putting in (1.1)

,from where we get

dividing the integral by two in the range from 0 to 1 and from 1 to

and applying the substitution to the second integral, we get

2. Gamma function

2.1 Definition

An exclamation point in mathematical works usually means taking the factorial of some non-negative integer:

n! = 1·2·3·...·n.

The factorial function can also be written as a recursion relation:

(n+1)! = (n+1)·n!.

This relationship can be considered not only for integer values of n.

Consider the difference equation

Despite the simple form of notation, this equation cannot be solved in elementary functions. Its solution is called the gamma function. The gamma function can be written as a series or as an integral. To study the global properties of the gamma function, the integral representation is usually used.

2.2 Integral representation

Let's move on to solving this equation. We will look for a solution in the form of the Laplace integral:

In this case, the right side of equation (2.1) can be written as:

This formula is valid if there are limits for the non-integral term. We do not know in advance the behavior of the image [(G)\tilde](p) for p®±¥. Let us assume that the image of the gamma function is such that the non-integral term is equal to zero. After the solution is found, it will be necessary to check whether the assumption about the non-integral term is correct, otherwise we will have to look for G(z) in some other way.

GAMMA FUNCTION, G-function, is a transcendental function T(z), propagating the values of the factorial z! for the case of any complex z ≠ 0, -1, -2, .... G.-f. introduced by L. Euler [(L. Euler), 1729, letter to X. Goldbach (Ch. Goldbach)] using an infinite product

from which L. Euler obtained an integral representation (Eulerian integral of the second kind)

true for Re z > 0. The function x z-1 is disambiguated by the formula x z-1 = e (z-1)ln x with real ln x. Designation Г(z) and name. G.-f. were proposed by A. M. Legendre (A. M. Legendre, 1814).

On the entire z plane with dropped points z = 0, -1, -2, ... for the G.-f. Hankel's integral representation is valid:

where s z-1 = e (z-1)ln s, and ln s is a branch of the logarithm, for which 0

Basic relationships and properties of geometric functions.

1) Euler's functional equation:

zГ(z) = Г(z + 1),

Г(1) = 1, Г(n + 1) = n!, if n > 0 is an integer, then count 0! = Г(1) = 1.

2) Euler's addition formula:

Г(z)Г(1 - z) = π/sin πz.

In particular,

if n > 0 is an integer, then

y - real.

3) Gaussian multiplication formula:

For m = 2 this is Legendre's doubling formula.

4) When Re z ≥ δ > 0 or |Im z| ≥ δ > 0 asymptotic. expansion of ln Г(z) into a Stirling series:

where B 2n are Bernoulli numbers. What does equality mean?

In particular,

More accurate is Sonin's formula:

5) In the real region Г(х) > 0 for x > 0 and takes the sign (-1) k+1 in sections -k - 1

GG"" > Г" 2 ≥ 0,

i.e., all branches of both |Г(x)| and ln |Г(x)| - convex functions. Property of logarithmic convexity is determined by G.-f. among all solutions of the functional equation

Г(1 + x) = xГ(x)

up to a constant factor.

Rice. 2. Graph of the function y = Г(х).

For positive x G.-f. has a single minimum at x = 1.4616321..., equal to 0.885603.... Local minima of the function |Г(х)| as x → -∞ form a sequence tending to zero.

Rice. 3. Graph of the function 1/Г(x).

6) In the complex region, for Re z > 0, the G.-f. decreases rapidly as |Im z| → -∞

7) The function 1/Г(z) (see Fig. 3) is an entire function of the 1st order of maximal type, and asymptotically as Г → ∞

ln M(r) ~ r ln r,

It can be represented by the infinite product of Weierstrass:

absolutely and uniformly convergent on any compact set of the complex plane (here the C-Euler constant). The Hankel integral representation is valid:

where the contour C * is shown in Fig. 4.

Integral representations for powers of G.-f. were obtained by G. F. Voronoi.

