10 Logrithm

z ∞ 1 n We have defined analytic functions. For examples, f(z) = e = n=0 n! z , sin z and cosz are analytic functions on C. P The definition of logarithmic function on ∆(1, 1) In Calculus, the logarithmic function, log x : (0, ) R, is defined to be the of ex : R (0, ) gievn by 42 ∞ → → ∞ elog x = x, x (0, ); log ex = x, x R. ∀ ∈ ∞ ∀ ∈ x ′ x ′ 1 Since (e ) = e , by the formula of inverse function, it implies (log x) = x and hence ′ 1 x 1 x 2 3 (log(1+ x)) = 1+x . By taking integral, we have log(1+ x)= 0 1+t dt = 0 [1 t + t t + 2 3 − − ...]dt = x x + x ... for 0 < x < 1, i.e., R R − 2 3 − | | ∞ ( 1)n log(1 + x)= − xn+1, x ( 1, 1). (41) n +1 ∈ − Xn=0 Here the radius of convergence of this power is 1. By replacing the real variable x with complex variable z, we can define

∞ ( 1)n log(1 + z) := − zn+1, z ∆(1). n +1 ∀ ∈ Xn=0 which is analytic. Replacing 1 + z with z, one gets

∞ ( 1)n log z = − (z 1)n+1, z ∆(1, 1). (42) n +1 − ∀ ∈ Xn=0

There is another formula for log z. Writing z = reiθ = z ei arg z and by using the addition property of , we obtain | |

log z = log z + log eiarg z = log z + i arg z, z ∆(1, 1). (43) | | | | ∀ ∈ The formulas (42) and (43) are identical. Extension of the logarithmic function outside of ∆(1, 1) We notice that the expo- nential function ez : C C 0 → −{ } 42If f : X Y is an one-to-one and onto map, it has inverse map f −1 : Y X satisfying f −1(f(x)) = → −1 → −1 ′ 1 x, x X and f(f (y)) = y, y Y . For differential case, we have the formula (f (y)) = ′ . ∀ ∈ ∀ ∈ f (x) 68 is not one-to-one. In fact, ez+2iπk = ez holds for any z C and for any integer k. In z ∈ other words, for the function f(z) = e , for any w0 C 0 with f(z0) = w0, its inverse −1 z+2iπk z ∈ −{ } z x+iy f (w0)= e k Z is an infinite set. e cannot take value 0 because e = e = exeiy and ex{ = 0. This| indicates∈ } that log z cannot extend across the point z = 0. 6 From (43), log z is defined on ∆(1, 1). For any point z C outside of ∆(1, 1), z = 0, 0 ∈ 6 we can take a curve from z = 1 to z = z0 in C 0 and extend the logarithm function log z along the curve. By the Identity Theorem, the−{ extension} is unique along the curve.

For example, if z0 = i, we can take a piece of the unit circle π γ = eiθ 0 [0, ] { | ∈ 2 } to get π π log i = log i + i = i. | | 2 2 If we replace this γ by γ + n∂∆(1), we obtain π log i = i +2iπn. 2 If we replace this γ by γ n∂∆(1), we obtain − π log i = i 2iπn. 2 − We see that such extension depends on the choice of the curve . In general, if D is a simply connected (i.e., there is no “hole”) domain in C 0 containing ∆(1, 1), then by extension along curves as above, we can extend log z as a single-− { } valued defined on D. For example, we can extend the domain of definition of log z to D := C negative real numbers and the origin . −{ } As a multi-valued function, we can defined on ∆(1, 1)

log z = log z + i arg(z)+2πni, n Z. | | ∈ For each integer n Z, the above defines a single-valued function log z, which is called the nth branch of log z∈. Therefore the is an infinite-to-one map, so that it does not have an “inverse function” as a function, but as a multi-valued function. Definition of ab for complex numbers For any a, b C with a = 0, ∈ 6 we define ab := eb log a.

69 [Example] We can define

1 1 1 2iπn z 2 = √z := e 2 log z = elog 2 + 2 on ∆(1, 1), n =0, 1, which is the inverse of the two-to-one map w w2. iz −iz 7→ e +e 2iz iz iz [Example] Recall cos z = 2 = w, i.e., e 2we + 1 = 0. Then e is a root of the Z2 2wZ +1 = 0 so that we obtain − − eiz = w √w2 1. ± − In other words, z = cos−1w = i log(w √w2 1). − ± − We can show z = cos−1w = i log(w + √w2 1) . In fact, ± − w2 (w2 1) i log(w √w2 1) = i log − − = i log(w + √w2 1). − − − − w + √w2 1 − − Also, we claim π sin−1w = cos−1w. (44) 2 − iz −iz In fact, since sin z = e −e = w, i.e., e2iz 2iweiz 1 = 0. Then eiz is a root of the 2i − − equation Z2 2iwZ 1 = 0 so that we obtain − − eiz = iw √ w2 +1. ± − In other words, z = sin−1w = i log(iw √ w2 + 1). − ± − Then π π sin−1w = i log i(w √w2 1) = i log[w √w2 1] + i = cos−1w. −  ± −  −  ± − 2  2 −

