Definite Integration in Mathematica V3.0

Transcription

1 Definite Integration in Mathematica V3. Victor Adamchi Introduction The aim of this aer is to rovide a short descrition of definite integration algorithms imlemented in Mathematica Version 3.. $Version Linu 3. HAril 5, 997L Proer Integrals All roer integrals in Mathematica are evaluated by means of the Newton-Leibni theorem a b f HL Ç = FHbL - FHaL where FHL is an antiderivative. It is well-nown that the Newton-Leibni formula in the given form does not hold any longer if the antiderivative FHL has singularities on an interval of integration Ha, bl. Let us consider the following integral + + þþ Ç where the integrand is a smooth integrable function on an interval H, L.

2 mier.nb + + PlotA þþþ, 8,, <E ú Grahics ú 3 As it follows from the Risch structure theorem the corresondent indefinite integral is doable in elementary functions int = + + þþþ Ç tan-i j þ - - y - - { tan-i j þþþþ + y - { If we simly substitute limits of integration into the antiderivative we get an incorrect result. Limit@int, D - Limit@int, D -tan -i j y - tan - J þþ 5 { N This is because the antiderivative is not a continuous function on an interval H, L. It has a jum at =, which is easy to see in the following grahic.

3 mier.nb 3 Plot@int, 8,, <D ú Grahics ú The right way of alying the Newton-Leibni theorem is to tae into account an influence of the jum Limit@int, ->, Direction -> D - Limit@int, ->, Direction -> -D + Limit@int, ->, Direction -> D - Limit@int, ->, Direction -> -D -tan -i j y - tan - J þþ 5 { N Mathematica evaluates definite integrals in recisely that way. + + þþþ Ç -tan -i j y - tan - J þþ 5 { N The origin of discontinuities of antiderivatives along the ath of integration is not in the method of indefinite integration but rather in the integrand. In the discussed eamle, the integrand has four singular oles that become branch oints for the antiderivative. NRoots@ ==, D == É Ã == É Ã == É Ã == É Connected in airs these oints mae two branch cuts. And the ath of integration crosses one of them. The following ContourPlot clearly eoses the roblem.

4 mier.nb ƒ. -> + I ydd, 8, -3, <, 8y, -3, <, ContourShading -> False, Contours ->, PlotPoints ->, Eilog -> 8Hue@D, Thicness@.5D, Line@88, <, 8, <<D<D ú ContourGrahics ú We see that in a comle lane of the variable the antiderivative has two branch cuts (bold blac vertical lines) and the ath of integration, the line (,), intersects the right branch cut. Obviously, by varying the constant of integration we can change the form of the antiderivative so that we would get various forms of branch cuts. Here df HL we understand the constant of integration as a function f() such that þþ is ero. As a simle eamle let us d consider the ste-wise constant function!!!!!! þþþ!!!!!! SimlifyA þ E Thining hard, we can built an antiderivative that does not have a branch cut crossing a given interval of integration. However, we can never get rid of branch cuts! Analysis of the singularities of antiderivatives is a time consuming and sometimes heuristic rocess, esecially if trigonometric or secial functions are involved in antiderivatives. In the latter case Integrate may not be able to detect all singular oints on the interval of integration, which will result in a warning message Integrate::gener : Unable to chec convergence

