2.4 Lecture 7: Exponential and trigonometric

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1 154 CHAPTER. CHAPTER II Figure.9: generalized elliptical domains; figures are shown for ǫ = 1, ǫ = 0.8, eps = 0.6, ǫ = 0.4, and ǫ = 0 the case ǫ = 0. is in Figure.7).4 Lecture 7: Exponential and trigonometric functions.4.1 The real exponential and some comments on rigor In discussing the derivative series ja j z j 1 j=1 we used the fact that j 1/j 1 and j ր, and we used Ahlfors direct proof to show this fact. Of course, we have assumed here the real j-th root is a well-defined real function with familiar properties. This function and its properties may be obtained rigorously using the ordering and completeness of the real numbers and techniques from calculus.) Another familiar proof of the fact that j 1/j as j ր runs as follows: Because ln j 1/j = ln j/j, which is an indeterminate form /, we can use L Hopital s rule to see lim ln j/j = lim 1/j = 0. Thus, by the continuity of the real exponential lim j 1/j = e 0 = 1. This argument uses a number of facts from the calculus of real functions. Notably, we needed the existence of the real exponential and logarithm), facts about derivatives, and L Hoptial s rule. While each of these may be understood and justified to hopefully) a satisfactory level of rigor, Ahlfors wishes to present the real exponential and trigonometric functions) as restrictions to the real line of their complex versions. Partially to highlight some differences between the real approach and the complex approach and partially because a rigorous proof of the needed version of) L Hopital s rule us usually not included in elementary calculus,

2 .4. LECTURE 7: EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS155 let us quickly present an outline of the elements used in the argument above regarding j 1/j. This also affords us the opportunity to present/review the real exponential. The real exponential may be defined by the series e x = j=0 1 j! xj..8) The convergence is absolute for all x R. Essentially, the same argument we used for complex series involving Hadamard s construction applies here based on the fact that ) 1/j 1 0 as j ր..9) j! To see this limit, it is enough to show j!) 1/j which follows from the estimate j! > j j/ = [ j] j or j!) > j j..30) To establish.30), note that j!) = j lj l + 1) = j1) j 1)) j )3) )j 1) 1)j). l=1 The j factors appearing here lj l + 1) each satisfy lj l + 1) j for 1 l j with strict inequality for 1 < l < j. The function fl) = lj l+ 1) = l +j+1)l is a quadratic polynomial with negative leading coefficient, f1) = l, fl) = l and a maximum at l = j + 1)/; see Figure ) Again, our theorem on the complex derivative applies in the real case, so d dx ex = j=1 1 j 1)! xj 1 = j=0 1 j! xj = e x. That is, y = e x satisfies the ordinary differential equation y = y.31) with the initial value y0) = 1. It is immediate from the initial value problem that y = e x is positive and increasing for non-negative x. If we assume the 3 It is interesting that this fact does not seem to allow a simple proof by induction.

3 156 CHAPTER. CHAPTER II Figure.10: quadratic dependece of factors in a particular expression for j! existence of some real x for which e x = 0, then we may take the the largest such value; call it x 0 < 0. This leads to a contradiction of the existence and uniqueness theorem for ODEs: Theorem 3 local existence and uniqueness for nonlinear ODEs) If x 0, y 0 ) R, and there is some δ 0 such that the function satisfies f : x 0 δ 0, x 0 + δ 0 ) y 0 δ 0, y 0 + δ 0 ) R f, f y C0 R) where R = x 0 δ 0, x 0 + δ 0 ) y 0 δ 0, y 0 + δ 0 ), then there is some δ > 0 such that the initial value problem { y = fx, y) yx 0 ) = y 0 has a unique solution on x 0 δ, x 0 + δ). We conclude that y = e x > 0 and y x) > 0 for x R. There is another version of the existence and uniqueness theorem for ODEs which applies here owing to the fact that y = y is a linear equation. Theorem 4 global existence and uniqueness for linear ODEs) If f, q C 0 a, b), then the initial value problem { y + qx)y = fx) yx 0 ) = y 0 has a unique solution on a, b) for any x 0 a, b) and any y 0 R.

