Chapter 9 Special Functions We want to look at some special functions that can arise, especially in trying to solve certain types of rather simple equations. 9.1 Hyperbolic Trigonometric Functions The usual trigonometric functions with which we are familiar are often calle circular functions because their values can be etermine by the geometry of a circle. While mathematicians were able to compute the area an circumference of a circle eactly, they were unable to o the same for an ellipse. If a + y b = 1 escribes an ellipse centere at the origin with major an minor aes of lengths a an b, then we can show that the area of the ellipse is πab. This is actually not too har using simple integration. However, the problem of fining the arclength of the ellipse, or the rectification of an ellipse, turne out to be much, much harer. The functions that were encountere were much harer. The integrals that arose in this stuy were known as elliptic integrals. These were foun to be of several kins. Nonetheless, they all arose basically in the search for fining the arclength of an ellipse. One reason that the ellipse was of such interest was that with Newton s theory of gravity, scientists an mathematicians began to unerstan that the earth wasn t a sphere but an ellipsoi. Also, with Kepler s Laws of Planetary Motion, the motion of the planets aroun the sun were not circular, but elliptical. Computing how long it takes for a planet to go aroun the sun, means that we nee to know how far it travels, which is the arclength of an ellipse. Thus, the stuy of functions efine by ifferent conics was not too surprising. One point of importance, though, is that the ellipse can be parameterize, just like the circle, with the usual trigonometric functions: (t) = a cos(t) y(t) = b sin(t). The stuy of a hyperbola woul not len itself to that technique, unless we allow 137
138 CHAPTER 9. SPECIAL FUNCTIONS comple numbers. Since early on comple numbers were not well receive, mathematicians looke for a ifferent way to approach this. Vincenzo Riccati (1707-1775) introuce the hyperbolic functions. Johann Heinrich Lambert (178-1777) further evelope the theory of hyperbolic functions in Histoire e l acamie Royale es sciences et es belles-lettres e Berlin, vol. XXIV, p. 37 (1768). Vincent Riccati, S.J. was born in Castel-Franco, Italy. He worke together with Girolamo Salaini in publishing his iscovery, the hyperbolic functions. Riccati not only introuce these new functions, but also erive the integral formulas connecte with them, an then, still using geometrical methos. He then went on to erive the integral formulas for the trigonometric functions. His book Institutiones is recognize as the first etensive treatise on integral calculus. The works of Euler an Lambert came later. Vincent Riccati evelope the hyperbolic functions an prove their consistency using only the geometry of the unit hyperbola y = 1 or y = 1. Riccati followe his father s interests in ifferential equations which arose naturally from geometrical problems. This le him to a stuy of the rectification of the conics in Cartesian coorinates an to an interest in the areas uner the unit hyperbola. He evelope the properties of the hyperbolic functions from purely geometrical consierations. Riccati evelope the hyperbolic functions in a manner similar to what follows. The algebra concerning sectors of a circle is relatively simple since the arc length is proportional to the area of the sector. l = rθ, A = r θ, so l = A r. y=k Hyperbolas, however, o not have this property, so a ifferent algebra is neee. If we look at the hyperbola y = k, an let O enote the origin, A, a point on the graph an A the foot of A on the -ais, then the area of OAA is given by B A = 1 bh = 1 y = k. A C Thus, all such right triangles have the same area, no matter which point A we choose on the hyperbola. This O B' A' C' property enables us to replace area measure by linear measure OA, thus introucing a kin of logarithm. Figure 9.1: Let us now consier the curve y = 1. It can be seen from Figure 9. that the following areas are equivalent: area( AOA ) = area( POP ) because of the property just mentione, area( OPQ) = area( QAA P ) by subtracting the common area OQP from each. Then area(app A ) = area(aop) by aing AQP to each. So for any point A on the MATH 6101-090 Fall 006
