Integrals of functions containing parameters
IINTEGRALS OF FUNCTIONS CONTAINING PARAMETERS 1
Integrals of functions containingparameters
Robert M. Corless, David J. Jeffrey and David R. Stoutemyer “ There is always a well–known solution to every human problem– neat, plausible, and wrong .”H. L. Mencken [15] The
Non Sequitur cartoon for 20 January 2016Calculus students are taught that an indefinite integral is definedonly up to an additive constant, and as a consequence generations ofstudents have assiduously added “ + C ” to their calculus homework.Although ubiquitous, these constants rarely garner much attention, andtypically loiter without intent around the ends of equations, feelingneglected. There is, however, useful work they can do; work which isparticularly relevant in the contexts of integral tables and computeralgebra systems. We begin, therefore, with a discussion of the context,before returning to coax the constants out of the shadows and assignthem their tasks.Tables of integrals are inescapable components of calculus text-books [2, 18], and there are well known reference books that publishvoluminous collections [1, 9, 16, 20]. A modern alternative to inte-gral tables is provided by computer algebra systems (CAS), which arereadily available on computing platforms ranging from phones to super-computers. These systems evaluate integrals using a mixture of integraltables and algebraic algorithms. A feature shared by tables and com-puter systems is the fact that the formulae usually contain parameters.No one would want a table of integrals that contained separate entriesfor x , x and x , rather than one entry for x n , and many tables includeadditional parameters for user convenience; for example, there will beentries for integrals containing sin ax , rather than the sufficient, butless convenient, sin x .Although parameters add greatly to the scope and convenience ofintegral tables, there can be difficulties and drawbacks occasioned bytheir use. We shall use the word specialization to describe the actionof substituting specific values (usually numerical, but not necessarily)into a formula. The specialization problem is a label for a cluster of The truth of this statement is reinforced by the fact that it is often misquoted. a r X i v : . [ m a t h . HO ] S e p THE MATHEMATICAL GAZETTE (submitted)issues associated with formulae and their specialization, the difficultiesranging from inelegant results to invalid ones. For example, in [12] anexample is given in which the evaluation of an integral by specializinga general formula misses a particular case for which a more elegantexpression is possible. The focus here, however, is on situations inwhich specialization leads to invalid or incorrect results. To illustratethe problems, consider an example drawn from a typical collection [19,ch8, p346, (5)]: I = (cid:90) (cid:0) α σz − α λz (cid:1) dz = 12 ln α (cid:18) α λz λ + α σz σ − α ( λ + σ ) z λ + σ (cid:19) . (1.1)Expressions equivalent to this are returned by Maple, Mathematica andmany other systems, such as the Matlab symbolic toolbox.Before we proceed, we acknowledge that some readers may questionwhether anyone at all competent would write the integral this way:surely there are better ways? Why not transform α σz into exp( pz ),where p = σ ln α , and thus reduce the number of parameters? Orscale the variable of integration to absorb, say, the λ ? Such actionsare possible for people who are free to recast problems in convenientways, for example, if (1.1) were an examination question devoid ofcontext. CAS, in contrast, are obliged to deal with expressions as theyare presented, either by users or by other components within the systemitself; and in the general case some of these “obvious” simplificationsand transformations are surprisingly difficult to discover automatically.Humans are still superior at simplification, we believe.Returning to the answer as returned by the CAS, it is easy to seethat the specialization σ = 0 leaves the left side of (1.1), the integrand,well defined, but the expression for its integral on the right-hand sideis no longer defined. If we pursue this further, we see that there aremultiple specializations for which (1.1) fails, viz. α = 0, α = 1, λ = 0, σ = 0, λ = − σ , and combinations of these. The question of howor whether to inform computer users of these special cases has beendiscussed in the CAS literature many times [6].This brings us to the second theme of this discussion: comprehensiveand generic results. A comprehensive result lists explicit expressionsfor each set of special parameter values, while a generic result is correctfor ‘most’ or ‘almost all’ values of the parameters. Let us consider howa comprehensive result for (1.1) would look.NTEGRALS OF FUNCTIONS CONTAINING PARAMETERS 3 I = λ ln