12 Logarithmic Function

Transcription

1 12 Logarithmic Function The definition of logarithmic function From Calculus, the logarithmic function, log x : (0, ) R, is defined to be the inverse function of e x : R (0, ) gievn by 42 e log x = x, x (0, ); log e x = x, x R. Since (e x ) = e x, by the formula of inverse function, it implies (log x) = 1 and hence x (log(1+x)) = 1. By taking integral, we have log(1+x) = x 1 dt = x 1+x 0 1+t 0 [1 t+t2 t ]dt = x x2 + x3... for 0 < x < 1, i.e., 2 3 log(1+x) = n=0 Here the radius of convergence of this power series is 1. ( 1) n n+1 xn+1, x ( 1,1). (45) By replacing the real variable x with complex variable z, we can define log(1+z) := n=0 which is analytic. Replacing 1+z with z, one gets log z = n=0 We notice that the exponential function ( 1) n n+1 zn+1, z (1). ( 1) n n+1 (z 1)n+1, z (1,1). e z : C C {0} is not one-to-one. In fact, e z+2iπk = e z holds for any z C and for any integer k. In other words, for the function f(z) = e z, for any w 0 C {0} with f(z 0 ) = w 0, its inverse f 1 (w 0 ) = {e z+2iπk k Z} is an infinite set. Therefore the exponential function is an infinite-to-one map, so that it does not have an inverse function as a function, but as a multi-valued function. At this moment, on (1, 1), we defined a single-valued function log z satisfying e log z = z, z (1,1), 42 If f : X Y is an one-to-one and onto map, it has inverse map f 1 : Y X satisfying f 1 (f(x)) = x, x X and f(f 1 (y)) = y, y Y. For differential case, we have the formula (f 1 (y)) = 1 f (x). 71

2 butlog e z = z maynotbetrueon (1,1). Infact, ifz 0 = z e iθ (1,1)where π 2 < θ < π 2, then log e z 0 = z 0 holds. If we replace z 0 by z 1 = z 0 + 2kπ for any integer k 0, then we still have z 1 (1,1) but loge z 1 = z 1 is no longer true. We may discusse many issues on logarithm funtion (continuation, extension, multi-valued, etc.) late. [Example] For any complex number a,b C with a 0, we define a b := e b log a. Therefore, we can define z := z := e 2 log z on (1,1). More generally, we can define for any complex number b, z b := e b log z on (1,1). [Example] Recall cos z = eiz +e iz = w, i.e., e 2iz 2we iz +1 = 0. Then e iz is a root of 2 the equation Z 2 2wZ +1 = 0 so that we obtain In other words, e iz = w± w 2 1. z = cos 1 w = i log(w± w 2 1), foranyw satisfyingw+ w 2 1 (1,1). Wecanshowz = cos 1 w = ±ilog(w+ w 2 1). Also, sin 1 w = π 2 cos 1 w. Appendix: History of Logarithm Archimedes and hyperbola Archimedes (287 B.c.-212 B.C.) was the first to apply the so-called method of exhaustion successfully to calculate area of a segment of parabola. He also found the areas of a variety of plane figures and the volumes of spaces bounded by all kinds of curved surfaces. These included the areas of the circle, parabola and many other cases. However, Archimedes could not make it work in the case of two other famous curves: the ellipse and the hyperbola, which, together with the parabola, make up the family of conic sections. He could only guess correctly that the area of the entire ellipse is abπ. In fact, these cases had to wait for the invention of integral calculus two thousand years later. We now know that the calculation of area of ellipse needs to calculate elliptical integrals and the calculation of area of hyperbola involves the concept of logarithm. Napier s invention of logarithmic function The invention of logarithms by Napier ( )isoneofvery fewevents inthehistoryofmathematics thereseemed tobeno visible developments which foreshadowed its creation. Its progress completely revolutionized arithmetic calculations. 72

3 Long before Napier, Michael Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table, which is in terms of modern notation: n n... (46) The rules 2 m 2 n = 2 m+n and 2 m /2 n = 2 m n were known for positive exponents. If we want to find A B where A = 2 m and B = 2 n, then we just need to do addition m+n and then get the result 2 m+n. Around the time of the 16th century, trigonometry functions such as sine and cosine were generally calculated to 7 or 8 digits, and these calculations were long so that occurring of errors was invertible. In order to decrease computational errors, astronomers realized that it would be greatly reduce the number of errors if the multiplication and divisions could be replaced by additions and subtractions. The following trigonometric identity, used by the 16th century astronomers, is a such example. 2 sin α sin β = cos(α β) cos(α+β). The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). This was half century before Calculus was discovered. Joost Bürgi independently invented logarithms but published six years after Napier. While the table (46) can be used only to calculate multiplication A B only for special A = 2 m and B = 2 n, Napier wants to make a table to calculate all sines and cosines for every minute between 0 and 90. At that time, the sine was not considered as a ratio, but as the length of a semi-chord in a circle of a certain radius. For Napier, sine x = 10 7 sin x where sin x is the modern sine and the sine of 90 is Napier considere Cq 1 Cq 2 Cq 3 Cq 4... Cq n n... (47) How to make a such table such that q 1,Cq 2,...,Cq n,... cover all numbers from sin 0 to sin 90? Napier spent several years to decide to choose C = 10 7 and q = Based on this idea with some more sophisticated technical treatment, 43 Napier spent 20 years, from , to finish his table of logarithm. Napier first named the exponent of 43 Denis Roegel, Napier s ideal construction of logarithms, part of the LOCOMAT project, 2012, p

