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Argument Principle Topic 11 Notes Jeremy Orloff 11 Argument Principle 11.1 Introduction The argument principle (or principle of the argument) is a consequence of the residue theorem. It connects the winding number of a curve with the number of zeros and poles inside the curve. This is useful for applications (mathematical and otherwise) where we want to know the location of zeros and poles. 11.2 Principle of the argument Setup. � a simple closed curve, oriented in a counterclockwise direction. f .z/ analytic on and inside �, except for (possibly) some finite poles inside (not on) � and some zeros inside (not on) �. p ; ; p f � Let 1 § m be the poles of inside . z ; ; z f � Let 1 § n be the zeros of inside . Write mult.zk/ = the multiplicity of the zero at zk. Likewise write mult.pk/ = the order of the pole at pk. We start with a theorem that will lead to the argument principle. Theorem 11.1. With the above setup f ¨ z (∑ ∑ ) . / dz �i z p : Ê f z = 2 mult. k/* mult. k/ � . / Proof. To prove this theorem we need to understand the poles and residues of f ¨.z/_f .z/. With this f z m z f z z in mind, suppose . / has a zero of order at 0. The Taylor series for . / near 0 is f z z z mg z . / = . * 0/ . / g z z where . / is analytic and never 0 on a small neighborhood of 0. This implies ¨ m z z m*1g z z z mg¨ z f .z/ . * 0/ . / + . * 0/ . / f z = z z mg z . / . * 0/ . / m g ¨.z/ = z z + g z * 0 . / g z g¨ z g z z z Since . / is never 0, . /_ . / is analytic near 0. This implies that 0 is a simple pole of f ¨ z f z . /_ . / and 0 1 f ¨.z/ Res ; z = m = mult.z /: f .z/ 0 0 1 11 ARGUMENT PRINCIPLE 2 z m f z z Likewise, if 0 is a pole of order then the Laurent series for . / near 0 is f z z z *mg z . / = . * 0/ . / g z z where . / is analytic and never 0 on a small neighborhood of 0. Thus, m m f ¨.z/ m.z * z /* *1g.z/ + .z * z /* g ¨.z/ = * 0 0 f z z z *mg z . / . * 0/ . / m g ¨.z/ = * z z + g z * 0 . / z f ¨ z f z Again we have that 0 is a simple pole of . /_ . / and 0 1 f ¨.z/ Res ; z = *m = *mult.z /: f .z/ 0 0 The theorem now follows immediately from the Residue Theorem: f ¨.z/ dz = 2�isum of the residues Ê f .z/ � ( ) ∑ ∑ = 2�i mult.zk/* mult.pk/ : Z f � P Definition. We write f ,� for the sum of multiplicities of the zeros of inside . Likewise for f ,� . So the Theorem 11.1 says, f ¨ () dz �i Z P : Ê f = 2 f ,� * f ,� (1) � Definition. Winding number. We have an intuition for what this means. We define it formally via � z Cauchy’s formula. If is a closed curve then its winding number (or index) about 0 is defined as �; z 1 1 dz: Ind. 0/ = �i Ê z z 2 � * 0 (In class I’ll draw some pictures. You should draw a few now.) 11.2.1 Mapping curves: f ◦� One of the key notions in this topic is mapping one curve to another. That is, if z = �.t/ is a curve and w = f .z/ is a function, then w = f ◦�.t/ = f .�.t// is another curve. We say f maps � to f ◦�. We have done this frequently in the past, but it is important enough to us now, so that we will stop here and give a few examples. This is a key concept in the argument principle and you should make sure you are very comfortable with it. Example 11.2. Let �.t/ = eit with 0 f t f 2� (the unit circle). Let f .z/ = z2. Describe the curve f ◦�. Solution: Clearly f ◦�.t/ = e2it traverses the unit circle twice as t goes from 0 to 2�. Example 11.3. Let �.t/ = it with −∞ < t < ∞ (the y-axis). Let f .z/ = 1_.z + 1/. Describe the curve f ◦�.t/. 