α (cid:0) α λz − α − λz − zλ ln α (cid:1) , (cid:34) λ + σ = 0, σ (cid:54) = 0 , α (cid:54) = 0 , α (cid:54) = 1 ; z + 12 λ ln α (cid:0) α λz ( α λz − (cid:1) , (cid:34) σ = 0, λ (cid:54) = 0 , α (cid:54) = 0 , α (cid:54) = 1 ; z + 12 σ ln α ( α σz ( α σz − , (cid:34) λ = 0 , σ (cid:54) = 0 α (cid:54) = 0 , α (cid:54) = 1 ;0 , (cid:104) α = 1 ;0 , (cid:104) α = λ = σ = 0 ;ComplexInfinity , (cid:34) α = 0 , (cid:60) ( λz ) (cid:60) ( σz ) < , (cid:34) α = 0 , (cid:60) ( λz ) (cid:60) ( σz ) ≥ α (cid:18) α λz λ + α σz σ − α ( λ + σ ) z λ + σ (cid:19) , otherwise (generic case)(1.2)Conditions are here shown as in printed tables; otherwise they couldbe presented using the logical ∨ and ∧ operators.To generalize, we denote a function depending on parameters by f ( z ; p ), with z being the main argument, here the integration variable,and p a list of parameters. The definition is then: Definition 1.1 A comprehensive antiderivative of a parametric func-tion f ( z ; p ) is a piecewise function F ( z ; p ) containing explicit conse-quents for each special case of the parameters. Designers of computer algebra systems are reluctant to return com-prehensive expressions by default, because they can quickly lead to un-manageable computations, and as well many users might regard themas too much information . Instead, tables and CAS commonly adopt theapproach of identifying a generic case, which is then the only expres-sion given; in the case of CAS, the generic case is typically returnedwithout explicitly showing the conditions on the parameters. In the consequent : following as a result or effect; the second part of a conditionalproposition, dependent on the antecedent. THE MATHEMATICAL GAZETTE (submitted)case of tables, any special-case values would be used to simplify the in-tegrand and then the resulting integrand and its antiderivative wouldbe displayed as a separate entry somewhere else in the table.
Definition 1.2 A generic antiderivative is one expression chosen froma comprehensive antiderivative that is valid for the widest class of con-straints. Remark 1.3
The above definition is an informal one, since the choiceof which result to designate as generic may include personal taste.
We now come to the third theme of the discussion: the treatment ofremovable discontinuities. Consider the improper integral (cid:90) ln ( x (1 − x )) dx = (cid:20) x ln ( x (1 − x )) − x − ln(1 − x ) (cid:21) . (2.1)The expression for the integral contains a removable discontinuity ateach end, and a computer system (and we hope students) would au-tomatically switch to limit calculations to obtain the answer. In thissection, by extending the handling of removable discontinuities to con-stants of integration, we introduce a new idea for handling the special-ization problem in integration . The idea is new in the sense that we donot know of any published discussion, but it originated with WilliamKahan [14] and was circulated informally possibly as early as 1959.The example (1.2) dramatically illustrates the potential size of com-prehensive antiderivatives, but is, unsurprisingly, too cumbersome forexplaining ideas. We turn to simpler examples. We begin with thecomprehensive antiderivative known to all students of calculus : (cid:90) z α d z = ln z , if α = − ,z α +1 α + 1 , otherwise (generic case) . (2.2)Substituting α = − / z . Oftenwhen a substitution fails, a limit will succeed, so we try the limit as The specialization problem is not confined to integration. Any formula whichuses parameters to cover multiple cases is likely to have some specialization prob-lems. The ideas presented here, however, apply specifically to integration. Most textbooks use x n , but we wish to emphasize continuity and use α insteadof n . We also use z and α because we are thinking in the complex plane, and weprefer ln z instead of ln | x | for the same reason. NTEGRALS OF FUNCTIONS CONTAINING PARAMETERS 5 α → −
1. Disappointingly, this also fails, but we can examine how thelimit fails by expanding the generic case as a series about the pole at α = −
1, that is, treating α + 1 = ε as a small quantity. z α +1 α + 1 = e ε ln z ε = 1 + ε ln z + O (cid:0) ε (cid:1) ε . (2.3)If we can remove the leading term of the series, namely 1 /ε , then thenext term gives us the desired ln z . But an integral needs a constant !So, an equally correct integral is (cid:90) z α d z = z α +1 α + 1 − α + 1 , and now the limit as α → − z . Thus the comprehensiveantiderivative, (cid:90) z α d z = ln z , if α = − ,z α +1 − α + 1 , otherwise . (2.4)is continuous with respect to α , and the generic antiderivative nowcontains the exceptional case as a removable discontinuity [14]. Definition 2.1