5 mier.nb 5 You should ay attention to the message since it warns you that the result of the integration might be wrong. Imroer Integrals It is quite clear that the above rocedure cannot cover the whole variety of definite integrals. There are two reasons behind that. First, the corresondent indefinite integral cannot be eressed in finite terms of functions reresented in Mathematica. For instance, Cos@Sin@DD Ç coshsinhll Ç However, the definite integral with the secific limits of integration is doable. Cos@Sin@DD Ç J HL Second, even if an indefinite integral can be done, it requires a great deal of effort to find limits at the end oints. Here is an eamle, þþþ Tan@D ƒ Ç seci j y { This is an imroer integral since the to limit is a singular oint of the integrand. The result of indefinite integration is Tan@D ƒ Ç F H þþþ + +,;+ þþþ ; -tan HLL tan + HL þþ þþ + To find the limit at = þþþþ one has to, first, develo the asymtotic eansion of the hyergeometric Gauss function at, and second, construct the asymtotic scale for the Puiseu series. Currently, Mathematica s Series structure is based on ower series that allow only rational eonents. Following we resent a short descrition of the algorithm for evaluating imroer integrals. The overall idea has been given in [] and [3]. Some ractical details regarding "logarithmic" cases are described in [] and [].

6 6 mier.nb The General Idea By means of the Mellin integral transform and Parseval s equality, a given imroer integral is transformed to a contour integral over the straight line Hg -É, g+él in a comle lane of the arameter : f HL f J N þþþ Ç = þþþ É where f * HL and f * HL are Mellin transforms of f HL and f HL g+é g-é f * HsL f * HsL -s Ç s () f * HsL = f HL s- Ç The real arameter g in the formula () is defined by conditions of eistence of Mellin transforms f * HL and f * HL. Finally, the residue theorem is used to evaluate the contour integral in the right side of the formula () þþþ É G m f HL Ç = Ç res = a = f HL where G is a closed contour, and a are oles of f() that lie in a domain bounded by G. The success of this scheme deends on two factors: first, the Mellin image of an integrand f HL f H þþþþ L must eist, and, second, it should be reresented in terms of Gamma functions. If these conditions are satisfied then the contour integral in (), called the Mellin-Barnes integral or the Meijer G-function, can almost always be eressed in finite terms of hyergeometric functions. This fact is nown as Slater s theorem (see [] and [5]). We said "almost always", since there is a secial case of the G-function when the latter cannot be reduced to hyergeometric functions but to their derivatives with resect to arameters. By analogy with linear differential equations with olynomial coefficients, such a singular case of the G-function is named a logarithmic case. The modified Bessel function K HL is such a classical eamle, since its series reresentation K HL =-I HL logj N + È = y H + L i þ! j þ y { cannot be eressed via hyergeometrics. Mellin-Barnes Integrals These integrals are defined by (see[6]) þþþ É L ghsl -s Ç s where g HsL = À n j= G Ha j + sl À m j= G Hb j - sl þ l GHc j + sl À j= GHd j - sl þþ É j=

7 mier.nb 7 and the contour L is a line that searates oles of G Ha j + sl from G Hb j - sl. Let us investigate when the integral eists. On a straight line L, = Hg -É, g+él the real art of s ³ L is bounded, and ImHsL. Using the Stirling asymtotic formula GH +ÉyL =!!!!!!! y - þþþþ È - þþþþ i y j + O i j y þþþ y y {{, y we can deduce that the integrand ghsl -s vanishes eonentially as Im(s) if m + n - - l >, and arghl < þþþþ Hm + n - - ll. If m + n - - l =, then must be real and ositive. Some additional conditions are required here (see details in []). It is clear that we can not simly imly the residue theorem to this contour integral. We need initially to transform the contour L to the closed one. There are two ossibilities: we can either transform L into the left loo L -, the contour encircling all oles of G Ha j + sl, or to the right loo L +, the contour encircling the oles of G Hb j - sl. The criteria of which contour should be chosen aears from the convergence of the integral along that contour. On the left-hand loo L -, the imaginary art of s ³ L - is bounded and ReHsL -. Assuming that argh-sl < þþþþ, and maing use of the Stirling formula GHL =!!!!!!! - þþþþ È - J + OJ NN,, arghl < and the reflection formula of the Gamma function, we find that the Mellin-Barnes integral over the loo L - eists, if n + l - m - >. If n + l - m - =, then must be within the unit dis <. If is on a unit circle =, the integral converges if i n Re j Ç j= m a j + Ç j= b j + Ç j= l c j + Ç j= y d j <- + n - { On the right-hand loo L +, the imaginary art of s ³ L + is bounded and ReHsL +. Proceeding similarly to the above, we find that the Mellin-Barnes integral over the loo L + eists, if n + l - m - <. If n + l - m - =, then must be outside of the unit dis. Additional conditions are required if is on a unit circle. After determining the correct contour, the net ste is to calculate residues of the integrand. Since the integrand contains only Gamma functions, this tas is more or less formal. We don t even need to calculate residues of the integrand, but go straightforwardly to the generalied hyergeometric functions. The only obstacle is the logarithmic case, which occurs when the integrand has multile oles. We have to searate two subclasses here: the integrand that has a finite number of multile oles, and the integrand that has infinitely many multile oles. The Mathematica integration routine has a full imlementation of the former case. In the latter, the integration artificially restricted by the second order oles, since for the higher order oles it would lead to infinite sums with higher order olygamma functions. This class of infinite sums are etremely hard to deal with, symbolically and numerically. If such a situation is detected Integrate returns the Meijer G-function. However, there are some very secial transformations of the G-function, which could avoid buly infinite sums with olygamma functions, and give a nice result in terms of nown functions. In [] I demonstrated a few transformations that reduce the order of the G-function and mae it ossible to handle secial class of Bessel integrals in terms of Bessel functions.

