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14Evaluation of the integrability of functions by means of mapping
Sykhomlinova A.A., Stabnikov P.A. (stabnik@nsk.che.su)
Nikolaev Institute of Inorganic Chemistry of Siberian Branch of RAS, Novosibirsk
Introduction
Differentiation and integration are mutually inverse operations. No special problems usually arise for differentiation, because differentiation procedures are rather easy, and the result is usually expressed by the functions the complexity of which is not higher than that of the expression under differentiation itself. However, integration is a much more complicated problem. Even in the cases when an integral is expressed through elementary functions (or, as they say, can be taken in the finite form), there are no unified instructions that could allow one to find such an expression. In addition, the result of integration of some functions can contain more complicated functions. For example, the result obtained by integrating some elementary functions cannot be expressed by elementary functions only. The functions that are represented by indefinite integrals that cannot be taken in a finite form are new transcendental functions. Some of them have already been well investigated: integral logarithm, integral sine, integral exponential, etc. In other words, either the choice of integration way or even a preliminary evaluation of integration possibility is rather difficult problem. It is possible to simplify it using the mapping procedure.
The foundations of the mapping method were described in [1], where xp(Ln(x))q and xp(x+c)q functions were chosen as an initial basis and two colored integrability maps were proposed for these functions (Fig. 1 and 2). The maps are the Cartesian systems with p and q axes in which a single point with the parameters p and q corresponds to an integrand. All points are marked with a color depending on the result of integration. If the solution is expressed only by power functions, the color is black; if a logarithmic function is present in the solution, the color is green; if special functions are to be drawn to express integration result, the color is turquoise or pink. In other words, the more complicated is integration result, the more light-colored is a point; the corresponding colors change from black to green and then to turquoise or pink (Fig. 1 and 2). The main advantage of these colored maps is the possibility to indicate what classes of functions will be present in the final result, without integrating. These maps are vivid and allow providing a general notion of the function integrability.
Selection of the basis
Diversity of mathematical functions is so tremendous that any geometrical constructions that would include all the possible functions can hardly be possible. Because of this, some basis including primitive functions was chosen in [1] from the set of mathematical functions. This basis included:
1. A power function xp, where p is a real number.
2. A product of power functions xp(x + a)q, where a is a complex number and p, q are real numbers.
3. Natural logarithmic function ln (ax), or ln (x) + ln (a), where a is a complex number.
4. Products of a power function and a logarithmic function.
Expressions of the xp(xs + c)q kind were not included into the basis, because a change of variables u = xs results in ur(u + c)q, and this kind has already been included into the basis. Rather simple integrals containing an exponential ex can be transformed by changing the variables x = ln (u) into the integrals containing only power and logarithmic functions, and vice versa. For instance, Jx2e2xdx ^ J(ln(u))2udu. Because of this, it is necessary that the basis includes only one of the functions: either ex or ln (x). The logarithmic function was included into the basis. It is known that trigonometric and hyperbolic functions can be expressed through the exponential ex, while antitrigonometric (inverse trigonometric) and antihyperbolic (inverse hyperbolic) functions are expressed through the logarithmic function ln(x) [2]. So, neither trigonometric nor hyperbolic functions, nor their inverse functions were included into the basis.
Since the connection of a antitrigonometric function with a logarithmic function is carried out with the help of an imaginary variable, for example arcsin(x) = -iln (ix + (1 - x2)12) [2], we shall accept in the present work that both the variable x and the functions can take on either real or imaginary values, that is, we shall consider them in a complex field.
The basis will be limited to these kinds of functions; however, such a basis is quite sufficient for the problem under consideration. It should be noted that such a basis is generally open, and it can be expanded for the analysis of more complicated integrals by introducing other functions, for example special ones.
Description of xp(Ln(x))q map
Integral tables [3,4] were used to choose colors for points. On the xp(Ln(x))q map, only one line q=0 is black; it corresponds to Jxpdx = x p+1/(p+1) integrals (Fig. 1). (two figures and attachment in Russian variant of the paper) There is a point corresponding to the Jx-1dx = ln(x) integral on this line. This point is green. A number of horizontal lines corresponding to Jxp(ln(x))ndx integrals, where n is a natural number, are marked with green color because logarithmic function is present in solutions. For example, Jxpln(x)dx = xp+1ln(x)/(p+1)-xp+1/(p+1)2, Jxp(ln(x))2dx = xp+1 (ln(x))2/(p+1)-2xp+1ln(x)/(p+1)2+2xp+1/(p+1)3, etc. A vertical line p=-1, which corresponds to Jx-1(ln(x))qdx integrals, is marked with green color because Jx-1(ln(x))qdx = (ln(x))q+1/(q+1). This line contains a point (-1,-1), which corresponds to the Jx-1(ln(x))-1dx = ln(ln(x)) integral. This point is light-green. For the other points remaining on the xp(Ln(x))q map, the solutions of the corresponding integrals can be written down only using transcendental functions, that is why the background of the map is turquoise.
