Defining Exponential and Trigonometric Functions Using Differential Equations Author(s): Harvey Diamond Source: Mathematics Magazine, Vol. 87, No. 1 (February 2014), pp. 37-42 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/10.4169/math.mag.87.1.37 . Accessed: 30/12/2014 14:59 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to Mathematics Magazine. http://www.jstor.org This content downloaded from 157.182.52.108 on Tue, 30 Dec 2014 14:59:51 PM All use subject to JSTOR Terms and Conditions NOTES Defining Exponential and Trigonometric Functions Using Differential Equations HARVEYDIAMOND West Virginia University Morgantown, WV 26506 [email protected] Students in science and mathematics are exposed to transcendental functions relatively early, in the sense that formal definitions of these functions must await years of further study. The formal definitions usually come in a standard calculus course. The first encounter with an “official” definition sometimes comes after the definite R x integral is available, when ln x is defined by ln x 1 .1=t/dt. Then exp.x/ is defined as the inverse function of ln x. This is the approach,D for instance, in [1]. It works as a definition, and provides a good opportunity for deriving the properties of ln x, but somehow it reverses the natural order of things and makes ln x depend on an inconve- nient exception (the power that doesn’t integrate as a power). We might, with a similar 1 R x p 2 approach [2, p. 160], define sin x by its inverse function sin− x 0 .1= 1 t /dt, which at least has the virtue of representing the arc length of a unitD circle from− .0; 1/ to .x; p1 x 2/. But this approach is rare, perhaps because proving the standard iden- tities is awkward.− A bit later in calculus, sin x, cos x, exp.x/ can be defined in a “cleaner” way us- ing power series. While power series have appealing simplicity, and provide a unify- ing framework for defining functions, they do not lend themselves directly to proving many fundamental properties, such as the periodicity of sin x and cos x, the multiplica- tive property exp.t/ exp.x/ exp.t x/, even the fact that ex 0 as x . Indeed, we can use all ofD these formalC definitions to show students! that! in −∞ mathe- matics, precise, simple, and useful definitions for simple concepts can be hard to come by. In this note we suggest that these functions—exp.x/, sin x, cos x—can be defined most naturally through their governing differential equations. These equations are beautifully simple, and all of the functions’ basic properties follow with almost no calculation, using the ideas of existence and uniqueness for initial value problems. This material can be easily presented as part of an introductory differential equa- tions course. (We would like to argue further, that with the computational and graphics tools now at our disposal, differential equations, initial value problems, and these ideas could be introduced much earlier in the calculus sequence, but that discussion is for another time.) This discussion would also show students that in mathematics, we never stop searching for the most natural ways to define objects, and that basic objects can look simpler through the use of more advanced concepts. Nothing presented here is new. The multiplicative property of the exponential func- tion, as well as analogous trigonometric identities, are special cases of the functional Math. Mag. 87 (2014) 37–42. doi:10.4169/math.mag.87.1.37. c Mathematical Association of America 37 This content downloaded from 157.182.52.108 on Tue, 30 Dec 2014 14:59:51 PM All use subject to JSTOR Terms and Conditions 38 MATHEMATICS MAGAZINE properties of the transition matrix for linear systems x0 Ax, and the analysis of solu- tions of differential equations through their governing equationsD (e.g., Sturm–Liouville theory) is similarly well-established. Our intended contribution is the packaging and presentation for undergraduates. The basic theorems We begin by stating the basic existence and uniqueness theorems for differential equa- tions, and some of their immediate consequences. These theorems hold in much greater generality, but we need only the simplest cases. THEOREM 1. If q, x0, and a are real numbers, then there is a unique differentiable function y R R such that V ! y0.x/ q y.x/ 0 (1) C D for all x R and 2 y.x / a: 0 D Any function y R R that satisfies (1) for all x R is called a solution of the V ! 