8 8 mier.nb An Eamle Consider the integral According to the formula (), we have sinhl W= þ H + L Ç W= þþþþ + sinj þþþ ƒ N þþþ Ç = þþþ É where f * HsL and f * HsL are Mellin transforms of È - and sinh þþþþ L: s- f = þþþ + Ç If i jrehsl > Ä ReHsL <, cscj þþþ s N, s- þþþþ + Ç y { g+é g-é f * HsL f * HsL Ç s Therefore, f = SinA þþþþ E s- Ç If i s jrehsl >- Ä ReHsL <, -GH-sL sinj þþþ N, s- sin i j y Ç y { { W=-þþþ É g+é g-é GH-sL Ç s where g is defined by conditions of the eistence of Mellin transforms <g=rehsl < As it follows from the revious section we can transform the integration contour Hg -, g+él into the right loo L +. Then, using the residue theorem we evaluate the integral as a sum of residues at simle oles s =,,.... W=-þþþþ È = H -ÈL - þ È H-L þþþþ!

9 mier.nb 9 Meijer G-Function We did not intend to give a comlete icture of the Meijer G-function here, but only necessary facts. In Version 3. the G-function is defined by MeijerG@88a, a,..., a n <, 8a n+, a n+,..., a <<, 88b, b,..., b m <, 8b m+, b m+,..., b q <<, D = þþþ É L m,n G i,q j a, a, b, b,..., a y =..., b q { À m n i= GHb i + sl À i= GH - a i - sl þþ þþ q þ þþþþ À i=n+ GHa i + sl À i=m+ GH - b i - sl -s Ç s and the contour L is a left (or right) loo L - (or L + ) searated oles of G Hb j + sl from G H - a j - sl. The current imlementation of the Meijer function has two noticeable features that differ it from the classical G- function. First, you are not allowed to choose or move the contour of integration, and, second, MeijerG has an otional arameter (the classical G-function does not) that regulates the branch cut. The Meijer function is suorted symbolically and numerically. MeijerG@88<, 8<<, 88<, 8<<, D þþþþ + MeijerGA88-<, 8, <<, 99, þþþþ =, 8<=, þþþþ E MeijerG i j88-<, 8, <<, ::, >, 8<>, y { N@%D.9893 Only in very trivial cases MeijerG is simlified automatically to the lower level secial functions. Beyond that all further transformations are assigned to FunctionEand: FunctionEand@%%D!!!! -!!!! I HL þþþþ -!!!! þ þ I HL È È MeijerG is interlaced with indefinite and definite integration, and with solving linear differential equations with olynomial coefficients. Here is an eamle related to definite integration:

10 mier.nb D þþþ!!!!!!!!!!!!!! - Ç!!!! MeijerG i j:8<, :- >>, 88-,, <, 8<<, y { The first element of the second argument of MeijerG, which is {-,,}, indicates that the integrand of the corresondent contour integral contains the roduct of three Gamma functions GHs - L GHsL GHsL and has an infinite number of trile oles at s=-, =,,,.... From the design oint of view it is definitely an advantage for Integrate to return a short object, MeijerG, rather than enormous infinite sums involving derivatives of the Gamma function. Hyergeometric Functions The essential art of integration is the generalied hyergeometric function, which is defined by HyergeometricPFQ@8a,..., a <, 8b,..., b q <, D = F i q j a, a, b, b,..., a y =..., b q { À j= Ha j L È þ q þ À j= Hb j L = þþ! where HaL is a Pochhammer symbol Ha j L = É l= Ha j + l - L = G Ha j + L þ G Ha j L Conditions of convergence can be easily obtained by alying the d Alembert test. It follows, F q converges for all finite if ˆ q, and for < if = q +. Additional conditions are required on the circle of convergence =. The above series definition of F q has an ecetional case, when = and q = - such HyergeometricPFQ is defined via the confluent hyergeometric function HyergeometricU HyergeometricPFQ@8a, b<, 8<, D a i j- y U i { ja, a - b +, - y { For r and = q +, the hyergeometric function is defined as an analytic continuation via the Mellin- Barnes integral, with the branch cut [, ): q+ F i q j a, a, b, b,..., a q+ y =..., b q { q À = GHb L þ þ q+ GHa L À = g+é þþþ É g-é G HsL À q+ j= G Ha j - sl þþ q þþþ G Hb j - sl À j= H-L -s Ç s

11 mier.nb where the straight line Hg -É, g+él searates oles of G(s) from G Ha j - sl. Almost all symbolic simlifications of F q to the lower level functions are done automatically HyergeometricFA þþþþ 3, þþþþ 3, þþþþ 3, þþþþ E 9!!!! 3 I!!!! 3 +3 loghlm HyergeometricPFQA9, þþþþ 3, þþþþ 3 =, 9 þþþþ 3, þþþþ 3 =,E 9 yhli j y 3 { HyergeometricPFQ@8,,, <, 8,, <, -D 7 þþþ H loghl + 3 H3LL The ecetion is the Gauss function HyergeometricF. Since it has so many different transformation and simlification rules some of them are assigned to FunctionEand: HyergeometricFA, þþþþ, þþþþ 5, þþþþ E F i j, ; 5 ; y { FunctionEand@%D cot - I!!!! M log I - þþþ!!!! M log I + þþþ!!!! M þ!!!! þ - þ!!!! + þ!!!! Princial-Value Integrals Consider the integral - þþþþ Ç Integrate::idiv : Integral of - Ç does not converge on 8-, <.

12 mier.nb which does not eist in the Riemann sense, since it has a nonintegrable singularity at =. However, if we isolate = by e > from the left and by e > from the right and tae a limit of corresondent integrals when e and e, we obtain lim e e i j - -e Ç + e Ç y { = lim HlogHL + loghe L - loghe LL The double limit eists, and so it is a given integral, if and only if e =e. Such understanding of divergent integrals is called the rincial-value or the Cauchy rincial-value. It is easy to see that if an integral eists in the Riemann sense, it eists in the Cauchy sense. Thus, the class of Cauchy integrals is larger than the class of Riemann integrals. In Version 3., definite integrals in the Riemann sense and rincial-value integrals are searated by the new otion PrincialValue. If you want to evaluate an integral in the Cauchy sense, set the otion PrincialValue to True (the default setting is False). Here is an eamle e e IntegrateA þþþþ, 8, -, <, PrincialValue TrueE loghl IntegrateA þ þ, 8, -, <, PrincialValue TrueE Sinh@5 + D loghcothhl tanhh3ll 5 IntegrateA Sin@3D, 9, - þþþþ 3 H3 L, þþþ =, PrincialValue TrueE C þ 9 If the integrand contains high-order olynomials, Integrate returns RootSum objects IntegrateA, 8, -, <, PrincialValue TrueE É þþ þþ RootH# 7 + # + &,L 6 RootSum i logh-# - L j#7 + # + &, þ þþþþ & y 7# 6 + { + RootSumi logh - #L j#7 + # + &, þþþ 7# 6 + &y { Here are eamles of integrals with movable singularities