In applications, the so-called polygamma functions that are derivatives of ln Г(z). Function (Gaussian ψ function)

is meromorphic, has simple poles at the points z = 0,- 1,_-2, ... and satisfies the functional equation

ψ(z + 1) - ψ(z) = 1/z.

From the representation of ψ(z) for |z|

this formula is useful for calculating Г(z) in the vicinity of the point z = 1.

For other polygamma functions, see. The incomplete gamma function is defined by the equality

The functions Г(z), ψ(z) are transcendental functions that do not satisfy any linear differential equation with rational coefficients (Hölder's theorem).

The exclusive role of G.-f. in mathematics analysis is determined by the fact that with the help of G.-f. a large number of definite integrals, infinite products and sums of series are expressed (see, for example, Beta function). In addition, G.-f. finds wide applications in the theory of special functions (hypergeometric functions, for which the geometric function is a limiting case, cylindrical functions, etc.), in analytic. number theory, etc.

Lit.: Whittaker E. T., Watson J. N., Course of modern analysis, trans. from English, vol. 2, 2nd ed., M., 1963; Bateman G., Erdelyi A., Higher transcendental functions Hypergeometric function. Legendre's functions, trans. from English, M., 1965; Bourbaki N., Functions of a real variable. Elementary theory, trans. from French, M., 1965; Mathematical analysis. Functions, limits, series, continued fractions, (Reference mathematical library), M., 1961; Nielsen N.. Handbuch der Theorie der Gamma-funktion, Lpz., 1906; Sonin N. Ya., Research on cylindrical functions and special polynomials, M., 1954; Voronoi G.F., Collection. soch., vol. 2, K., 1952, p. 53-62; Janke E., Emde F., Lesch F., Special functions. Formulas, graphs, tables, trans. from German, 2nd ed., M., 1968; Ango A., Mathematics for electrical and radio engineers, trans. from French, 2nd ed., M., 1967.

L. P. Kuptsov.

Sources:

Mathematical Encyclopedia. T. 1 (A - D). Ed. board: I. M. Vinogradov (chief editor) [and others] - M., “Soviet Encyclopedia”, 1977, 1152 stb. from illus.