Let a function F ( z ; p ) be an indefinite integral of anintegrand f ( z ; p ) . That is, F ( z ; p ) = (cid:90) f ( z ; p ) d z . If a point p c in parameter space exists at which F ( z ; p ) is discontinuous with respect to one or more members of p , and if a function C ( p ) , whichserves as a constant of integration with respect to z , has the propertythat F ( z ; p ) + C ( p ) has only a removable discontinuity and thus can bemade continuous with respect to p at p c , then C ( p ) is called a Kahanianconstant of integration . Remark 2.2
The definition does not guarantee the existence of C ( p ) .If, for some value of p , the integral does not exist, then there will beno Kahanian. The lack of a constant in (2.2) betrays our CAS allegiance: it is a rare CASthat adds a constant, because the user can easily add one (and name the constant)by typing, for instance, “ int(f,x)+K ”. Since it is a function of p , one can question whether it should be called aconstant. It is constant with respect to z , and this seems a useful extension ofcalculus terminology. THE MATHEMATICAL GAZETTE (submitted)Figure 1: A parametrically discontinuous integral. The real part of eachconsequent of the comprehensive integral (2.5) is plotted. The surfaceshows the generic expression plotted as a function of the integrationvariable z and the parameter α . The surface curves up to becomesingular all along the line α = 0. The detached curve hovering belowthe surface in the plane α = 0 is the special case integral.A second example shows the effect of Kahanian constants graphi-cally. To begin with, consider the comprehensive antiderivative (cid:90) dzz √ z − α = − √ z , α = 0 , α arccot α √ z − α , generic . (2.5)We write √ z rather than | z | so that the expression is valid for non-real z . Figure 1 shows this antiderivative as a three-dimensional plottreating both z and α as real variables. The generic expression is thenthe surface shown in the plot; it becomes singular along the line α = 0(as α − ). The special case α = 0 is shown as a detached curve confinedto the plane α = 0. It can be seen hovering forlornly underneath thesurface of the generic integral, dreaming of gaining an invitation to theparty. For | z | < | α | , the values of the integral are non-real, but only thereal part is plotted, because that displays the properties of interest . The definition of arccot varies between computer systems and amongst refer-ences, and even between different printings of the same reference work [7]. Theplots shown here were made with Mathematica, and other systems such as Mapleand Matlab may create different plots.
NTEGRALS OF FUNCTIONS CONTAINING PARAMETERS 7Figure 2: A parametrically continuous integral. The real part of eachconsequent of the comprehensive integral (2.6) is plotted. The surfaceshows the generic expression plotted as a function of the integrationvariable z and the parameter α . The surface curves down, and is sin-gular only at z = α = 0. The special case curve lies happily within thesurface in the plane α = 0.To achieve continuity, we now add Kahanian constants to each con-sequent of the comprehensive antiderivative. We show the new con-stants in bold print. (cid:90) dzz √ z − α = − √ z + , α arccot α √ z − α − α arccot α √ − α . (2.6)Figure 2 shows the real part of the new expressions. The extra termin the generic (the lower) consequent in (2.6) subtractively cancels aparametric pole asymptotic to 1 /α as α →
0, which can be seen inthe generic consequent of result (2.5). This makes the parametricallycontinuous expression now approach the same values from both sidesof α = 0, converting that parametric pole to an indeterminate slit in anotherwise parametrically continuous surface—a removable singularity.Moreover, the extra term 1 in the α = 0 consequent of (2.6) raises thespace curve exactly the right amount to make it contiguous with thesurface on both sides, thus removing that removable singularity. THE MATHEMATICAL GAZETTE (submitted) We have defined the specialization problem as the failure of a genericformula when particular values are substituted for parameters, with thegeneric integrals above being examples. One way to avoid specializationproblems completely would be to specify all parameters in advance, thatis, delay starting a calculation until the parameters are known. This,however, negates the very power and generality that algebra extendsto us. As well, it is not always the case that one can know parametersin advance; for example, a parameter might depend on the outcome ofan intermediate computation.The specialization problem can be looked at another way, a way per-haps more suitable for computer algebra. There are variations possiblein the order in which operations are applied. For example, in Maplesyntax, where as usual operations take precedence from the inside out,it is the difference between subs( n=-1, int( x^n, x ) ) ;int( subs( n=-1, x^n ), x ) .