Shifts over the map
Let us write down the known expressions [3,4]: Jup(ln(u))qdu = J(ln(x))qdx/(p+1)q1 up=xp/(p+1) p * -1. (1a)
J(ln(x))qdx = x(ln x)q - qJ(ln(x))q-1dx. (1b)
J(ln(x))qdx = [x(lnx)q+ 1 - J(ln(x))q+1dx]/(q+1) q * -1. (1c)
Equations (1) will be called translations or shifts. These equations, with a small exception, do not lead to the solutions of integrals, but they allow us to pass from one integrand to another. On the map, this means the transition from one point to another. As it follows from equations (1), additional items composed of primitive functions
can arise during shifts. These additional items are not determinative for the color of the two points. In other words, a shift can bring together points having the same color. Some shift formulas may contain conditions indicating which points cannot be connected by a given translation.
It follows from equation (1a) that all the points on the map except the points on the line p = -1 can be brought to the vertical line p = 0. Further on, due to equations (1b) and (1c), all the points on the p = 0 line at unit intervals, except the points (0,0) and (0,-1), are connected with each other. Such an interconnection allows choosing an internal basis on this line, which will be a segment -1 < q < 0. It will be referred to as the pivotal segment. At the boundary of the pivotal segment, at the point (0,-1), the solution of the integral corresponds to the integral logarithm Li(x). In order to make it possible to write down the result of integration of any point on the segment -1 < q < 0, let us introduce the designation for new special functions on the analogy of the integral logarithm in the form of Lir(x), where -1 < r < 0. So, Li-1(x) = J(ln(x))-1dx = Li(x) is a classic function of the integral logarithm; Li-05(x) = J(ln(x))-0'5dx, Li-02(x) = J(ln(x))-0'2dx, etc.. It should be noted that special functions are introduced in this case not for convenience of solving the applied tasks but on the basis of the internal connection between xp(ln(x))q dx expressions.
So, with the help of the formulas of translation, or shift (1), one can write down the result of integration of any expression xp(ln(x))qdx, using a power, logarithmic functions and the proposed special ones Lir(x). The corresponding points are marked on the map with black, green and turquoise colors.
Using the change of variable, one can transform xp(ln(x))q dx into many other kinds: ex(p+1)xqdx, (ln(x))p(ln(ln(x)))qx^1dx, (Li-1(x))px(ln(Li-1(x)))q/ln(x)dx, (sin(x)/(ln(sin(x)))qcos(x)dx, etc. The proposed map is suitable for all these expressions. At the same time, the map of functions xp(ln(x))qdx itself is a plane section of a three-dimensional structure corresponding to more general expressions xp(ln(x))qx(ln(ln(x)))^dx or xp(ln(x))q(x + a)sdx.
Description of x^x + c)q map
At first, let us write down the shift formulas:
along the vertical line downward \xp(x + c)qdx = [x^+1(x + c)q + qc\xp(x + c)q-Xdx]/(p + q + 1) p + q * -1, (2a)
along the vertical line upward
Jxp(x + c)qdx = [—xp+1(x + c)q+1 +
+ (p + q + 2)Jxp(x + c)q+1dx]/(q + 1)/c
along the horizontal line to the left
Jxp(x + c)qdx = [xp(x + c)q+1 - pcJxp-1(x + c)qdx]/(p + q + 1)
along the horizontal line to the right
Jxp(x + c)qdx = [xp+1(x + c)q+1 -
- (p + q + 2)Jxp+1(x + c)qdx]/(p + 1)/c
along the diagonal downward Jxp(x + c)qdx = [xp+1(x + c)q - qJxp+1(x + c)q-1dx]/(p + 1)
q Ф -1, (2b)
p + q * -1, (2c)
p * -1, (2d)
p * -1, (2e)
along the diagonal upward
\xp(x + c)qdx = [x?(x + c)q+1 - pjxp-1(x + c)q+1dx]/(q + 1) q * -1. (2f)
The formulas (2) are simplified versions of the integrals J xp (axr + b)qdx, listed in [4]
under the number of 1.2.2.