The domain of definition of the gamma function Г(х) In the integral (1) there are singularities of two types: 1) integration along a half-line 2) at a point the integrand goes to infinity. To separate these features, we represent the function Г(х) as the sum of two integrals. The Gamma function is called an integral. The domain of the gamma function. Some properties of the gamma function. The Beta function and its properties. The domain of the beta function. Application of Euler integrals in the calculation of definite integrals and Let's look at each of them separately. Since then the integral converges at (by comparison). The integral converges for any x. In fact, taking an arbitrary one, we find that for any x At the integral converges, therefore, the integral converges for any x. Thus, converges at and we have proved that the domain of definition of the gamma function Г(х) is a half-line. Let us show that integral (1) converges uniformly in x on any interval Let. Then, when we have, the integrals on the right-hand sides of formulas (2) and (3) converge, and according to the Weierstrass criterion, the integrals on the left-hand sides of inequalities (2) and (3) uniformly converge. Consequently, by virtue of equality, we obtain uniform convergence of Γ(x) on any interval [c, d], where. The uniform convergence of Г(х) implies the continuity of this function for Some properties of the gamma function 1. (the gamma function for x > 0 has no zeros). 2. For any x > 0, the reduction formula for the gamma function holds 3. For x = n, the formula holds For x = 1, we have Using formula (4), we obtain Applying the formula n times, for we obtain 4. Curve y = Г( x) convex downwards. In fact, it follows that the derivative on the half-line can have only one zero. And since, according to Rolle’s theorem, this zero x0 of the derivative Γ"(x) exists and lies in the interval (1.2). Since, then at the point x0 the function Γ(x) has a minimum. It can be shown that on (0, +oo) the function Г(х) is differentiable any number of times. From the formula for it is continuous for 6. The complement formula has the form shown in Fig. 4. § 4. The beta function and its properties is called the beta function. integral depending on parameters 4.1. Domain of definition of the beta function B(x) The integrand at has two singular points. To find the domain of definition, we represent integral (7) as the sum of two integrals, the first of which (at) has a singular point, and the second (at - singular point t = 1. The integral is an improper integral of the 2nd kind. It converges provided that for, and the integral is called the integral. The domain of the gamma function. Some properties of the gamma function. The Beta function and its properties. definitions of the beta function The use of Euler integrals in the calculation of definite integrals converges at Thus, the beta function B(x) y) is defined for all positive values of hnu. It can be proven that the integral (7) converges uniformly in each region x^a>0, Y>b>Oy so that the beta function is continuous for Some properties of the beta function 1. For the formula The beta function is symmetric with respect to xn. This follows from formula (9). §5. Application of Euler integrals in the calculation of definite integrals Let's consider several examples. Example 1. Calculate the integral 4 Let us introduce the replacement and get Therefore Example 2. Calculate the integral Let us assume then that the limits of integration remain the same, so that the given integral is reduced to the beta function: Example 3. Based on the equality, calculate the integral Here we used the definition of the beta function and formulas Exercises Calculate the limits: Find the derivatives F "(y) for the following functions: o. Based on the equality, calculate the integral 7. Using the equality, by differentiating with respect to the parameter, obtain the following formula: 8. Prove that the integral converges uniformly in y on the entire real axis. 7 dx 9. Prove that the integral converges uniformly with respect to the parameter s on any segment 10. Using equality, calculate the integral by differentiation with respect to the parameter. Using Euler integrals, calculate the following integrals: Express in terms of Euler integrals: The Gamma function is the integral Domain of gamma -functions Some properties of the gamma function Beta function and its properties Domain of definition of the beta function Application of Euler integrals in the calculation of definite integrals positive integer) Let us prove that the integral converges uniformly on the entire real axis: 1) the relation of any as A(e) ), mentioned in the definition of an improper integral uniformly converging with respect to the parameter y, we can take For B > A we will have We prove that the integral f(α) = / uniformly converges for a Since the integral converges for O 1, then by Weierstrass’s sufficient criterion we conclude that this integral converges uniformly. 10. We have Differentiating n times

46. Nature, origin and properties of gamma and x-ray radiation. Mechanisms of interaction of gamma and x-ray quanta with atoms of matter. The probability of different ways of interaction of quanta with atoms depending on the energy of the quanta.

The most important characteristic of any ionizing radiation is the phenomenon. Its ionizing ability. A quantitative measure of this ability is the linear ionization density (LID). It is equal to the number of pairs of ions created by a particle (quanta) per unit path in a substance. The ABI depends on the nature and energy of the particle and on the properties of the substance. In the literature, the ABI is usually indicated for a standard substance - dry air, and one centimeter is taken as a unit of distance. It is easy to understand that the greater the ABI, the greater the damaging effect on the body. As quanta pass through matter, they gradually lose energy, which is spent on ionizing molecules and atoms. The rate of energy loss determines the penetrating ability of a given ionizing radiation. The measure of penetrating power for particles is the distance at which the particle slows down to an energy close to the average energy of thermal motion. For quanta of X-rays or gamma rays, the distance at which the radiation power drops by a factor of “e” is taken as a measure of penetrating power. The higher the ABI, the lower the penetrating power of radiation in a given substance. Radiations with a high PV are called hard; if the PS is small, such radiation is called soft. But these terms are relative. Alpha particles have a very small PS; even in air their range is several cm. Denser substances are impenetrable to alpha particles with a thickness of a fraction of a mm. The stream of alpha particles falling on a person is completely absorbed in the upper layers of the skin. Due to their low PS, alpha particles are almost completely safe for humans during external irradiation. But if the alpha-active isotope gets inside the body, then the danger will be very great, because The particles emitted by the isotope inside the tissues will cause very strong ionization, damaging living structures. The PS of beta particles is approximately 100 times greater; in air they travel several meters, in solid media - several mm (depending on energy). X-rays and gamma rays, which have a low ABI, penetrate deeply even into dense media. High-energy gamma rays can pass through a layer of earth or concrete several meters deep.