Claim (44) is proved. So far, we clearly understand: All elementary transcendental functions can be expressed through ez and its inverse log z. In other words, there is essentially only one elementary transcendental function. and 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

70 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 by Napier (1550-1617) is one of very few events in the history of mathematics —– there seemed to be no visible developments which foreshadowed its creation. Its progress completely revolutionized arithmetic calculations. It has been done long before calculus was invented. Around the time of the 16th century, trigonometry functions such as 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). − − Napier’s great success Napier’s work was greeted with great enthusiasm. Henry Briggs (1561-1631), 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.” Hyperbola Let us consider a hyperbola given by the equation

x2 y2 =1. a2 − a2 71 By changing coordinates, it becomes

(x + y)2 (x y)2 − = a2, 2 − 2

a2 1 1 i.e., xy = 2 . The problem is essentially about the function y = x or y = x+1 . In 1647 Gr´egoire de Saint-Vincent related logarithms to the of the hyperbola 1 y = 1+x , by pointing out that the area f(t) under the hyperbola from x =1to x = t satisfies f(tu)= f(t)+ f(u). In fact, he discovered

x 1 dt = x x2/2+ x3/3 x4/4+ ... Z1 1+ t − − Although Vincent did not mention logarithm, it is indeed the modern definition of logarithm. Now we can understand why Archimedes was unable to find the area of sector of hyperbola: it involves logarithm! In 1668, Nicholas Mercator(1620-1687) also wrote the equation of the hyperbola in the form 1 y = =1 x + x2 x3 + ... 1+ x − − He gives a crude explanation of the process of which we now indicate by xndx = xn+1 n+1 , and then integrates the terms of the above series to get R

log(1 + x)= x x2/2+ x3/3 x4/4+ ... (45) − − Since the logarithm with the base e has not been introduced yet, Mercator did not relate it to log(1 + x) directly. 43 Now people call (45) 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 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. 44 The union between ex and log x concepts took place 1685, Wallis(1616-1703) in his developed the theory of logarithms, beginning with progressions 1, 2, 4, 8, ... and 0, 1, 2, 3, .... Then he generalizes by taking 1,r,r2,r3,r4, ..., and 1, 2, 3, 4, ..... He marked that

43F. Cajori, History of the exponential and logarithmic concepts, American Math Monthly, vol.20(1913)1, p.12. 44e: The Story of a Number, Eli Maor, Princeton University Press, 1994, p.86.

72 “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. A similar point of view was reached by John Bernoulli I in a letter of May, 1694, addressed to Leibniz. The process shows that Bernoulli passed from xx = 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 xx = y and x log x = log y. Therefore Leibniz and J. Bernoulli had a grasp at this time of the exponential function. Around 1730, defined the exponential function and the by ex = lim (1 + x/n)n, logx = lim n(x1/n 1). n→∞ n→∞ − Euler also showed that the two functions are inverse to one another. The controversy on complex logarithms between Leibniz and J. Bernoulli It was Johann Bernoulli who noted that in 1702 1 1 1 1 = + . 1+ x2 2 1 ix 1+ ix − dx 1 −1 Since 1+ax = a ln(1 + ax)+ C, the above equation involves the real function tan x and complexR 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. dx −dx John Bernoulli considered x = −x , by integration, he claimed log x = log( x). We 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 1716. This correspondence between Leibniz and John Bernoulli during the years of 1712 and 1713 was not published until 1745. 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 1727-1731 have been in the possession of the Stockholm academy of sciences and have for the first time been published in full by G. Enestr¨om in 1902. 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.

73 x dy 01/09/1728 J. Bernoulli to Euler: If y = ( n) , then log y = x log( n) and y = • dz− − log( n)dx = log(+n)dx for d log( z) = −z = d log z. Integrating, log y = x log n and−y = nx. Hence y =( 1)x becomes− 1x−=1 or y = 1. ± 12/10/1728 Euler to J. Bernoulli: I have arguments both for and against log x = • 2 1 √ 2 √ 2 log( x). If log(x ) = z, we have 2 z = log x . But x is as much x as +x. − 1 2 − Hence 2 z = log x = log x. It may be objected that x 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 ab = c and define b = logac, 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. The end of story —– Euler’s contribution 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:

eiφ = 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. 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 dis- cussion 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,

74 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 imagina- ress” to the Berlin Acadamy. In the same year, Eulder wrote to another famous French 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 1747. Euler’s original paper of 1747 was published in 1862. (96 years later!) Now we know, by Euler’s formula z = z ei arg(z) = z (cos arg(z) + i sin arg(z)), | | | | log z = log z + i arg(z) has infinitely many values. | |

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