Formulas (2) together wil the conditions form the structure of the entire + c)q map. If p * n, q * n and p + q * n, where n is an integer number, such a point can be freely translated all over the map. As a result, all the points with a period of unity are interconnected. Let us introduce an internal basis for them: a square -1 < p,q > 0, which will be called a pivotal square. The pivotal square is marked in Fig. 2 with pink color, while the general background is light-pink.
Let us consider in more detail the conditions (2) q * -1, p * -1 u p + q * -1. They mean that there are three pairs of lines on the + c)q map the shift between which is prohibited. It follows from the condition q * -1 (2b) and (2f) that all the points of the horizontal line q = -1 cannot be shifted upward. That is, a shift up to the q = 0 line is impossible, while the translation downward till the q = -2 line is possible. In other words, the q = -1 and q = 0 lines cannot be connected; therefore, they should be of different colors. The same is true for the p = -1 and p = 0 lines (from (2d) and (2e)), and for the p + q = -1 and p + q = -2 lines (from (2a) and (2c)).
The p = 0, q = 0 and p + q = -2 lines are black because in agreement with formulas (2) solutions of the integrals in points located on these lines are written down with power functions. However, a unique exception is present on each of these lines. For the p = 0 line, it is the point (0,-1) J(x + c)-1dx = ln(x+c), for the q = 0 line, (-1,0) jx-1dx = ln(x) and for the p + q = -2 line it is (-1,-1) Jx-1(x + c)-1dx =ln(x) -ln(x+c). These three points are marked with green color. Each of these black lines can be shifted in agreement with conditions (2). For example, the entire horizontal line p=0 can be shifted upward (2b) and along the diagonal upward (2f). Both these shifts lead to the p=l line. The green point (0-1) is drawn by these two shifts to two points (1,-1) and (1,-2). Because of this, a new black line p=1 contains already two green points. Such upward shifts can be continued without limit; one more green point is found on each resulting black line. The same is true also for the shift of the vertical black line q=0 to the right and along the diagonal downward, and a slanting line p+q=-2 both downward and to the right. So, a number of horizontal, vertical and slanting black lines with separate green points on them can be marked on the Xp(x + c)q map.
Let us now consider q = -1, p = -1 and p + q = -1 lines which cannot be black. We may not move the horizontal line q = -1 corresponding to Jxp(x+c)-1dx integrals upward to the black line q = 0, but translation along the line itself is possible (conditions (2c) and (2d)). As a result, all the points on this line at a distance of unity from each other are interconnected. Mutual translation along the horizontal line is prohibited only for two points (0,-1) u (-1,-1). Because of this, we shall choose -1 < p > 0 segment, which includes these two points and is adjacent to the reference square, as an internal basis on this line. Similarly, the basis segment -1 < q > 0 can be chosen along the vertical line q = -1 (conditions (2a) and (2b)), and the basis segment -1 < p,q > 0 along the slanting line p+q=-1 (conditions (2e) and (2f)). It is shown in Supplements 1 and 2 that it is possible to solve the integrals corresponding to the points of the basis segments only if p and q are rational numbers (p=m/n, where m, n are natural numbers):
Jx-m/n(x+c0)-1dx = (-1)n-m-1 £ct"-m ln x1'n + ct \/cc (3a)
Jx-1(x+c0)-m/ndx = (-1)™ 2 cM ln (x + c0fn + c |/c0 (3b)
Jx-m/n(x+c0)m/n-1dx = (-1)m-1ln ((x + co)/x)1n +St |, (3c)
However, if p, q are irrationals, then the solutions are not developed; because of this, we shall mark these three basis segments with a dotted line of green color because the logarithmic function is present in the solutions of the corresponding integrals. Formulas and conditions (2) allow us to spread the structure and color of the basis segments for all the three lines under consideration: q = -1, p = -1 and p + q = -1. They can also be shifted in agreement with conditions (2), which allows establishing the color and structure of other lines.