Interaction with matter of alpha and beta particles

Individual alpha and beta particles can penetrate the nuclei of atoms and cause certain nuclear reactions there. But the overwhelming number of particles interact only with electron shells. Having a large mass, alpha particles practically do not deviate from a straight trajectory when colliding with the electrons of an atom. Electrons are torn away from atoms and molecules, i.e. ionization occurs. For a given isotope, all alpha particles have approximately the same energy, so all alpha particles of a given isotope have the same range in a substance. Beta particles are light, so they change their direction of motion significantly when they collide with an atom. This process is called scattering. Scattered beta particles fly in all directions and can be a source of injury to people located close to the body on which the stream of beta particles falls, even if this stream does not directly hit the person. A source of danger may be X-ray bremsstrahlung, which occurs during braking ** in solid substances. Due to the existence of bremsstrahlung, even pure beta emitters require fairly serious protection during storage or transportation. Finally, in things with positron activity, annihilation occurs, i.e. When positrons collide with electrons of a substance, the particles turn into two gamma quanta with an energy of 0.51 MeV each, therefore all positron-active isotopes of the phenomenon. Simultaneously sources of gamma radiation.

Practically important effects due to scattering

A. Scattered radiation spread. In all directions. This requires the adoption of additional Precautionary measures. For example, in an x-ray, the direct beam of rays is directed downwards, but the radiation scattered in the patient’s body goes to the sides and upwards, which forces measures to be taken to protect neighboring and even higher-lying rooms. Similarly, gamma radiation generated by a submarine's reactor is scattered in seawater, and some of it returns to the submarine's compartments, increasing the background radiation.

B. If, when measuring ionizing radiation, the measuring device is located next to massive objects or walls, the radiation scattered in them can significantly distort the measurement results.

B. Scattered radiation spoils the x-ray image. Quanta that deviate from the original direction end up in random places on the screen or film, “exposing” it and making the image less clear and contrasty.

Abstract

The purpose of this course work is to study the special properties of the Euler Gamma function. During the work, the Gamma function, its main properties were studied and a calculation algorithm was compiled with varying degrees of accuracy. The algorithm was written in a high-level language - C. The result of the program is checked against the table. No discrepancies in values were found.

To write the course work, 7 sources were used.

Introduction

There is a special class of functions that can be represented in the form of a proper or improper integral, which depends not only on the formal variable, but also on the parameter.

Such functions are called parameter-dependent integrals. These include Euler's gamma and beta functions.

Beta functions can be represented by the Euler integral of the first kind:

The gamma function is represented by the Euler integral of the second kind:

Euler integrals are well-studied non-elementary functions. The problem is considered solved if it leads to the calculation of Euler integrals.

1. Beta features I am Euler

Beta functions are determined by the Euler integral of the first kind:

It represents a function of two variable parameters and : function B. If these parameters satisfy the conditions and , then the integral (1.1) will be an improper integral depending on the parameters and , and the singular points of this integral will be the points and

Integral (1.1) converges at . Assuming we obtain:

= - =

i.e. argument and enter symmetrically. Taking into account the identity

by the honors integration formula we have

Where do we get it from?

For integer b = n, successively applying (1.2)

for integers = m,= n, we have

but B(1,1) = 1, therefore:

Let us put in (1.1). Since the graph of the function is symmetrical about a straight line, then

and as a result of substitution, we get

putting in (1.1) , from where , we get

dividing the integral by two in the range from 0 to 1 and from 1 to and applying the substitution to the second integral, we obtain

2. Gamma function

2.1 Definition

An exclamation point in mathematical works usually means taking the factorial of some non-negative integer:

n! = 1·2·3·...·n.