4 each power its artificial number but later he decided on the term logarithm, which means ratio number. Napier s great success Napier s work was greeted with great enthusiasm. Henry Briggs ( ), an English mathematician in Gresham College in London, wrote: I never saw a book that pleased me better, or made me wonder more. Johannes Kepler was an enthusiastic user of the newly invented tool, because it speeded up many of his calculations. He used logarithm tables extensively to compile his Ephemeris and therefore dedicated it to Napier. Early resistance to the use of logarithms was muted by Kepler s successful work. Friedrich Gauss had been able to predict the trajectory of the dwarf planet Ceres by surprising accurate mathematics. Reputedly when asked how he can do that, Gauss replied, I used logarithms. The eighteenth century French mathematician Pierre-Simon de Laplace wrote that the invention of logarithms, by shorting the labors, doubled the life of the astronomer. Logarithm with base 10 Immediately after Napier s invention, Henry Briggs in 1617 visited Napier. Both decided to improve table of logarithms by putting the base b = 10. Briggs first table contained the common logarithms of all integers in the range 11000, with a precision of 8 digits. Since then, it has been used for the next 300 years. Initially, it might seem that since the common numbering system is base 10, this base would be more natural than base e. But mathematically, the number 10 is not particularly significant. The most natural number turns out to be e = lim n (1+ 1 n )n discovered by Euler. Mercator s series hyperbola in the form In 1668, Nicholas Mercator( ) wrote the equation of the y = 1 1+x = 1 x+x2 x Hegivesacrudeexplanationoftheprocessofsummationwhichwenowindicateby x n dx =, and then integrates the terms of the above series to get x n+1 n+1 log(1+x) = x x 2 /2+x 3 /3 x 4 /4+... (48) Since the logarithm with the base e has not been introduced yet, Mercator did not relate it to log(1+x) directly. 44 Now people call (48) the Mercator s series. By the way, although Newton found it first, he did not publish it until Nicolaus Mercator published in 1668 in his work entitled Logarithmotechnia. When learned of Mercator s 44 F. Cajori, History of the exponential and logarithmic concepts, American Math Monthly, vol.20(1913)1, p

5 publication, Newton was bitterly disappointed and felt that his credit had been deprived. However, Newton continued his style: always confide his work only to a close circle of friends and colleagues. 45 Saint-Vincent s formula and hyperbola In 1647 Grégoire de Saint-Vincent related logarithms to the quadrature of the hyperbola y = 1, by pointing out that the area f(t) 1+x under the hyperbola from x = 1 to x = t satisfies f(tu) = f(t)+f(u). In fact, he discovered x t dt = x x2 /2+x 3 /3 x 4 /4+... Although Vincent did not mention logarithm, it is indeed the modern definition of logarithm (see (45) and (48)). 46. Now we can understand why Archimedes was unable to find the area of sector of hyperbola: it involves logarithm! The union between e x and log x concepts took place 1685, Wallis( ) in his Algebra developed the theory of logarithms, beginning with progressions 1, 2, 4, 8,... and 0,1,2,3,... Then he generalizes by taking 1,r,r 2,r 3,r 4,..., and1,2,3,4,... He marked that these exponents they call logarithms, which are artificial numbers,.... And yet, Wallis does not come out, resolutely, with the modern definition of a logarithms and use it. AsimilarpointofviewwasreachedbyJohnBernoulliIinaletterofMay, 1694,addressed to Leibniz. The process shows that Bernoulli passed from x x = y to x log x = log y, though he did not actually write down this last equation. In June, 1694, Leibniz sent J. Bernoulli a letter in reply, in which he writes both x x = y and x log x = log y. Therefore Leibniz and J. Bernoulli had a grasp at this time of the exponential function. The controversy on complex logarithms between Leibniz and J. Bernoulli was Johann Bernoulli who noted that in x = 1 ( ix + 1 ). 1+ix Since dx 1+ax = 1 a ln(1+ax) +C, the above equation involves the real function tan 1 x and complex logarithms. John Bernoulli did not evaluate the integral, but he seriously began to consider logarithm of an imaginary. It then leads a controversy on logarithms Leibniz and J. Bernoulli. Started from March 1712, two great mathematicians Leibniz and John Bernoulli had debated for this problem for 16 months. 45 e: The Story of a Number, Eli Maor, Princeton University Press, 1994, p F. Cajori, History of the exponential and logarithmic concepts, American Math Monthly, vol.20(1913)1, p.12. It 75