So, using the formulas and conditions (2), it was possible to reveal three pairs of lines in the xp(x + c)q map the shifts between which are prohibited; one line of each pair is black with a green point, while another line of each pair is green but dotted. These six lines will be called pivotal lines. Formulas (2) allow spreading the color and structure of these six lines over the whole xp(x + c)q map. It should be noted that, taking account of the color of the points, the obtained lines are symmetrical with respect to the slanting line A—A in Fig. 2.
Let us consider the conclusions which can be made about the color of node points in the xp(x + c)q map (p and q are integers). The marked lines intersect at these points (see Fig. 2). Three lines intersect in any node point; the color of a node point is the same as the color of the majority of lines intersecting at it. For example, a black horizontal line and two green lines (vertical and slanting) intersect at the point (-2,1). This point is green. Two black lines (horizontal and slanting) and one green (vertical) line intersect at point (-3,1). This point is black.
Now let us consider the reference square -1 <p, q < 0. As we have shown above, solutions of the integrals corresponding to points on p + q = -1 line are written down as a sum of logarithmic functions ifp u q are rational numbers. There are no solutions if p or q are irrationals. Separation of the points in the square -1 <p, q < 0 depending on whether p and q are rational or irrational is quite reasonable in our opinion. We may state that if at least one of the numbers p and q is irrational, then, the integrals corresponding to the points of the reference square are insoluble. Because of this, we shall not consider such integrals in the present work.
In the pivotal square -1 < p, q < 0, let us consider integrals Jx-p(x+c)-qdx, where p and q are rational numbers and p + q ^ -1. Solutions for some of these integrals are listed in tables [4], but these solutions are expressed with the help of elliptic integrals. For example, the integral No. 1.2.68.2 [4] J(x3+c)-1/2dx = -F(^,k)/31/4 can
3 f 2/3 1/2
be transformed using the u = x substitution into integral 1/3ju" (u+c) dx, which corresponds to the point (-2/3,-1/2). Similarly, starting from 1.2.71.2 [4] one may obtain the value of an integral for the point (-3/4,-1/2), from 1.2.76.1 [4] — for the point (-5/6,-1/2), from 1.2.79.1 [4] — for the point (-1/2,-1/4). Some other integrals are also listed in [4]; all of them correspond to the points of the xp(x + x)q map for which either p = -1/2 or q = -1/2.
It may be concluded that inside the pivotal square-1 < p, q < 0 the integrals corresponding only to the points with rational p and q have been solved by present. This is the p + q = -1 line drawn with a green dotted line, and separate points on lines p= -1/2 and q = -1/2, which will be marked with red, because the solutions of the corresponding integrals are expressed with the help of elliptic integrals of the 1st
and the 2nd kind.
In order to have a possibility to write the solutions of integrals in the pivotal square -1 < p, q < 0, let us introduce new designations in the form of special
functions FpFq (x). According to these designations, the solution of the integral Jx-1/2(x+c)-12 can be written down as F "1/2 Fc"1/2( x), and the solution for the integral Jx-3/4(x+c)-1/4dx asF-3/4F"1/4(x), etc. Two letters F mean that this integral corresponds to the xp(x + c)q map or to the integrals of Jxp(x+c)qdx kind. It should be noted that these designations are true if p and q are rational numbers.
So, with the help of formulas (2) we may write the result of integration of any expression xp(x + c)qdx, if p or q is natural number, if p+q is an integer negative number or if p and q are rational numbers, using a power, logarithmic function and the proposed special functions FpFq (x). The corresponding points are marked on the map with black, green or red color.
By changing variables, one may transform the expression xp(x + c)q dx into many other kinds: e^+V + c)qdx, (ln(x))^(ln((x) + c)qx~1dx, (sin(x))p(sin(x) + c)qcos(x)dx, etc. The proposed map is suitable for all these expressions. At the same time, the map of functions xp(x + c)qdx itself is a plane section of a three-dimensional structure which corresponds to more general expressions xp(x + c)qx(ln(x))sdx or xp(x + c)q(x + a)sdx.
Application of the proposed maps
A. Evaluation of the integrability of functions. Using the proposed map, it is possible to state beforehand, without integrating, basing on the position and color of the corresponding point, how many independent terms (items) and what functions will be present in the final result: power, logarithmic or special. Preliminary analysis with the help of the proposed maps may simplify search for solutions of integrals in the existing tables and predict the possibility of integration if functions using special software, for example Mathcad-2001. Tables of integrals cannot embrace all the functions the number of which is infinite. The proposed maps can be broadened without any limit increasing p and q parameters; it is also possible to introduce additional system axes, which is equivalent to the transition from flat maps to multidimensional structures. This will allow one to evaluate integrability of much larger function array. Tables can help in finding separate integrals or at best solutions for related integrals, while the proposed maps throw light upon the problem of integrating simple functions in general. In other words, these maps allow forming a new notion of the integrability of functions.