4 .4. LECTURE 7: EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS157 Notice that the theorem for linear equations is much simpler and has a much stronger conclusion. In particular, notice the δ determining the interval of existence in the theorem for nonlinear equations may be much smaller than the δ 0 determining the interval of definition for the equation. In fact, it might be that δ is very small when δ 0 =. This is a consequence of the possible nonlinearity of the equation. Basic example: y = y.) Both of the existence and uniqueness theorems generalize to systems of ODEs, and we will use the generalization to systems below. Nexercise 30 State versions of the existence and uniqueness theorems for ODEs for systems. The ordinary differential equation.31) and the theorems on existence and uniqueness of solutions may be used to verify many useful properties of the function e x defined by the series.8). For example, because we see the functions fx) = e x+b and gx) = e x e b satisfy f = f, f0) = e b and g = g, g0) = e b we know by the existence and uniqueness theorem e a+b = e a e b for all a, b R..3) As mentioned, the real function e x is monotone increasing, and for this reason, it has a well-defined real inverse lnx. The derivative d/dx)e x is at least 1 for x > 0 and, therefore, the range of the exponential clearly includes all nonnegative numbers. By the exponent rule e x = 1 ց 0 as x ր. ex Therefore, the range of e x is precisely 0, ) and ln : 0, ) R. The exponential rule.3) leads immediately to the rule lnαβ) = ln α + ln β for all α, β > 0. With some additional effort one also obtains lnx p = p lnx for all x > 0 and p R

5 158 CHAPTER. CHAPTER II which, in view of the continuity of the logarithm, puts one in a position to consider ln j 1/j = ln j j. The fact that ln j ր as j ր follows simply from the fact that e x is defined for all x R and takes arbitrarily large values for positive x; remember d/dx)e x = e x > 1 for x > 0. This means we are in a position, at least nominally, to apply L Hopital s rule. Nexercise 31 Show d dx ln x = 1 x. Let us briefly consider the version of L Hopitals rule required here: Theorem 5 L Hopital s rule for the indeterminate form / ) If lim fx) = lim gx) =, xր xր there is some A for which g x) > 0 for x > A, and then f x) lim = L R,.33) xր g x) fx) lim xր gx) Proof: For any ǫ > 0, there is some N 1 so that = L..34) x > N 1 implies ǫ < f x) g x) L < ǫ. That is, L ǫ ) g x) < f x) < L + ǫ ) g x). This implies, for example, that and, hence, that fx) f x) L + ǫ ) g x) < 0 L + ǫ ) gx) is decreasing as x ր.

6 .4. LECTURE 7: EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS159 In particular, fx) or fx) gx) < L + ǫ + fn 1) For x large, say x > N > N 1, so Similarly, and therefore f x) fx) L + ǫ ) gx) < fn 1 ) L + ǫ ) gn 1 ) L + ǫ ) gn1 ) gx) fn 1 ) + L + ǫ ) gn1 ) gx) < ǫ, fx) gx) < L + ǫ + ǫ = L + ǫ. L ǫ ) g x) > 0 for x > N 1, for x > N 1. L ǫ ) gx) is increasing as x ր. We have fx) L ǫ ) gx) > fn 1 ) L ǫ ) gn 1 ), fx) gx) > L ǫ + fn 1) L ǫ ) gn1 ), gx) and there is some N > N such that x > N implies, It follows that for x > N we have We have established.34). fn 1 ) L ǫ ) gn1 ) gx) L ǫ < fx) gx) < L + ǫ. < ǫ. Nexercise 3 Modify the proof of L Hopital s rule above to treat the cases L = and L = +. Note: The proof of L Hopital s rule for the indeterminate form 0/0 is fundamentally different and is usually obtained using the Cauchy mean value theorem, i.e., the mean value theorem for parametric curves in R.

7 160 CHAPTER. CHAPTER II Complex differential equations The complex exponential function is defined by the same series e z 1 = j! zj. k=0 j=0 We have already verified that j!) 1/j, so the radius of convergence is and the function is entire. Similarly, we define 1) k 1) k cosz = k)! zk and sin z = k + 1)! zk+1 each of which is easily seen to be entire: k=0 [j + 1)!] 1/j > [j)!] 1/j = { [j)!] 1/j)}. From these series the following familiar facts are easy to verify: e iz = cosz + i sin z,.35) cosz = eiz + e iz and sin z = eiz e iz. i The expression in.35) is known as Euler s formula. It is also straightforward to verify directly from the series representations that fz) = e z satisfies while cz) = cosz and sz) = sin z satisfy f = f, f0) = 1.36) c = s, c0) = 1, and s = c, s0) = 0..37) Consequently, both cos z and sin z satisfy the familiar differential) equation f = f..38) It is important, however, to realize that the differential equations in.36),.37), and.38) are not ordinary differential equations. The existence and uniqueness theorems given above for real ODEs have generalizations to complex functions of a single real varaible y : x 0 δ, x 0 + δ) C, but we cannot expect a comparable theorem for the initial value problems in.36) and.37). In particular, it should be noted that we have not yet established an analogue of.3) for fz) = e z defined on C. Here is the result with which we must work:

8 .4. LECTURE 7: EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS161 Theorem 6 If f z) 0 in some open connected set U C and fz 0 ) = w 0 for some z 0 U, then fz) w 0 on U. Fortunately, standard rules of differentiation like the chain rule and the product rule are still at our disposal. If we set then Since g0) = e β, we again have fz) = e z+β e z, f z) = e z+β e z e z+β e z 0. e α+β = e α e β for α, β C..39) Here are some important consequences of.39): Lemma 13 e z 0 for z C. Proof: e z e z = e 0 = 1. Lemma 14 e iθ = 1 for θ R, and as a corollary) Proof: cosθ+i sin θ = e iθ e iθ = e iθ cos θ + sin θ = 1 for θ R..40) j=1 1 j! iθ)j = e iθ j=1 1 j! iθ)j = e iθ e iθ = 1. In fact, since e z = e x for z = x R, all the properties obtained for x R using ordinary differential equations and calculus hold for the complex exponential function restricted to R); they are the same function. The identity.40) is usually not obtained rigorously, i.e., it is asserted based on a geometric definition of cosine and sine rather than an analytic definition via series or differential equations though the identity can be obtained in the real case using ODEs). Lemma 15 cos z + sin z = 1 for z C. Proof: ) e iz + e iz ) e + iz e iz = eiz + + e iz i 4 + eiz + e iz 4 = 1.

9 16 CHAPTER. CHAPTER II Lemma 16 e z = e Re z. Proof: e z = e Re z e iim z = e Re z. Lemma 17 For all α, β C Nexercise 33 Prove Lemma 17. cosα + β) = cosαcosβ sin α sin β sinα + β) = sin α cosβ + sin β cos α. It is natural at this point to consider the functions fz) = e z, fz) = cosz and fz) = sin z as mappings f : C C. The first step is to determine the real and imaginary parts u and v of f = u + iv as functions of z = x + iy. See, for example, Exercise II We will return to this consideration in detail in Chapter III Complex periodicity Definition 8 Given f : C C, we say τ C\{0} is a period of f if fz + τ) = fz) for all z C. We will now proceed to show that fz) = e z is periodic and so are sin z and cosz). A period τ of e z must satisfy e τ = e Re τ cos Im τ + i sin Im τ) = 1. We know e τ = e Re τ, and we know the real exponential is increasing with e 0 = 1, so we must have Re τ = 0. Writing θ 0 = Im τ, we have shown 1. If fz) = e z has a complex period τ, than τ is purely imaginary τ = iθ 0, and. cosθ 0 = 1 while sin θ 0 = 0. Conversely, if we can find θ 0 0 for which cosθ 9 + i sin θ 0 = 1, then iθ 0 is a period for fz) = e z. Since we are looking for a real value θ 0, we may turn our attention to the restrictions of cosine and sine to R.

10 .4. LECTURE 7: EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS163 Properties of c = cosθ and s = sin θ, θ R Recall that c0) = 1, s0) = 0, c = s and s = c. Clearly, there is an initial interval to the right of θ = 0 upon which cosine is positive and decreasing while sine is positive and increasing. Lemma 18 There is a first positive θ 1 for which cosθ 1 = 0, and cosθ decreases to cosθ 1 = 0, sin θ increases to sin θ 1 = 1. Proof: Fix θ > 0 in the initial interval with cos θ > sin θ > 0. Then for θ > θ as long as cosθ > 0 so that sin θ is increasing) we have cosθ < cosθ sin θ θ θ ) ց as θ ր..41) If this assertion needs further justification, simply notice that since c = s Figure.11: behavior of the real trigonometric functions on the quarter period and s = sin θ is increasing, the function c = sin θ is strictly decreasing. On the other hand, if there were some θ for which were violated, then the mean value theorem would give us some θ > θ on the same interval) for which c θ) sin θ = cθ ). This contradicts the strict monotonicity of c. The inequality.41) shows there is a first θ 1 > 0 for which cosθ 1 = 0. Since sin θ is increasing for 0 θ < θ 1 and sin θ + cos θ = 1, the remaining assertions follow as well.