6 John Bernoulli considered dx = dx, by integration, he claimed log x = log( x). We x x know that this is not correct because it missed constant. Leibniz replied that if log( 2) would hold, then log( 2) = 1 log( 2). However, 2 is an impossible number, which 2 implies log( 2) is also an impossible number. Leibniz died in This correspondence between Leibniz and John Bernoulli during the years of 1712 and 1713 was not published until Euler s early touch on logarithms The theory of logarithms of negative numbers was incidentally touched by Euler very early, in his correspondence with John Bernoulli I. The letters which passed between these men in have been in the possession of the Stockholm academy of sciences and have for the first time been published in full by G. Eneström in Euler was then 20 years old; John Bernoulli was 60. The following is a synopsis of the correspondence on logarithms. 11/05/1727 Euler to J. Bernoulli: The equation y = ( 1) x is difficult to plot, since y is now positive, now negative, now imaginary. It cannot represent a continuous line. 01/09/1728 J. Bernoulli to Euler: If y = ( n) x, then log y = x log( n) and dy = y log( n)dx = log(+n)dx for d log( z) = dz = d log z. Integrating, log y = xlog n z and y = n x. Hence y = (±1) x becomes 1 x = 1 or y = 1. 12/10/1728 Euler to J. Bernoulli: I have arguments both for and against log x = log( x). If log(x 2 ) = z, we have 1 2 z = log x 2. But x 2 is as much x as +x. Hence 1 2 z = log x = log x. It may be objected that x2 has two logarithms, but whoever claims two, ought to claim an infinite number. Argument against: From the equality of the differentials we cannot infer the equality of the integrals. Moreover, log x = log x + log( 1); hence log x = log x only if log( 1) = 0. Again, if log x = log x, then x = x and 1 = 1, but I rather think the conclusion from the equality of the logarithms to the equality of the numbers cannot be drawn.... Most celebrated Sir: what do you think of these contradictions? Here Euler touched for the first time the truth that log n has an infinite number of values. But he does not pursue this matter further at this time. When we write a b = c and define b = log a c, a and c are taken to both single-value. Euler dropped the restriction on c. Also Euler gave a death blow to log x = log x. Euler touches for the first time the truth that log n has an infinite number of values. by Around 1730, Leonhard Euler defined the exponential function and the natural logarithm e x = lim n (1+x/n) n, log x = lim n n(x 1/n 1). 76

7 Euler also showed that the two functions are inverse to one another. The Cotes-Euler formula In 1714, Roger Cotes( ) published a paper in the Philosophical Transactions of London, in which he found a formula, in terms of modern language, iφ = log(cos φ+i sin φ). (49) We now know that the above equation is true modulo integer multiples of 2πi, but Cotes missed the fact that a complex logarithm can have infinitely many values due to the periodicity of the trigonometric functions. It was Euler (presumably around 1740) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now named after him: e iφ = cos φ+i sin φ. It was published in 1748, and his proof was based on the infinite series of both sides being equal. Neither of these mathematicians saw the geometrical interpretation of the formula: the view of complex numbers as points in the complex plane arose only some 50 years later. The end of story Euler s contribution It is desirable to look back, for a moment, over the 35 years of history of logarithms of negative and complex numbers. Thus far only three mathematicians have attempted to unravel the mysteries of this subject, namely Leibniz, John Bernoulli I and Euler. Their discussion on entirely by letters; these letters were not published at the time. No articles or memoirs on this controversy has reached the press. The question has not been brought to the attention of the mathematical public. In 1745, the correspondence between Leibniz and John Bernoulli I was published. The reading of that correspondence acted as a tremendous stimulus upon Euler. As a boy of 20 he himself, as we have seen, had corresponded on this subject with his revered master, John Bernoulli I. That correspondence had set bare serious difficulties of the subject, but had not removed them. Since the time Euler had discovered the exponential expressions for sin x,cos x, and cos x+sin x; he had acquired a deeper insight into the properties of imaginary numbers. It was in 1745 that he completed his manuscript on the Introductio, which was issued from the press three years later. In his Introductio, 1748, Chap. VI, 102, Euler gives the definition involving exponents. In this same chapter Euler gives an exposition of negative and fractional exponents and calls attention to the multiple values of a number having a fractional exponent, an explanation seldom found in mathematical treatises of that time. In 1747, Euler sent a manuscript Sur les logarithmes des nombres negatifs et imaginaress to the Berlin Acadamy. In the same year, Eulder wrote to another famous French 77

8 mathematician D Alembert and tried to convin D Alembert his theory on exponential and logarithm functions. However, in 1748, Euler makes the astonishing statement that he is not able to reply rigorously to some of D Alembert s arguments. Because of this, Euler published his paper on 1749 in which the main theorem is: there is an infinitely of logarithms of every number. But this paper is a reduced version of his 1747 paper, which does not contain all the good thins found in his original manuscript of Euler s original paper of 1747 was published in (96 years later!) Now we know, by Euler s formula z = z e i arg(z) = z (cos arg(z) + i sin arg(z)), log z = log z +i arg(z) has infinitely many values. 78