The factorial function can also be written as a recursion relation:

(n+1)! = (n+1)·n!.

This relationship can be considered not only for integer values of n.

Consider the difference equation

2.2 Integral representation

Let's move on to solving this equation. We will look for a solution in the form of the Laplace integral:

In this case, the right side of equation (2.1) can be written as:

The left side of equality (2.1) is written as follows:

Then equation (2.1) for the image of the gamma function has the form:

This equation is easy to solve:

It is easy to notice that the found function [(Г)\tilde](p) is in fact such that the out-of-integral term in formula (2.2) is equal to zero.

Knowing the image of the gamma function, it is easy to obtain an expression for the prototype:

This is a non-canonical formula; in order to bring it to the form obtained by Euler, it is necessary to replace the integration variable: t = exp(-p), then the integral will take the form:

The constant C is chosen so that for integer values of z the gamma function coincides with the factorial function: Г(n+1) = n!, then:

therefore C = 1. Finally, we obtain Euler’s formula for the gamma function:

This function is very common in mathematical texts. When working with special functions, perhaps even more often than an exclamation mark.

You can check that the function defined by formula (2.3) actually satisfies equation (2.1) by integrating the integral on the right side of this formula by parts:

2.3 Domain and poles

In the integrand of the integral function (2.3) with the exponential exp( -tz) with R( z) > 0 decreases much faster than the algebraic function increases t(z-1) . The singularity at zero is integrable, therefore the improper integral in (2.3) converges absolutely and uniformly for R (z) > 0. Moreover, by successive differentiation with respect to the parameter z it is easy to verify that Г( z) is a holomorphic function for R ( z) > 0. However, the unsuitability of the integral representation (2.3) for R ( z) 0 does not mean that the gamma function itself is not defined there - the solution to equation (2.1).

Let us consider the behavior of Г(z) in the vicinity of zero. To do this, let's imagine:

where is a holomorphic function in the neighborhood z = 0. From formula (2.1) it follows:

that is, Г(z) has a first-order pole at z = 0.

It's also easy to get:

that is, in a neighborhood of the point the function Г( z) also has a pole of first order.

In the same way you can get the formula:

From this formula it follows that the points z = 0,-1,-2,... are simple poles of the gamma function and this function does not have other poles on the real axis. It is easy to calculate the residue at the point z = -n, n = 0,1,2,...:

2.4 Hankel representation via loop integral

Let's find out whether the gamma function has zeros. To do this, consider the function

The poles of this function are the zeros of the function Г(z).

Difference equation for I( z) can be easily obtained by using the expression for Г( z):

The expression for solving this equation in the form of an integral can be obtained in the same way as the integral expression for the gamma function was obtained - through the Laplace transform. Below are the calculations. They are not the same as in step 1). And the integral will be points _______________________________________________________________________________

After separating the variables we get:

Integrating we get:

Passing to the Laplace preimage gives:

In the resulting integral, we make a replacement of the integration variable:

Then

It is important to note here that the integrand for non-integer values z has a branch point t= 0. On the complex plane of the variable t Let's make a cut along the negative real semi-axis. Let us represent the integral along this semi-axis as the sum of the integral along the upper bank of this cut from to 0 and the integral from 0 to along the lower bank of the cut. To prevent the integral from passing through the branch point, we will arrange a loop around it.

Fig1: Loop in Hankel integral representation.

As a result we get:

To find out the value of the constant, remember that I(1) = 1, on the other hand:

Integral representation

is called the Hankel representation along a loop.

It is easy to see that the function 1/Г( z) has no poles in the complex plane, therefore the gamma function has no zeros.