9.1. HYPERBOLIC TRIGONOMETRIC FUNCTIONS 139 curve, the area of the sector AOP equals the area uner the curve 1/ from A to P. This area is u = ln a where (0, a) correspons to the point A. Now, we can use this parameter, u, to parameterize the original hyperbola, y = 1. For any point A on the hyperbola, let u/ be the area of the region boune by the -ais, the hyperbola, an the line from the origin to A. The coorinates of the point A can be y = 1 given by (cosh u, sinh u). This is very similar to the situation for circular functions. If P is a point on the unit circle, then the area of the sector of the circle from D P the positive -ais to the raius given by P is given by A θ/ an the coorinates of the point P are (cos θ, sin θ). Q This shows that the coorinates of the point on the unit circle can be etermine by the area of the sector swept O P' A' out by the raius. Figure 9.: The coorinates of the circle are then etermine by the angle. This is not quite true for the hyperbola, because the hyperbola is asymptotic to the line y =. This means that the angle etermine by the - +y = 1 ais, the origin, an a point on the hyperbola must A always be less than π/4, or 45. Likewise, as mentione above the coorinates of a point on the circle can be O P D parameterize by the arclength. For the hyperbola, the arclength is not u, but is bigger than that. We efine the hyperbolic trigonometric functions by Figure 9.3: sinh = e e cosh = e + e Since e = n n=0 converges for all R, the power series epansions of the n! hyperbolic trigonometric functions are sinh = cosh = n=0 n=0 n+1 (n + 1)! n (n)! an converge for all real. In fact, using comple analysis an letting i = 1, we MATH 6101-090 Fall 006
140 CHAPTER 9. SPECIAL FUNCTIONS can show that ( ) sinh = i sin(i) = i sin ( i ) cosh = cos(i) = cos i There are also the usual collections of hyperbolic trigonometry ientities: cosh sinh = 1 cosh( + y) = cosh cosh y + sinh sinh y sinh( + y) = sinh cosh y + cosh sinh y To unerstan the graphs of the hyperbolic sine an cosine functions, we first note that, for any value of, an sinh( ) = e e cosh( ) = e + e = sinh(), = cosh(). Now for large values of, e 0, in which case sinh() = 1 (e e ) 1 e an sinh( ) = sinh() 1 e. Thus the graph of y = sinh() appears as in Figure 9.4. Similarly, for large values of, cosh() = 1 (e + e ) 1 e an cosh( ) = cosh() 1 e. The graph of y = cosh() is shown in Figure 9.5. 10 10 8 y 5 6 4 0 4 y 4 5 10 Figure 9.4: y = sinh 4 0 4 Figure 9.5: y = cosh The erivatives of the hyperbolic sine an cosine functions follow immeiately from their efinitions. Namely, sinh() = 1 (e e ) = 1 (e + e ) = cosh cosh() = 1 (e + e ) = 1 (e e ) = sinh MATH 6101-090 Fall 006
9.1. HYPERBOLIC TRIGONOMETRIC FUNCTIONS 141 Since sinh() = cosh() > 0 for all, the hyperbolic sine function is increasing on the interval ( 1, 1). Thus it has an inverse function, calle the inverse hyperbolic sine function, with value at enote by sinh 1 (). Since the omain an range of the hyperbolic sine function are both (, ), the omain an range of the inverse hyperbolic sine function are also both (, ). The graph of y = sinh 1 () is shown in Figure 9.6. As usual with inverse functions, y = sinh 1 () if an only if sinh(y) =. This means that = sinh(y) = 1 (ey e y ) e y = e y 1 e y e y 1 = 0 (e y ) e y 1 = 0 e y = 1 ( + 4 + 4 ( y = ln + ) + 1 where we chose the positive branch of the square root because e y is never negative. Note that 1 + = 1 + sinh (u) = cosh (u) = cosh(u), This means that many integrals take the form of the following: 1 = 1 1 + cosh u cosh u u = u + C = sinh 1 () + C, instea of 1 ( = ln + ) + 1 + C 1. 1 + Similarly, since cosh() = sinh() > 0 for all > 0, the hyperbolic cosine function is increasing on the interval [0, ), an so has an inverse if we restrict its omain to [0, ). That is, we efine the inverse hyperbolic cosine function by the relationship y = cosh 1 () if an only if = cosh(y). where we require y 0. Note that since cosh() 1 for all, the omain of of the inverse hyperbolic cosine function is [1, ). The graph of y = cosh 1 () is shown in Figure 9.7. It will come as no surprise then that we can show that cosh 1 = ln ( + 1 ). Likewise, it shoul come as no surprise that we can efine the other four hyperbolic MATH 6101-090 Fall 006