2 differential equation y0 qy 0. The equation has many solutions, but only one sat- isfying any particular initialC conditionD such as y.x / a. 0 D THEOREM 2. If p, q, x0, a, and b are real numbers, then there is a unique differ- entiable function y R R such that V ! y00.x/ p y0.x/ q y.x/ 0 (2) C C D for all x R and 2 y.x / a; y0.x / b: 0 D 0 D A function satisfying (2) is called a solution of the differential equation y00 py0 qy 0. C C WeD state the following immediate consequences in the case of second-order equa- tions, as in Theorem2. They hold as well for first-order equations. THEOREM 3. Let p and q be real numbers. (a) If u R R and v R R are solutions of the differential equation y00 py0 qy V 0 and! α and βVare! any real constants, then y αu βv is also a solution.C C D D C (b) If φ is a solution of y00 py0 qy 0 and c is a real number, and y R R is C C D V ! defined by y.x/ φ.x c/ for all x R, then y is a solution. D − 2 (c) If φ is a solution, then its derivative φ0 is a solution. Each part follows by direct verification. The exponential function: definition and properties DEFINITION. The function exp R R is the unique solution of the differential equation V ! y0 y D satisfying the initial condition y.0/ 1. D This content downloaded from 157.182.52.108 on Tue, 30 Dec 2014 14:59:51 PM All use subject to JSTOR Terms and Conditions VOL. 87, NO. 1, FEBRUARY 2014 39 From the definition, we know that exp is its own derivative and that exp.0/ 1. If c is any real number, we can substitute directly to see that y.x/ c expD.x/ is D the solution of y0 y with y.0/ c. For any t R, we can let c exp.t/, so that D D 2 D exp.t/ exp.x/ is a solution of y0 y with y.0/ exp.t/. From Theorem3, part (b), the function y.x/ exp.x t/ isD a solution of theD same equation satisfying the same initial condition, andD so theC two solutions must be identical; that is, we must have exp.x t/ exp.t/ exp.x/ for all real x and t—the multiplicative property of the ex- ponentialC function.D In particular, exp.x/ exp. x/ exp.0/ 1 and for any integer n, we have exp.nx/ exp.x/ n. We define e − expD.1/ and theD multiplicative property then shows us thatDTexp.p=qU/ q exp.p/ De p so that exp.p=q/ e p=q for any in- tegers p, q. For rationalT r, weU thereforeD haveD exp.r/ er . The functionD exp.x/ then gives us the unique continuous extension of ex from theD rational numbers to the real numbers. Since exp.x/ is clearly positive and increasing (we cannot have exp.a/ 0, lest D we obtain the contradiction exp.x/ 0 as the solution of y0 y; y.a/ 0) we can ≡1 D D then define the inverse function exp− .x/ ln x and develop its properties in the usual way (e.g., if u exp.a/ and v exp.b/, then≡ uv exp.a b/, from which ln.uv/ a b ln u Dln v follows). D D C D CFromD the differentialC equation, it is clear that the nth derivative of exp.x/ is equal to exp.x/, and we can expand in the usual Taylor series from there. The limit formula for e From the fact that exp.1/ e .exp.1=n//n, we would like to obtain the famous limit D D lim .1 1=n/n e: n !1 C D The traditional approach via the derivative of ln x requires the machinery of inverse functions. We take a more direct approach. From the mean value theorem (MVT), we know that exp.1=n/ exp.0/ − exp.c/ 1=n D for some c satisfying 0 < c < 1=n. Since exp.x/ is increasing, we have exp.1=n/ exp.0/ exp.0/ < − < exp.1=n/: 1=n From the left inequality, we have 1 1=n < e1=n, and from the right, we have C 1 e1=n 1 < 1; − n so that 1 1 1 1 1 e1=n < 1 : C n − n C n Combining the two inequalities and raising both sides to the n power, 1 n 1 n 1 e < 1 < e: − n2 C n This content downloaded from 157.182.52.108 on Tue, 30 Dec 2014 14:59:51 PM All use subject to JSTOR Terms and Conditions 40 MATHEMATICS MAGAZINE Now, using the MVT, when 0 < x < 1 and k > 1, we have k .1 x/ 1 k 1 − − k .1 c/ − > k x D − − − so that .1 x/k > 1 kx and in particular − − 1 n 1 1 > 1 : − n2 − n We obtain the inequality .1 1=n/e < .1 1=n/n < e and the limit formula follows from the squeeze theorem.