Using this integral representation, we can obtain a formula for the product of gamma functions. To do this, we make a change of variable in the integral, then:

2.5 Euler's limit form

The gamma function can be represented as an infinite product. This can be seen if we represent in integral (2.3)

Then the integral representation of the gamma function:

In this formula we can change the limits - the limit of integration in the improper integral and the limit at inside the integral. Here's the result:

Let's take this integral in parts:

If we carry out this procedure n times, we get:

Passing to the limit, we obtain the Euler limit form for the gamma function:

2.6 Formula for the product

Below we will need a formula in which the product of two gamma functions is represented through one gamma function. Let us derive this formula using the integral representation of gamma functions.

Let us represent the repeated integral as a double improper integral. This can be done using Fubini's theorem. As a result we get:

The improper integral converges uniformly. It can be considered, for example, as an integral over a triangle limited by the coordinate axes and the straight line x+y = R for R. In the double integral, we make a change of variables:

The Jacobian of this replacement

Integration limits: u varies from 0 to ∞, v in this case it changes from 0 to 1. As a result, we get:

Let us rewrite this integral again as an iterative one, and as a result we obtain:

where R p> 0, R v > 0.

2. Derivative gamma function

Integral

converges for every , since , and the integral at converges.

In the region where is an arbitrary positive number, this integral converges uniformly, since and we can apply the Weirstras criterion. The entire integral is convergent for all values since the second term on the right side is an integral that obviously converges for any. It is easy to see that the integral converges in any region where is arbitrary. Valid for all specified values and for all , and since converges, then the conditions of the Weierstrass test are satisfied. Thus, in the area integral converges uniformly.

This implies the continuity of the gamma function for. Let us prove the differentiability of this function for . Note that the function is continuous for and, and we will show that the integral:

converges uniformly on each segment, . Let's choose a number so that ; then at .Therefore, there is a number such as on.But then on the inequality is true

and since the integral converges, then the integral converges uniformly with respect to . Similarly, for there is a number such that the inequality holds for all . With such and all we get , from which, due to the comparison criterion, it follows that the integral converges uniformly with respect to . Finally, the integral

in which the integrand is continuous in the region

Obviously, converges uniformly with respect to . Thus, on the integral

converges uniformly, and, therefore, the gamma function is infinitely differentiable for any and the equality

Regarding the integral, we can repeat the same reasoning and conclude that

It is proved by induction that the Γ-function is infinitely differentiable and its i-th derivative satisfies the equality

Let us now study the behavior of the function and construct a sketch of its graph. (see Appendix 1)

From the expression for the second derivative of the function it is clear that for all . Therefore, it increases. Since , then by Role’s theorem on a segment the derivative at and at , i.e., decreases monotonically on and monotonically increases on . Next, because , then at . When it follows from the formula that when .

Equality , valid for , can be used when extending the function to a negative value.

Let us assume that . The right side of this equality is defined for (-1,0) . We find that the function extended in this way takes on negative values at (-1,0) for both , as well as for function .

Having thus defined on , we can use the same formula to extend it to the interval (-2,-1). On this interval, the continuation will be a function taking positive values and such that for and . Continuing this process, we define a function that has discontinuities at integer points (See Appendix 1.)

Note again that the integral

defines the G-function only for positive values; we formally continued to negative values using the reduction formula .

4. Calculation of some integrals.

Stirling formula

Let's apply the gamma function to calculate the integral:

where m > -1,n > -1. Assuming that , we have

and based on (2.8) we have

In integral

Where k > -1,n > 0, it is enough to put

Integral

Where s > 0, expand into a series

where is the Riemann zetta function

Let's consider incomplete gamma functions (Prym functions)

bound by inequality

Expanding in a series we have

Moving on to the derivation of the Stirling formula, which gives, in particular, an approximate value of n! for large values of n, let us first consider the auxiliary function

(4.2)

Continuous on the interval (-1,) increases monotonically from to when changing from to and turns to 0 at u = 0. Since