14 CHAPTER 9. SPECIAL FUNCTIONS 5 4 4 y 3 10 5 0 5 10 1 4 Figure 9.6: y = sinh 1 0 4 6 8 10 Figure 9.7: y = cosh 1 functions as: tanh = sinh cosh = e e e + e = e 1 e + 1 coth = cosh sinh = e + e e e, 0 sech = 1 cosh = e + e csch = 1 sinh = e e, 0 These functions have the following properties: sinh = cosh cosh = sinh tanh = sech coth = csch sech = sech tanh csch = csch coth Each one of these functions has an inverse function. We will look only at the inverse hyperbolic tangent function. As usual, efine y = tanh 1 to mean that = tanh y. The tangent function an its inverse are graphe in Figure 9.8. As with MATH 6101-090 Fall 006
9.1. HYPERBOLIC TRIGONOMETRIC FUNCTIONS 143 the other inverse hyperbolic functions, we fin that tanh 1 = 1 ( ) 1 + ln, 1 < < 1. 1 We nee to note that e e lim tanh = lim = 1. e + e Also, lim tanh = 1. 1 0.5 10 5 0 5 10 1 0.5 0.5 1 0.5 1 Figure 9.8: y = sinh 1 Figure 9.9: y = cosh 1 The hyperbolic trigonometric functions mirror many of the same relationships as o the circular functions. A collection of the ientities for the hyperbolic trigonometric functions follows: MATH 6101-090 Fall 006
144 CHAPTER 9. SPECIAL FUNCTIONS cosh sinh = 1 (9.1) 1 tanh = sech (9.) sinh( ± y) = sinh cosh y ± cosh sinh y (9.3) cosh( ± y) = cosh cosh y ± sinh sinh y (9.4) tanh + tanhy tanh( + y) = 1 + tanhtanh y (9.5) sinh = cosh 1 cosh = cosh + 1 tanh = cosh 1 tanh = cosh + 1 sinh cosh + 1 = cosh 1 (9.6) (9.7) (9.8) (9.9) sinh (9.10) sinh = sinh cosh (9.11) sinh ± sinh y = sinh 1( ± y) cosh 1 ( y) (9.1) cosh + cosh y = cosh 1( + y) cosh 1 ( y) (9.13) cosh cosh y = sinh 1 ( + y) sinh 1 ( y) (9.14) 9. Lambert W function Johann Heinrich Lambert was born in Mulhouse on the 6th of August, 178, an ie in Berlin on the 5th of September, 1777. His scientific interests were remarkably broa. The self-eucate son of a tailor, he prouce funamentally important work in number theory, geometry, statistics, astronomy, meteorology, hygrometry, pyrometry, optics, cosmology an philosophy. Lambert was the first to prove the irrationality of π. He worke on the parallel postulate, an also introuce the moern notation for the hyperbolic functions. In a paper entitle Observationes Variae in Mathesin Puram, publishe in 1758 in Acta Helvetica, he gave a series solution of the trinomial equation, = q + m, for. His metho was a precursor of the more general Lagrange inversion theorem. This solution intrigue his contemporary, Euler, an le to the iscovery of the Lambert function. Lambert wrote Euler a corial letter on the 18th of October, 1771, epressing his hope that Euler woul regain his sight after an operation; he eplains in this letter how his trinomial metho etens to series reversion. MATH 6101-090 Fall 006
9.. LAMBERT W FUNCTION 145 The Lambert function is implicitly elementary. That is, it is implicitly efine by an equation containing only elementary functions. The Lambert function is not, itself, an elementary function. It is also not a nice function in the sense of Liouville, which means that it is not epressible as a finite sequence of eponentiations, root etractions, or antiifferentiations (quaratures) of any elementary function. The Lambert function has been applie to solve problems in the analysis of algorithms, the sprea of isease, quantum physics, ieal ioes an transistors, black holes, the kinetics of pigment regeneration in the human eye, ynamical systems containing elays, an in many other areas. When we run into the equation y = e we can solve this by using the inverse function y = ln. A similar equation that we fin is y = e. The graph of this function is in Figure 9.10. Note that this function is not one-to-one, but is almost one-to-one. The minimal value is 1/e which 15 occurs at = 1. If we restrict the omain to [ 1, ) 10 or (, 1] then there is an inverse on either branch. This function, the solution to the equation y = e, is the LambertW function name in honor of Lambert who first stuie it, or one of its relative. Since 5 the graph of the inverse is the reflection of the original 3 1 0 1 graph across the line y =. This looks like the graph in Figure 9.11. This nomenclature was introuce by the evelopers of Maple when they foun it to be very useful in solving a wie variety Figure 9.10: y = e of problems. 1 3 4 5 Consier for eample the equation y = e. The graph of this equation looks like the graph of y = e flippe across the y-ais an the -ais. Thus, the solution woul be W( ). Note that y = W() means that = ye y. Thus, y = W( ) means y y = W( ) y = W( ) = ( y)e y = ye y 4 Figure 9.11: y = W() MATH 6101-090 Fall 006
