52 CHAPTER 2. COMPLEX-VALUED FUNCTIONS 2.5 Exponential Functions and Trigonometric Func- tions Exponential Function and Its Properties By the theory of power series, we can define ∞ zn ez := n! Xn=0 which is called the exponential function. Here 0! = 1 and n!=1 2 ... (n 1) n. · · · − · ez has the following properties: |an| 1. The radius of convergence R = + . In fact, R = limn = limnn = + . ∞ |an+1| ∞ Therefore, ez is an analytic function defined on C. 2. When z = x R, the restriction ex is the usual real exponential function. This is ∈ x ∞ 1 n because the Taylor series of e is n=0 n! x . P 3. ez+w = ez ew, z,w C. In fact, ex+u = ex eu forall x, u R. Then by the uniqueness theorem 2.4.6,· ∀ez+w ∈= ez ew holds for any ·z,w C. ∈ · ∈ 4. As a function from C to C, ez is not a one-to-one function. Consider solutions of the equation ez = 1. It is equivalent to z =2πki for any integer k. s66π 6 s 4π s 2π ez s - - s - 0 s 2π 1 − s 4π − s 6π − ′ 5. We have ez = ez by the power series definition. 2.5. EXPONENTIAL FUNCTIONS AND TRIGONOMETRIC FUNCTIONS 53 n x x 6. e = limn 1+ , x R. →∞ n ∀ ∈ 7. Euler’s formula: eix = cos x + i sin x, x R. (2.10) ∀ ∈ From this, we have ez = r(cos θ + i sin θ) for any z C. ∈ Trigonometric Functions 1. From Euler’s formula, taking conjugate: e−ix = cos x i sin x so that − eix + e−ix eix e−ix cos x = , sin x = − . 2 2i which can be used to define complex-valued function cos z and sin z. eiz + e−iz eiz e−iz cos z := , sin z := − . 2 2i Hence eiz = cons z + isin z, and ∞ ( 1)n ∞ ( 1)n cos z = − z2n, sin z = − z2n+1. (2n)! (2n + 1)! Xn=0 Xn=0 n ∞ (−1) 2n Here the formulas are similar to the real case: cos x = n=0 (2n)! x and sin x = n ∞ (−1) 2n+1 P n=0 (2n+1)! x . P 2. We shall see the difference between the real case and the complex case. In the real case, cos x 1 and sin x 1; in the complex case, cosz 1 and sin z 1 are | | ≤ | | ≤ | | ≤ | | ≤ no longer true. In fact, take z = iy where y R, we have ∈ ei(iy) e i(iy) e−y ey sin z = − − = − , as y . | | 2i 2 →∞ →∞ 3. We can verify: cons2z + sin2z =1 cos(z + w)= cos z cos w sin z sin w, sin(z + w)= sin z cos w + cos z sin w. − 4. Consider sin z =0 eiz e−iz =0 s2iz =1 2z =2πk where k Z z = πk where k Z⇐⇒. Similarly,− cos z =0⇐⇒ z = π ⇐⇒+ πk where k Z. ∈ ⇐⇒ ∈ ⇐⇒ 2 ∈ 54 CHAPTER 2. COMPLEX-VALUED FUNCTIONS 5. We can also define sin z cos z tan z := , cotz := . cos z sin z Logarithm Function 1. Review Calculus: the inverse function of ex is log x. In this course we denote log x to be the natural log ln x. By the inverse function property, we have 12 elog x = x, x (0, ); log ex = x, x R. (2.11) ∀ ∈ ∞ ∀ ∈ x −1 ′ 1 Denote y = e . By the derivative formula of inverse function ((f ) = f ′ , we have 1 1 1 (log x)′ = = = . (2.12) (ey)′ ey x x ′ Then by the fundamental theorem of calculus f(x)= a f (t)dt, we obtain R x 1 log x = dt (2.13) Z1 t which is regarded as definition of log x in some textbook. 2. We notice that ez is not one-to-one function on C (see above) so that we cannot define the inverse function log z of ez. However, ez could be one-to-one function on some domains and log z could be defined on some domain. Recall the logarithm function has power series expansion: ∞ ( 1)n log(1 + x)= − xn+1, x ( 1, 1). n +1 ∈ − Xn=0 Here the radius of convergence of this power series is 1. Then we have ∞ ( 1)n log(1 + z) := − zn+1, z ∆(1). n +1 ∀ ∈ Xn=0 and, by replacing 1 + z with z, ∞ ( 1)n log z = − (z 1)n+1, z ∆(1; 1). n +1 − ∀ ∈ Xn=0 12If 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 . → ∀ ∈ ∀ ∈ 2.5. EXPONENTIAL FUNCTIONS AND TRIGONOMETRIC FUNCTIONS 55 Since elog x = x, x (0, ), by applying the uniqueness theorem 2.4.6, it implies ∀ ∈ ∞ elog z = z, z ∆(1; 1). ∀ ∈ 3. On the other hand, write z = reiθ = z ei arg z, by the addition property of logarithm, | | we obtain log z = log z + log eiarg z = log z + iarg z, z ∆(1; 1). | | | | ∀ ∈ 4. The addition theorem of exponential functions implies log(z1z2)= log(z1)+ log(z2), arg(z1z2)= arg(z1)+ arg(z2). 