And so the derivative is continuous and positive throughout the entire interval, satisfying the condition

From the previous it follows that there is an inverse function defined on a continuous interval and monotonically increasing in this interval,

Turns to 0 at v=0 and satisfies the condition

We derive the Stirling formula from the equality

assuming we have

Assuming in the end, we get

in the limit at i.e. at (see 4.3)

where does Stirling's formula come from?

which can be taken in the form

where, at

for large enough it is assumed

the calculation is made using logarithms

if the integer is positive, then (4.5) turns into an approximate formula for calculating factorials for large values of n

Let's give a more precise formula without derivation

where in brackets there is a non-convergent series.

5. Examples of calculating integrals

Formulas required for calculation:

G()

Evaluate integrals

PRACTICAL PART

To calculate the gamma function, an approximation of its logarithm is used. To approximate the gamma function on the interval x>0, use the following formula (for complex z):

Г(z+1)=(z+g+0.5) z+0.5 exp(-(z+g+0.5))

This formula is similar to the Stirling approximation, but it has a correction series. For values g=5 and n=6, it is verified that the error is ε does not exceed 2*10 -10. Moreover, the error does not exceed this value over the entire right half of the complex plane: z > 0.

To obtain the (real) gamma function on the interval x>0, the recurrent formula Г(z+1)=zГ(z) and the above approximation Г(z+1) are used. In addition, you can notice that it is more convenient to approximate the logarithm of the gamma function than the gamma function itself. Firstly, this will require calling only one mathematical function - the logarithm, and not two - the exponent and the power (the latter still uses the logarithm call), secondly, the gamma function is rapidly growing for large x, and approximating it with a logarithm removes overflow issues.

To approximate Ln(Г(х) - the logarithm of the gamma function - the following formula is obtained:

log(Г(x))=(x+0.5)log(x+5.5)-(x+5.5)+

log(C 0 (C 1 +C 2 /(x+1)+C 3 /(x+2)+...+C 7 /(x+8))/x)

Coefficient values C k- tabular data (see in the program).

The gamma function itself is obtained from its logarithm by taking the exponent.

Conclusion

Gamma functions are a convenient tool for calculating certain integrals, in particular many of those integrals that cannot be represented in elementary functions.

Due to this, they are widely used in mathematics and its applications, mechanics, thermodynamics and other branches of modern science.

References

1. Special functions and their applications:

Lebedev I.I., M., Gostekhterioizdat, 1953

2. Mathematical analysis part 2:

Ilyin O.A., Sadovnichy V.A., Sendov Bl.Kh., M., “Moscow University”, 1987

3. Collection of problems on mathematical analysis:

Demidovich B.P., M., Nauka, 1966

4. Integrals and series of special functions:

Prudnikov A.P., Brychkov Yu.A., M., Nauka, 1983

5. Special Features:

Kuznetsov, M., “Higher School”, 1965

6.Asymptotics and special functions

F. Olver, M., Science, 1990.

7. Zoo of monsters or introduction to special functions

O.M. Kiselev,

APPLICATIONS

Appendix 1 - Graph of the gamma function of a real variable

Appendix 2 – Gamma function graph

Table – a table of gamma function values for certain values of the argument.

Appendix 3 is a program listing that draws a table of gamma function values for certain values of the argument.

Appendix 4 – listing of a program that draws a graph of the gamma function

Abstract................................................. ............ .....................3

Introduction........................................................ .......... .....................4

Theoretical part…………………………………………………….5

Euler's beta function……………………………………………………….5

Gamma function................................................... ..................................8

2.1. Definition……………………………………………………………...8

2.2. Integral representation……………………………8

2.3. Domain and poles…………………………..10

2.4. Hankel representation through loop integral………..10

2.5. Euler's limit form……………………………...12

2.6. Formula for the product……………………………..13

Derivative gamma function........................ ....................... ..........15