The first gives a division by zero error; the second gives ln x . Thechallenge, then, is to retain the generality of algebra, while findingways to react to exceptional values of parameters. There are at least two ways to evaluate a special case of a knownparametric definite integral, especially in the computer-algebra setting.The first way is the typical human way: by working on each limitseparately. Thus to evaluate (cid:90) ba f ( x, p ) d x , one first finds an indefinite integral F ( x, p ), then makes any substitu-tions F ( x, p c ), then evaluates and simplifies F ( b, p c ) − F ( a, p c ). Forthis approach, the Kahanian term is important, because it allows a sub-stitution to be performed correctly, possibly as a limit lim p → p c F ( x, p ).For the second way, let us explicitly notate the presence of a Ka-hanian term, so that the indefinite integral is F ( x, p ) + C ( p ), wherenow F is any function that satisfies F (cid:48) = f and C ( p ) is the KahanianNTEGRALS OF FUNCTIONS CONTAINING PARAMETERS 9constant. We now perform the definite integral before specializing. (cid:90) ba f ( x, p ) d x = [ F ( b, p ) + C ( p ) ] − [ F ( a, p ) + C ( p ) ]= F ( b, p ) − F ( a, p ) . (3.1)Now we must evaluate this expression as p → p c . If we simplify eachterm separately, the calculation may fail, but keeping the terms to-gether , the calculation succeeds.For example,lim n →− (cid:90) ba x n d x = lim n →− (cid:18) b n +1 n + 1 − a n +1 n + 1 (cid:19) = ln b − ln a , using the methods of (2.3). Again we point out that separately thelimits of each term need not exist. Computer systems, like people, cansometimes succeed when asked one way and not another. With theKahanian form, either approach succeeds. We now give an example of Kahanian terms used in the solution of adifferential equation. A standard topic in physics and engineering isresonance. The equation of a forced, frictionless, harmonic oscillator is d x ( t ) dt + k x ( t ) = cos ωt , (3.2)It has the generic general solution x ( t ) = C cos kt + C sin kt + cos ωtk − ω , (3.3)where C and C depend on initial conditions, and the last term isthe particular integral. The phenomenon of resonance occurs when ω = k and the particular integral becomes invalid. We shall nowderive a particular integral containing Kahanian terms that enable theparticular integral to have a valid limit in the resonant case. We shalluse the method of variation of parameters [4, 13]. We start from thesolutions to the homogeneous equation: x = cos kt and x = sin kt .Then the particular integral is given by x p = u x + u x , where u = − (cid:90) x cos ωtW d t , u = (cid:90) x cos ωtW d t , W = x x (cid:48) − x x (cid:48) is the Wronskian. Evaluating the integrals inthe usual way, we obtain u = cos(( k − w ) t )2 k ( k − w ) + cos(( k + w ) t )2 k ( k + w ) , (3.4) u = sin(( k − w ) t )2 k ( k − w ) + sin(( k + w ) t )2 k ( k + w ) . (3.5)When the expression for x p is simplified, we are led to (3.3). If, however,we change to Kahanian antiderivatives, we obtain u = cos(( k − w ) t ) − k ( k − w ) + cos(( k + w ) t ) − k ( k + w ) , (3.6)and no Kahanian is needed for u , so it is still given by (3.5). The newparticular integral is x p ( t ) = cos ωt − cos ktk − ω , (3.7)and now the limits ω → ± k give x p ( t ) = t k sin kt , (3.8)showing an oscillation that increases with time—a hallmark of resonantbehaviour .Both Mathematica and Maple return (3.3) without provisos, and toobtain (3.8), one must substitute ω = k and rerun the solution. Thelesson from this example is that, because of their black-box automation,computer algebra systems should implement and exploit comprehensiveresults from the most basic operations on up through the most sophisti-cated. For example, it would greatly help to have comprehensive limitsand comprehensive series, as well. The derivation used in (2.4) appears ad hoc , but a more systematicprocedure is possible. Instead of computing an indefinite integral, wecalculate a “semi-definite” integral. This is an example of a computation that ought to be routine, for a humanusing familiar trigonometric identities. Instead of thinking back to how you solvedit in your first course in differential equations, correctly accounting for resonance,imagine that you are a computer subroutine, having to turn out a good answer by amechanical algorithm. In that context, the Kahanian approach makes automationeasier.
