10 Logrithm Function 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 inverse function 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 series 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 logarithm, 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 analytic function 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 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. Definition of ab for complex numbers For any complex number 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 equation 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. 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 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 logarithms 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 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). − − 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 quadrature 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 summation 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.