5. At this moment, the analytic function log z is defined on the disk ∆(1, 1). We’ll show later that the domain of definition for log z can be extended. Historic Remark Logarithm function is closely related to its inverse function: exponential function by y x = b y = logbx. ⇐⇒ In 1614, John Napier (1550-1617) published a book entitled Mirifici Logarithmorum Canonis Descriptio in which logarithm was introduced. The table of logarithms made by Napier contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible (it turns multiplication into addition). 13 Early resistance to the use of logarithms was muted by Kepler’s enthusiastic support and his publication of a clear and impeccable explanation of how they worked. By the way, Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to 1 base e . In 1668, Nicholas Mercator (1620-1687), a Germany mathematician, first mentioned the natural logarithm ln x in his work Logarithmo-technia 14 In this treatise Mercator described the Mercator series, also independently discovered by Gregory Saint-Vincent: 1 1 1 log(1 + x)= x x2 + x3 x4 + ...... − 2 3 − 4 13Joost B¨urgi independently discovered logarithms; however, he did not publish his discovery until four years after Napier. 14en.wikipedia.org: natural logarithm. The mathematics teacher John Speidell had already in 1619 com- piled a table on the natural logarithm. 56 CHAPTER 2. COMPLEX-VALUED FUNCTIONS x 1 x+1 1 which was treated as the area of the hyperbola 0 1+t dt = 1 t dt (see (2.13)). It was also in this treatise that the first known use of the termR naturalR log for the natural log appears, in the Latin form log naturalis; his use of this term is somewhat surprising, since it predates the development of calculus, in which the most natural properties of this logarithm appear. 1 n In 1697, Jacob Bernoulli studied limn→∞(1+ n ) , which had been implicit in earlier work on natural logarithms. In 1748, Euler defined the two functions:15 x n ex = lim 1+ , logx = lim n(√n x 1). (2.14) n→∞ n n→∞ − and also proved his famous formula: eix = cos x + i sin x (2.15) Euler proved that the infinite series (2.15) of both sides being equal. 50 years later, the view of complex numbers as points in the complex plane arose. Until Euler the trigonometrical quantities sine, cosine, tangent, etc., were regarded as lines connected with the circle rather than functions. Even the derivation of the series expansion for the sine in dependence of the arc by Newton and Leibniz did not change this view. It was Euler who introduced the functional point of view. 16 By the way, in 1988, readers of the Mathematical Intelligencer voted it “ the Most Beautiful Mathematical Formula Ever.” In total, Euler was responsible for three of the top five formulas in that poll. 17 It was a difficulty problem how to define log a when a is a negative number. Between d(−x) dx 1712 and 1713, Bernoulli held the view that log a = log( a) because −x = x , while Leibniz believed that log( a) must be imaginary. Between− 1727 and 1731, the question − was taken up by Bernoulli and Euler without at a solution. It was only in 1749/1751, Euler developed ideas far enough to study logarithm so that it leads to a satisfying solution. Euler’s approach is to deal with logarithm of general complex numbers. Euler’s argument n is as follows: by the definition (2.14), log z = limn→∞ n(√z 1). For each positive integer n, n(√n z 1) has n different roots (Euler realized it!) so that− the limit indeed has infinitely − many different values. When z = 1, he got log 1= 2πki, k =1, 2, 3, ....18 ± 15Morris Kline, Mathematical Thought from Ancient to Modern Times, volume 2, New York Oxford, Oxford University Press, 1972, p.404. 16Hans Niels Jahnke (editor), A History of Analysis, AMS, 2003, p.115-116. 17Here is the list of the top five: 1. Euler’s formula; 2. Euler’s formula for a polyhedron: V + F = E + 2. 1 1 1 π2 3. The number of primes is infinite. 4. There are 5 regular polyhedrons. 5. 1+ 22 + 32 + 42 + ... = 6 (Euler). cf., David Wells, Are these the most beautiful? Mathematical Intelligencer 12(3)(1990), 37-41.