NTEGRALS OF FUNCTIONS CONTAINING PARAMETERS 11
Definition 4.1 A parametric semidefinite integral is one of theform P ( z ; p ) = (cid:90) zA f ( τ ; p ) d τ, (4.1) where the lower limit A is called the anchor point , and is constant. Remark 4.2
Some people use the term indefinite integral to describe (4.1) , allowing one point (usually the lower limit) to be fixed and theother to vary. We, however, have earlier used the term indefinite in-tegral to mean any antiderivative or primitive, as is common parlancewhere we work. To avoid confusion and to fix attention on the anchorpoint, we have introduced the term “semi-definite”, although we aresomewhat indefinite (that is, semi-indefinite) about the hyphen in theterm.
Lemma 4.3 In (4.1) , let z be finite and let f ( τ ; p ) be continuous withrespect to p and with respect to τ in the domain of interest containing z , τ and A ; A is a fixed finite numeric constant. Then P ( z ; p ) iscontinuous with respect to all finite values of its parameters p , exceptperhaps for removable singularities. Proof 4.4
This theorem is a consequence of classical theorems aboutthe interchange of limits when functions are uniformly continuous: seefor instance [3]. As a conceptual alternative, consider the following.Let G ( z ; p ) be a generic antiderivative of f ( z ; p ) . Then P ( z ; p ) = G ( z ; p ) − G ( A ; p ) , (4.2) and all discontinuities whose locations depend only on the parametersoccur both in G ( z ; p ) and in G ( A ; p ) . Therefore they cancel in thesemidefinite integral (4.2), leaving at worst an expression that is inde-terminate at the locations of those discontinuities, making the discon-tinuities removable. Remark 4.5
The finiteness of the interval in the hypotheses is neces-sary. If the interval of integration is unbounded, then the lemma neednot be true. For example, for real x we have (the signum function is − for negative arguments, +1 for positive arguments, and zero for zeroargument) (cid:90) x −∞ sin ptt dt = π p ) + (cid:90) px sin uu du (4.3) which has a jump discontinuity at p = 0 although the integrand sin( pt ) /t is continuous there. We are indebted to a referee for this example. Remark 4.6
The complexity of the Kahanian depends on the choiceof anchor point A . We want to avoid values of A that make G ( A ; p ) indeterminate or take an infinite magnitude. The least complex Kaha-nian is . Therefore it is worth comparing the complexities of Kahanianconstants corresponding to different anchor points, and preferring anythat yield a Kahanian of . The example (2.4) is obtained from the semidefinite integral (cid:82) z τ α d τ .Our opening example (1.1) can be modified by using an anchor point A = 0 to calculate a Kahanian constant: we obtain (cid:90) (cid:0) α σz − α λz (cid:1) dz = − ( λ − σ ) λσ ( λ + σ ) ln α + 12 ln α (cid:18) α λz λ + α σz σ − α ( λ + σ ) z λ + σ (cid:19) . (4.4)It is straightforward to verify that by taking the limits α → λ → λ → − σ , each of the special cases in (1.2) is reproduced. That is,the use of a Kahanian makes each of the special cases of the comprehen-sive integral into a removable discontinuity. Therefore, the conceptualadvance of replacing evaluation using substitution by evaluation usinglimits , as discussed previously, can be usefully applied. The danger of exceptional values is as old as algebra. Early enthusiasts,amazed at algebraic power, often overlooked exceptions. Eventually,though, experts such as Cauchy and Weierstrass worked to check theenthusiasm, pointing out that care was needed. Out of this care, mod-ern analysis was born; out of the enthusiasm, modern algebra. Hawkins[10] writes of Cauchy“Nevertheless, Cauchy did not accept the particular al-gebraic foundation used by Lagrange . . . Cauchy, however,had well-founded doubts about the automatic general in-terpretation of symbolic expressions. He had warned that“most (algebraic) formulas hold true only under certain con-ditions, and for certain values of the quantities they con-tain.”NTEGRALS OF FUNCTIONS CONTAINING PARAMETERS 13The dichotomy between analysis and algebra survives to this day,and can be seen in the present discussion of integration. Within com-puter algebra, integration is based on modern algebraic algorithms,such as the Risch algorithm and its generalizations [17, 5], for solvingthe elementary antidifferentiation problem. Algebraic algorithms ex-plicitly exempt the constant of integration from consideration. Indeed,when verifying an integration formula by differentiation, all piecewiseconstants are given zero derivatives.