Calculation of integrals. Stirling formula........................18

Examples of integral calculations................................................................... ......23

Practical part…………………………………………………….24

Conclusion................................................. .....................................25

References………………………………………………………………26

Applications………………………………………………………..27

APPENDIX 1

Graph of the gamma function of a real variable

APPENDIX 2

Gamma function graph

TABLE

X	g(x)

APPENDIX 3

#include

static double cof=(

2.5066282746310005,

1.0000000000190015,

76.18009172947146,

86.50532032941677,

24.01409824083091,

1.231739572450155,

0.1208650973866179e-2,

0.5395239384953e-5,

double GammLn(double x) (

log1=log(cof*(cof+cof/(x+1)+cof/(x+2)+cof/(x+3)+cof/(x+4)+cof/(x+5)+cof /(x+6))/x);

log=(x+0.5)*log(x+5.5)-(x+5.5)+lg1;

double Gamma(double x) (

return(exp(GammLn(x)));

cout<<"vvedite x";

printf("\n\t\t\t| x |Gamma(x) |");

printf("\n\t\t\t__________________________________________");

for(i=1;i<=8;i++)

x=x[i]+0.5;

g[i]=Gamma(x[i]);

printf("\n\t\t\t| %f | %f |",x[i],g[i]);

printf("\n\t\t\t__________________________________________");

printf("\n Dlia vuhoda iz programmu najmite lybyiy klavishy");

APPENDIX 4

#include

Double gam(double x, double eps)

Int I, j, n, nb;

Double dze=(1.6449340668422643647,

1.20205690315959428540,

1.08232323371113819152,

1.03692775514336992633,

1.01734306198444913971};

Double a=x, y, fc=1.0, s, s1, b;

Printf (“you entered incorrect data, please try again\n”); return -1.0;

If(a==0) return fc;

For (i=0;i<5;i++)

S=s+b*dze[i]/(i+2.0);

Nb=exp((i.0/6.0)*(7.0*log(a)-log(42/0)-log(eps)))+I;

For (n=1;n<=nb;n++)

For(j=0; j<5; j++)

Si=si+b/(j+1.0);

S=s+si-log(1.0+a/n);

Double dx,dy, xfrom=0,xto=4, yto=5, h, maxy, miny;

Int n=100, I, gdriver=DETECT, gmode, X0, YN0, X, Y, Y0,pr=0;

Initgraph(&gdriver,&gmode, “ ”);

YN0=getmaxy()-20;

Line(30, getmaxy()-10,30,30);

Line(20, getmaxy()-30, getmaxx()-20, getmaxy()-30);

)while (Y>30);

)while (X<700);

)while (X<=620);

)while (y>=30);

X=30+150.0*0.1845;

For9i=1;i

Dy=gam(dx,1e-3);

X=30+(600/0*i)/n;

If(Y<30) continue;

X=30+150.0*308523;

Line(30,30,30,10);

Line(620,450,640,450);

Line(30,10,25,15);

Line(640,450,635,445);

Line(640,450,635,455);

Line(170,445,170,455);

Line(320,445,320,455);

Line(470,445,470,455);

Line(620,445,620,455);

Line(25,366,35,366);

Line(25,282,35,282);

Line(25,114,35,114);

Line(25,30,35,30);

Outtexty(20,465,"0");

Outtexty(165,465, "1";

Outtexty(315,465, "2";

Outtexty(465,465, "3";

Outtexty(615,465, "4";

Outtexty(630,465, "x";

Outtexty(15,364, "1";

Outtexty(15,280, "2";

Outtexty(15,196, "3";

Outtexty(15,112, "4";

Outtexty(15,30, "5";

Course work: Special properties of Euler's Gamma function.

Introduction

Introduction

2. Derivative gamma function

Let's apply the gamma function to calculate the integral:

Conclusion

References

1. Special functions and their applications:

Lebedev I.I., M., Gostekhterioizdat, 1953