11 164 CHAPTER. CHAPTER II Lemma 19 There is a first θ > θ 1 for which sin θ = 0, and for θ 1 < θ < θ sin θ decreases to cosθ = 0, cos θ decreases to sin θ = 1. Proof: The same approach works in this case: sin θ < sin θ + cos θ θ θ ) ց as θ ր Figure.1: the real trigonometric functions on the second quarter period Lemma 0 θ = θ 1 and for 0 θ θ 0 Proof: Calculating we find sinθ 1 + θ) = sinθ 1 θ) cosθ 1 + θ) = cosθ 1 θ). d sinθ dθ 1 + θ) = cosθ 1 + θ), sinθ 1 + 0) = 1 d cosθ dθ 1 + θ) = sinθ 1 + θ), cosθ 1 + 0) = 0. and d sinθ dθ 1 θ) = cosθ 1 + θ), sinθ 1 0) = 1 d [ cosθ dθ 1 θ)] = sinθ 1 θ), cosθ 1 0) = 0.

12 .4. LECTURE 7: EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS165 This means the pairs of functions above both satisfy the IVP { f = g, f0) = 1 g = f, g0) = 0. By the existence and uniqueness theorem for systems of real ODEs, the pairs of functions must be the same. This establishes the symmetry identities. We then have sinθ 1 +θ 1 ) = sin0) = 0, and θ 1 is the smallest argument θ > θ 1 for which sin θ = 0. This means θ 1 = θ. Evidently this procedure may be repeated until we find a first positive value 4θ 1 for which cos4θ 1 ) = 1 and sin4θ 1 ) = 0. We define the number π Figure.13: the smallest positive period of cosine and sine to be half the period 4θ 1 : π = θ 1. We have shown except for a few minor details, e.g., the careful consideration of negative θ) that the unique periods of e z are ±πi, ±4πi, ±6πi,... The unique periods of e iz, cos z, and sin z are all real and are given by ±π, ±4π, ±6π,... Since cos θ+sin θ = 1, we have justified the geometric definition/interpretation of cosθ and sin θ as the coordinates of a point on S 1. Nexercise 34 Show cos[θ] = sin[θ 1 θ] for 0 θ θ 1. Nexercise 35 Show there is a unique value solution of cosθ = sin θ with 0 < θ < θ 1 = π/. Show the solution is θ = π/4.

13 166 CHAPTER. CHAPTER II The complex logarithm and Riemann surfaces 4.3) Here we consider the complex logarithm log : L C where L is the logarithmic Riemann surface which will be described soon. For now, if you don t have any idea what this Reimann surface might be, then just think of it as the complex plane or, more precisely, the punctured complex plane C\{0}.) Let us distinguish the complex logarithm log from the real logarithm ln which is the restriction of log). Our starting point is the following equation for log w = z = x + iy: e log w = w e x = w, e iy = w w. This equation makes sense for w 0, and because it does not make sense for w = 0 the exponential does not take the value w = 0) there is no complex logarithm of w = 0. For w 0, the equation e x = w has a unique solution given by the real logarithm: Relog w) = x = ln w. Notice that the second equation e iy = w/ w follows from the first one and the addition of exponents. Our discussion of the real trigonometric functions gives us infinitely many solutions y for which e iy = w w S1 with exactly one on each interval θ y < θ+π. This provides another and technically more rigourous) definition for the argument argw) for w 0, namely, the argument is the imaginary part of the complex logarithm. On the other hand, the choice of interval i.e., the choice of θ above) is required. In summary, a choice of interval θ argw) < θ + π determines a branch of the logarithm in precisely the same way one makes a choice of the interval for the argument to determine a branch of the square root or the n-th root); having made such a choice we have log w = ln w + i arg w. The bookkeeping for the argument is kept in the Riemann surface L.