In the early days of computer algebra, implementers and users wereequally delighted at the ability of systems to obtain generic results.Now, however, the low-hanging fruit has been harvested, and imple-menters can and should make another pass through their fundamentalfunctionalities such as integration, the solving of systems of equationsand inequalities, limits and series to make as many results as possiblecomprehensive. The robustness of high-level functionality demands it.
Computer algebra and the internet make it decreasingly likely thatthere will be a completely new printed table of integrals. There will,however, probably be new editions of existing tables, because there issomething to be learned by scanning a table of closely-related integrals,as opposed to seeing results one at a time from a computer algebrasystem, with no organizing principle.Printed integral tables would be impractically bulky if every entrywere a piecewise result of the kind in this article. However, many ofthe special cases could be listed once in the most appropriate place,then explicitly cross referenced from more generic cases. New editionscan also make the relevant domains as general as possible and moreexplicitly obvious.Moreover, an on-line version could assemble each piecewise resultas needed. There could even be a computer algebra system involved.The difference of such a mathematical knowledge base from a barecomputer algebra system is that a user can learn by browsing throughrelated examples that follow a clear organizing principle.4 THE MATHEMATICAL GAZETTE (submitted) “ A lot of times, people don’t know what they want until you show itto them .”–Steve JobsMathematicians are increasingly frequent users of computer algebraand other educational or research mathematical software. We suspectthat when users start encountering more comprehensive results, theywill become disappointed in software that does not provide them. Asa side benefit, users might become more careful about not overlookingspecial cases or relevant issues such as domain enforcement and conti-nuity issues with their manually derived results. Perhaps editors andreferees will also pay more attention to such details in articles they arereviewing.Meanwhile we hope that this article serves as a warning that whena computer algebra system returns a generic parametric antiderivative,the user should ponder the result, to determine whether there existspecial cases, and if so, compute them with separate integrands.We think that the idea of comprehensive antiderivatives will bewelcomed by many mathematicians, perhaps after they are exposedto it through using future versions of computer algebra systems thatoffer built-in comprehensive antiderivatives. We admit, however, that parametrically continuous antiderivatives will be adopted only slowly,because they are not always necessary, and they are usually more com-plicated than the simpler, incorrect answer. In addition, from a nu-merical point of view, they can suffer from catastrophic numerical can-cellation, which requires higher precision (or perhaps some numericalanalysis experience) to overcome. Nonetheless, the point remains thatwhen they are necessary, they are crucial to obtaining a correct andcomplete result.
The calculus curriculum is already quite full, and it seems unlikely thatthe textbook examples and the expected exercise or test results couldall be comprehensive results. However it does seem worthwhile to intro-duce students to the concept and have them do some simple examplesso that those who proceed into mathematical careers (or careers thatuse mathematics) are more thorough and careful. The concept andideas behind a Kahanian antiderivative also seem worthwhile, but wedo not expect that to be as prevalent.NTEGRALS OF FUNCTIONS CONTAINING PARAMETERS 15
We are well aware that an abrupt transition to Kahanian parameterizedantiderivatives is impossible. One of us implemented the non-piecewisegeneric portion of Kahan’s example (2.4) in an early version of a com-puter algebra program. The complaints were immediate, numerous andstrong. Even many mathematics teachers said that it was incorrecteven though it differed only by a constant from the traditional an-tiderivative. Perhaps the space-saving omission of a generic integrationconstant C from most integral tables led some mathematics teachers toforget what they taught, or perhaps they were simply concerned thatit would alarm, and hence intimidate, many of their students. Anotherof us taught the concept to their beginning calculus students. Theyhated it. The Wikipedia entry on
Lie-to-children quotes [11], “The pedagogicalpoint is to avoid unnecessary burdens on the students first encounterwith the concept.” Asking students, on their first encounter with an-tiderivatives, to worry about the continuity of their answer, in additionto other worries, might seem unreasonable. Asking computer algebrasystems to cater to the needs of first-year students as well as the needsof people who solve differential equations with parameters might alsoseem unreasonable, without some sort of switch to “expert mode”, say.On the other hand, if only experts learn to be careful with parametriccontinuity, then both education and software have done a disservice totheir audiences.