13 mier.nb 3 IntegrateA a - Tan@D, 9,, þþþþ =, PrincialValue TrueE If i j a >, þ a +, þþþ a - tan HL Ç y { IntegrateA l þ - a, 8,, <, PrincialValue TrueE If i ja > Ä ReHlL >- Ä ReHlL <, -a l cothll, - a Ç y { Here Integrate detects that the integrand has a singular oint along the ath of integration if the arameter a is real ositive. l New Classes of Integrals In this section we give a short overview of secific classes of definite integrals, which were essentially imroved in the new version. Mathematica now is able to calculate almost all indefinite and about half of definite integrals from the well-nown collection of integrals comiled by Gradshteyn and Ryhi. Moreover, the Version 3. maes it ossible to calculate thousands of new integrals not included in any ublished handboos. integrals of rational functions The RootSum object has been lined to Integrate to dislay the result of integration in a more elegant and shorter way - þþ Ç RootSum i j# 7 + # 3 log H - #L # - log H - #L + &, þþ þ & y - 7# 6 + 3# { RootSum i j#7 + # 3 log H-#L # - log H-#L + &, þþþþ & y 7# 6 + 3# { logarithmic and Polylogarithm integrals Log@D Log@ + D þþþþ + Ç 8 H loghl -H3LL

14 mier.nb - D 3 H - L Ç - 6Li 3H þþþ - þþþ L H - L ellitic integrals The acage, conducted an ellitic integration, is now autoloaded (as are as all other integration acages), so you don t need to worry about its loading anymore þþþ - þþþ þþ!!!!!!!!!!!!!!!!! Cos@D FJ 8 N Ç þþþ þþ!!!!!!!!!!!!!!!!! Sin@D Ç 3 F i j, 3,; 3, 3 ;y { integrals involving Bessel functions t E@-t D BesselI@, td BesselK@, td Çt -3 +!!!! i È K j y { BesselJ@, D E@- þþþ þþþ D Ç J HL K HL AiryAi@D Ç GH- þþþþ 6 - þþ L þþþ!!!! 3!!!! 6 3 3ƒ

15 mier.nb 5 integrals involving non-analytic functions + I Sin@Abs@DD Ç --I H3 + 3 ÉL i!!!! j þþþ 3 - cosi!!!!!!!! M cosi M y þþþþ 3!!!! - þþþ 3!!!! { 3 þ Ma@Sin@D, Cos@ DD Ç 3!!!! 3 þ Additional Features Mathematica s caability for definite integration gained substantial ower in the new version. Other essential features of Integrate not discussed in the revious sections are convergence tests and the assumtions mechanism. Conditions In Version 3. Integrate is "conditional." In most cases, if the integrand or limits of integration contains symbolic arameters, Integrate returns an If statement of the form If[ conditions, answer, held integral ] which gives necessary conditions for the eistence of the integral. For eamle l- E@-a D Ç IfJReHaL > Ä ReHlL >, a -l GHlL, È - a l- Ç N Setting the otion GenerateConditions to False revents Integrate from returning conditional results (as in Version.): Integrate@ l- E@-a D, 8,, <, GenerateConditions FalseD a -l GHlL If a given definite integral has symbolic arameters, then the result of integration essentially always deends on certain secific conditions on those arameters. In this eamle the restrictions ReHaL > and ReHlL >