146 CHAPTER 9. SPECIAL FUNCTIONS The solution of e = a is = W(a) by efinition, but Lémeray note that a variety of other equations can be solve in terms of the same transcenental function. For eample, the solution of b = a is = W(a ln b)/ lnb. The solution of a = b is e W(aln b)/a, an the solution of a = + b is = b W( a b ln a)/ ln a. Another eample is rewriting the iterate eponentiation h() =.... Euler was the first to prove that this iteration converges for real numbers between e e an e 1/e. When it converges, it can be shown to converge to... = W( ln) ln. It appears as if we o not know much about this function. However, we can learn quite a bit about it quickly. First, even though we can t write own the function eplicitly, we can fin it s erivative using implicit ifferentiation. but e W() = /W(), so = W()e W() = ( ) W()e W() 1 = W ()e W() + W()e W() W () W 1 () = e W() (1 + W()) W () = W(), 0, 1/e (1 + W()) We can fin a power series epansion for W(), but not in the usual way. What we know is a power series for e. We woul have y = n=0 n+1 n! = + + 3! + 4 3!.... We want a power series for in terms of y. The mathematician Lagrange was the first one to introuce the technique of reversion of a series. This is an algorithm that will allow for the computation of as a series in y. It has been implemente in Maple an Mathematica. Instea of oing this, suffice it to say for now that the MacLaurin series for W() is given by W() = ( n) n 1 n. n! n=1 MATH 6101-090 Fall 006
9.3. GLOG 147 (cf. Appeni H.) Using integration by parts, we can show that W() = (W () W() + 1) W() + C. This was iscovere while working on a ifferential equation of the form = pe p y = p so y = 1 p If we ifferentiate the first equation with respect to y, we get This can be easily integrate to get that p = p y ep + pe pp y 1 = (p + 1)epp p y y = p(p + 1)ep p y = (p p + 1)e p + C. Since y is an antierivative for W() we get the above formula. 9.3 glog Dr. Dan Kalman stuie a generalize logarithm evelope to solve linear-eponential equations, equations of the form a = b(+c) which o occur in Calculus an Numerical Analysis. His paper can be foun in The College Mathematics Journal, Volume 3, Number 1, January 001. His function, which he calls glog for generalize logarithm, is efine to be the solution to y = e. This is very close to the Lambert W function an, in fact, he shows that his function: glog() = W( 1/). The graph reveals at once some of the gross features of glog. For one thing, glog is not a function because for > e, it has two positive values. For < 0 glog is well efine, an negative, but it is not efine for 0 < e. When we nee to istinguish between glog s two positive values, we will call the larger glog + an smaller glog. MATH 6101-090 Fall 006
148 CHAPTER 9. SPECIAL FUNCTIONS 3 y 1 4 0 4 1 3 Figure 9.1: Graph of y = glog These inequalities can be erive analytically. For eample, suppose y = glog is greater than 1. By efinition, e y /y = so e y = y >. Thus y > ln. The secon graphical representation epens on the fact that glog c is efine as the solution to e = c. The solutions to this equation can be visualize as the - coorinates of the points where the curves y = e an y = c intersect. That is, glog c is etermine by the intersections of a line of slope c with the eponential curve. When c < 0 the line has a negative slope an there is a unique intersection with the eponential curve. Lines with small positive slopes o not intersect the eponential curve at all, corresponing to values of c with no glog. There is a least positive slope at which the line an curve intersect, that intersection being a point of tangency. It is easy to see that this least slope is at c = e with = 1. For greater slopes there are two intersections, hence multiple solutions an the nee for the two branches. MATH 6101-090 Fall 006