In one sense this paper merely offers a minor technical correction tothe current practice of computing indefinite integrals. However, thetotal impact of this minor correction is potentially large because thecurrent practice is taught early at the university level and to very manystudents—most of whom do not go on to become mathematics majors.Moreover, computer algebra systems have become widespread, includ-ing good free ones, some of which are available for smartphones. Mostcurrent computer algebra systems apply current textbook rules andamplify the effects of fundamental “minor” errors such as the error incontinuity that we address in this article. So in practice, the correctionwe present is important.6 THE MATHEMATICAL GAZETTE (submitted)In order to promote the ideas of comprehensive antiderivatives andKahanian constants, we have developed a Mathematica program thatcomputes comprehensive anti-derivatives and Kahanian forms. It canbe used as an enhancement to Mathematica’s own
Integrate
References [1] Milton Abramowitz and Irene A. Stegun.
Handbook of Mathemat-ical Functions: with Formulas, Graphs, and Mathematical Tables .Dover, 1964.[2] Robert A. Adams and Christopher Essex.
Calculus, a completecourse . Pearson, 8th edition, 2014.[3] Tom M. Apostol.
Mathematical Analysis: A Modern Approach toAdvanced Calculus . Addison-Wesley, 1957.[4] Carl M. Bender and Steven A. Orszag.
Advanced MathematicalMethods for Scientists and Engineers . McGraw-Hill, New York,1978.[5] Manuel Bronstein.
Symbolic Integration I. Transcendental Func-tions . Algorithms and Computation in Mathematics. Springer,2nd edition, 2005.[6] R. M. Corless and D. J. Jeffrey. Well... it isn’t quite that simple.
SIGSAM Bulletin , 26(3):2–6, 1992.[7] Robert M. Corless, David J. Jeffrey, Stephen M. Watt, andJames H. Davenport. “According to Abramowitz and Stegun”or arccoth needn’t be uncouth.
ACM SIGSAM Bulletin: Commu-nications in Computer Algebra , 34(2):58–65, 2000.[8] Richard Courant and Fritz John.
Introduction to Calculus andAnalysis I . Springer, 1999.[9] Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik.
Ta-ble of Integrals, Series, and Products . Academic Press, 6th edition,2007.NTEGRALS OF FUNCTIONS CONTAINING PARAMETERS 17[10] Thomas Hawkins. Weierstrass and the theory of matrices.
Archivefor History of Exact Sciences , 17(2):119–163, 1977.[11] D. J. Jeffrey and Robert M. Corless. Teaching linear algebra withand to computers. In Wei-Chi Yang, editor,
Proceedings of ATCM2001 , pages 120–129, 2001.[12] D. J. Jeffrey and A.D. Rich. Reducing expression size using rule-based integration. In S. Autexier, editor,
Intelligent ComputerMathematics , volume 6167 of
LNAI , pages 234–246. Springer,2010.[13] Harold Jeffreys and Bertha Swirles.
Methods of MathematicalPhysics . Cambridge University Press, 3rd edition, 1972.[14] W. M. Kahan. Integral of x n . Personal communication, circa Prejudices. Second Series ,chapter IV, pages 155–171. Alfred A. Knopf, New York, 1920.[16] Anatolii Platonovich Prudnikov, Yuri A. Brychkov, and Oleg Ig-orevich Marichev.
Integrals and Series . CRC Press, 1992.[17] Robert H. Risch. The solution of the problem of integration infinite terms.
Bull. Amer. Math. Soc. , 76(3):605–608, 1970.[18] James Stewart.
Calculus . Brooks Cole, 7th edition, 2012.[19] Adrian Fedorovich Timofeev.
Integrirovanie funktsii (Integrationof Functions) . Publishing House tehniko-teoreticheskoj literature,Leningrad, 1948.[20] Daniel Zwillinger.