B. Integration of functions. Integration of functions with the help of the proposed maps becomes a universal and simple method, since successful integration implies the ability to change variables and correct application of the shift formulas (1) and (2), and sometimes also formulas (3). In addition, this method is vivid and purposeful because it is confirmed by the transition from one point on the map to another. For example, integrating with the help of the x?(x + c)q, if the point corresponding to the integral is black, one should consequently make shifts to one of three pivotal lines. The result of all shifts will be expressed as a recurrent formula comprising power functions. At the pivotal line, a solution of an integral in power functions is provided according to formulas (2). Shifts to the pivotal square can be made for the points of another color. The result of these translations will also be expressed as a recurrent formula consisting of power functions. A solution of an integral in the pivotal square will be expressed either with logarithmic functions according to formulas (3) or with special functions, for which designation FpFq (x) is proposed.
If a point is located far from the pivotal square, several ways of translation will be possible; correspondingly, several solutions will exist. Different solutions of one and the same integral were referred to as expressions «differing by a constant» in [3]. However, not more than two solutions of this type were given in [3].
As a practical application, let us consider the solutions of two integrals Ju-3(ln(u))-2du and Jx-1/2(x+c0)-1dx.
The Ju-3 (ln(u))-2du integral corresponds to the point (-3,-2) which is turquoise. Hence, the solution of this integral will contain at least one special function. Let us introduce the change of variable according to the formula (1a) up = xp(p+1) u-3 = x32, u = x-1/2 , du = -1/2x-3/2dx Ju-3(ln(u))-2du = -2j(ln(x))-2dx. Now let us use the formula (1c) -2j(ln(x))-2dx = -2[x(ln(x))-2 + 2j(ln(x))-1dx]/(-1). Further shifts are impossible; the result may be written down as
Ju-3(ln(u))-2du = -2j(ln(x))-2dx = -2[x(ln(x))-1 +Li-1(x)]/(-1) = -u-2(ln(u))-1 + Li-1(u-2). The correctness of the solution can be easily checked by differentiation: [-u-2(ln(u))-1 + Li-1(u-2)]' = 2u-3(ln(u))-1 + u-3(ln(u))-2 -2u-3(ln(u))-1
The Jx- 12(x+c0)-1dx integral corresponds to the point (-1.2, -1), which is located on the q=-1 line. Since p=-1/2 is a rational number, let us apply formula (3a): Jx-1/2(x+c0)-1dx = (-1)2-1-1(c1ln| x1/2+c1| + c2ln| x1/2+c2| )/c0. This is a general solution which depends on c0 value. Two solutions for which 1) c0 = b, 2) c0 = -b, where b is a positive number, are as follows:
1) Jx-1/2(x+b)-1dx = i(ln | x1/2 + i 4b | - ln | x1/2 - i 4b | )/4b .
2) Jx-1/2(x+b)-1dx = (ln | x1/2 + 4b | - ln | x1/2 - 4b | )/4b . The obtained solutions coincide with that in [3] 1'2'15'11'
C. Development of new algorithms to record functions. The designations proposed in the present work for Li-r(x) and FpFq (x) functions are in fact the
attempts to develop new nomenclature of functions. Some special functions can also be recorded with the help of these designations, for example elliptic integrals, Eulerian integrals, etc.
Conclusion
The proposed maps are vivid and form a new graphical representation of the integrability of functions. These maps allow one to carry out preliminary evaluation
of the possibility of integration, so they will be a convenient supplement to the
existing manuals of the tables if integrals. Translation formulas (1) and (2) depict periodicity and symmetry of the maps. In addition, formulas (1), (2) and (3) allow one to perform integration of functions. The proposed method can be spread without limit by supplementing the basis and introducing new independent system axes, which is
equivalent to the transition from flat to multi-dimensional maps. The proposed approach allows developing new designations of mathematical functions. The maps of integration of functions are vivid, simple and especially useful for those who only proceed to the study of higher mathematics.
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