14 .4. LECTURE 7: EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS167 complex powers If we specify a branch of the logarithm, we may define α β for α C\{0} and β C by α β = e β log α..4) Technically, α L in a certain copy of C\{0} determined by the choice of interval for arglog)). For example, say we choose a branch of log with π < arglog) π, then If we write β = β 1 + iβ, then α β = e β log α = e βln α +i arg α). α β = e β 1 ln α β arg α e iβ ln α +β 1 arg α). In particular, if α, β > 0, then arg α = 0 and α β = e β ln α e iβ arg α = e β lnα..43) We could, however, take α in a sheet of L on which α > 0 has argα) = π. Then α β = e β lnα e πiβ. If β is an integer, then e πiβ = 1 and this is the same number given in.43). Nexercise 36 Show that in general, no matter which branch of log is chosen, if α > 0 and β Z, then α β is single valued. If, on the other hand, β = 1/, then e πiβ = 1 and α 1/ = e ln α/ = α whereas the other branch represented by.43) gives α 1/ = α. Thus, picking a branch of log and using definition.4) with β = 1/ is equivalent to choosing a branch of the complex squrae root. Nexercise 37 Show that a choice of branch for the logarithm with β = 1/3 is equivalent to a choice of branch for the cube root. Convention: If α > 0, we use the branch of log with arg α = 0 unless stated otherwise).

15 168 CHAPTER. CHAPTER II logarithm of a product Recall e α+β = e α e β. If each of the complex numbers e α e β and e α+β are in a single branch of the logarithm, i.e., they have the same argument arg e α+β = arg e α + arg e β, then with respect to this branch of logarithm That is, log e α+β = loge α e β ) = α + β. logzw) = log z + log w as long as argzw) = arg z + arg w;.44) see Figure.14. Considering the identity in.44) under no unifying as Figure.14: exponential mapping into a sheet of L sumption we can write formally ln zw + i argzw) = ln z + i arg z + ln w + i arg w. We know ln zw = ln z +ln w, but the equality argzw) = argz)+argw) can be ambiguous. More precisely logzw) = log z + log w + πki for some k Z. We have probably covered enough to introduce the exercises following 3.4. I am now going to jump to the last section in Chapter III 4.3). Picking

16 .4. LECTURE 7: EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS169 up on Figure.14, we can put the branches of log together as indicated in Figure.15. Picking up on Figure.14, we can put the branches of log together as indicated in Figure.15. The Riemann surface L is the union of all the copies Figure.15: The exponential is globally one-to-one from C to L = j Z Lj ; the logarithm is a well-defined analytic function log : L C. of C\{0} on the right with identifications as indicated. This is the natural domain of log. Allowing ourselves to stretch and bend C conformally) as we did with stereographic projection to obtain the Riemann sphere, we can

17 170 CHAPTER. CHAPTER II also realize L as a single surface in R 3 as indicated in Figure.16. Notice the Figure.16: spiral staircase model of L sheets are all connected seamlessly. The conformal map l : L C\{0} which covers C\{0} countably many times is called the logarithmic projection. Any connected 4 set Ω L for which l : Ω C\{0} is one-to-one and onto is called a sheet; the image of a sheet is called a fundamental domain. The Riemann surface Σ of the squre root has two sheets: The origin lies on Σ and represents a single point because 0 1/ = 0 is well-defined; thus the sheets are entire copies of C. The origin is an example of a branch point. The origin is also a branch point for L or for the logarithm represented by the vertical dashed line in Figure.16) though 0 / L. If two sheets come together at a branch point, we say the branch point has order one. The branch point at the origin has order infinity for the logarithm. Finally, let us consider fz) = cosz. 4 We will say what it means to be connected later.

18 .5. ADDITIONAL TOPICS AND QUESTIONS Figure.17: the Riemann surface Σ, the image of fz) = z.5 Additional topics and questions.5.1 Translation in the Reimann sphere In Nexercise we asked about the effect of the translation f : C C by fz) = z +z 0 on the Riemann sphere. Naturally, this is a degree one rational function polynomial even) with fixed point at, but the geometry of what happens in the sphere is quite interesting. Let us start with the case z 0 = 1. We can, of course, assume z 0 is real and positive) by a rotation; then we may wish to consider the more general case of z 0 = α > 0. It is analytically evident that the north pole remains fixed, and it is geometrically evident that the other points execute a kind of generalized rotation in which one portion of S, namely {x 1, x, x 3 ) S : x 3 > 1 x 1 }, is stretched out to cover the back hemisphere {x 1, x, x 3 ) : x 1 < 0} with motion away from the north pole and the complementary portion {x 1, x, x 3 ) S : x 3 < 1 x 1 }, is compressed into the front hemisphere {x 1, x, x 3 ) : x 1 > 0} with motion, generally, toward the north pole. This description is, of course, rather vague, and we would like to say something more precise. To this end, we introduce some special coordinates on the