16 6 mier.nb came from conditions of the convergence. Even when a definite integral is convergent, some other conditions on arameters might aear. For instance, the resence of singularities on the integration ath could lead to essential changes when the arameters vary. The net section is devoted to the convergence of definite integrals. Convergence The new integration code contains criteria for the convergence of definite integrals. Each time Integrate eamines the integrand for convergence Cos@D þ - Ç Integrate::idiv : Integral of coshl does not converge on 8-, <. þþþ þþþ þþþþ coshl þþþ Ç - This integral has a nonintegrable singularity at =. Thus, Integrate generates a warning message and returns unevaluated. However, the integral eists in the Cauchy sense. Setting the otion PrincialValue to True, we obtain IntegrateA þ Cos@D, 8, -, <, PrincialValue TrueE ƒƒfullsimlify Consider another integral with a symbolic arameter a r- ArcTan@D Ç If i j ReHrL >-, y HL I þþþ r+ M -yhl I þþþ r+3 M + þþþ, r- tan - HL Ç y r { The integral has a singular oint at =, which is integrable only if ReHrL >-. If you are sure that a articular integral is convergent or you don t care about the convergence, you can avoid testing the convergence by setting the otion GenerateConditions to False. It will mae Integrate return an answer a bit faster. Setting GenerateConditions to False also lets you evaluate divergent integrals IntegrateA þþþþ, 8,, <, GenerateConditions FalseE loghl

17 mier.nb 7 Assumtions The new otion Assumtion is used to secify articular assumtions on arameters in definite integrals. Consider the integral with the arbitrary arameter y Here we set the otion Assumtions to ReHyL > È - - y þ!!! þ Ç IntegrateA E@- - yd þ, 8,, <, Assumtions Re@yD > E!!!! È þþ!!!! i y 8 yk þþþþ j þþ y y 8 { Setting Assumtions to ReHyL <, we get a different form of the answer IntegrateA E@- - yd þ, 8,, <, Assumtions Re@yD < E!!!! þþ È y 8!!!!!!! - y II - þþþþ I y 8 M + I þþþþ I y 8 MM þþ þþ!!!! though the integral is a continuous function with resect to the arameter y PlotANIntegrateA E@- - yd þ, 8,, <E, 8y, -, <E!!!! ú Grahics ú The net integral is discontinuous with resect to the arameter g

18 8 mier.nb IntegrateA þþ H - Cos@g þ DL þ, 8,, <, Assumtions g>e Hg -L Cos@D H - Cos@g DL IntegrateA þþþ, 8,, <, Assumtions - <g<e Cos@D H - Cos@g DL IntegrateA þþþ, 8,, <, Assumtions g<-e i y þ - j "####### g { If the given assumtions eactly match the generated assumtions, then the latter don t show u in the outut; otherwise, Integrate roduces assumtions comlementary to ones given: Integrate@ n- H - L m-, 8,, <, Assumtions Re@nD > D If i jrehml >, GHmL GHnL GHm+nL, H - L m- n- Ç y { References. O.I. Marichev, Handboo of integral transforms of higher transcendental functions: theory and algorithmic tables, Ellis Horwood, V.S. Adamchi, K.S. Kolbig, A definite integral of a roduct of two olylogarithms, SIAM J. Math. Anal. (988), V.S. Adamchi, O.I. Marichev, The algorithm for calculating integrals of hyergeometric tye functions and its realiation in REDUCE system, Proc. Conf. ISSAC 9, Toyo (99), -.. V. Adamchi, The evaluation of integrals of Bessel functions via G-function identities, J. Com. Alied Math. 6(995), L.J. Slater, Generalied Hyergeometric Functions, Cambridge Univ. Press, Y.L. Lue, The Secial Functions and Their Aroimations, Vol., Academic Press, 969.