11 ARGUMENT PRINCIPLE 3 Solution: f .z/ is a fractional linear transformation and maps the line given by � to the circle through the origin centered at 1_2. By checking at a few points: 1 1 + i 1 1 * i f .*i/ = = ; f .0/ = 1; f .i/ = = ; f (∞) = 0: *i + 1 2 i + 1 2 We see that the circle is traversed in a clockwise manner as t goes from −∞ to ∞. Im(z) Im(w) 1 γ(t) = it w = f(z) = z+1 f(−i) i = γ(1) f ◦ γ(t) = 1=(1 + it) Re(z) Re(w) 0 f(1) 1=2 1 = f(0) −i = γ(−1) f(i) The curve z = �.t/ = it is mapped to w = f ◦�.t// = 1_.it + 1/. 11.2.2 Argument principle You will also see this called the principle of the argument. Theorem 11.4. Argument principle. For f and � with the same setup as above f ¨ z . / dz �i f◦�; �i Z P Ê f z = 2 Ind. 0/ = 2 . f ,� * f ,� / (2) � . / Proof. Theorem 11.1 showed that f ¨ z . / dz �i Z P Ê f z = 2 . f,� * f,� / � . / So we need to show is that the integral also equals the winding number given. This is simply the change of variables w = f .z/. With this change of variables the countour z = �.t/ becomes w = f ◦�.t/ and dw = f ¨.z/ dz so f ¨ z dw . / dz �i f◦�; Ê f z = Ê w = 2 Ind. 0/ � . / f ◦� The last equality in the above equation comes from the definition of winding number. Note that by assumption � does not go through any zeros of f , so w = f .�.t// is never zero and 1_w in the integral is not a problem. Here is an easy corollary to the argument principle that will be useful to us later. Corollary. Assuming that f ◦� does not go through *1, i.e. there are no zeros of 1 + f .z/ on � then f ¨ �i f◦�; �i Z P : Ê f = 2 Ind. *1/ = 2 . 1+f ,� * f ,� / (3) � + 1 11 ARGUMENT PRINCIPLE 4 Proof. Applying the argument principle in Equation 2 to the function 1 + f .z/, we get f ¨ z .1 + / . / dz �i f◦�; �i Z P Ê f z = 2 Ind.1 + 0/ = 2 . 1+f ,� * 1+f ,� / � 1 + . / Now, we can compare each of the terms in this equation to those in Equation 3: f ¨ z f ¨ z .1 + / . / dz . / dz f ¨ f ¨ Ê f z = Ê f z (because .1 + / = ) � 1 + . / � 1 + . / Ind.1 + f ◦�; 0/ = Ind.f ◦�; *1/ (1 + f winds around 0 ⇔ f winds around -1) Z Z 1+f ,� = 1+f ,� (same in both equations) P P f f 1+f ,� = f ,� (poles of = poles of 1 + ) Example 11.5. Let f .z/ = z2 +z Find the winding number of f ◦� around 0 for each of the following curves. � 1. 1 = circle of radius 2. � 2. 2 = circle of radius 1/2. � 3. 3 = circle of radius 1. answers. f .z/ has zeros at 0; *1. It has no poles. f � f◦� ; Z So, has no poles and two zeros inside 1. The argument principle says Ind. 1 0/ = f,� * P 1 f ,� = 2 f � f◦� ; Likewise has no poles and one zero inside 2, so Ind. 2 0/ = 1 * 0 = 1 � f f For 3 a zero of is on the curve, i.e. .*1/ = 0, so the argument principle doesn’t apply. The � image of 3 is shown in the figure below – it goes through 0. Im(w) Im(z) γ 1 w = f(z) = z2 + z γ3 γ2 Re(z) Re(w) 1 1 2 2 The image of 3 different circles under f .z/ = z2 + z. 11.2.3 Rouché’s theorem. Theorem 11.6. Rouchés theorem. Make the following assumptions: � is a simple closed curve 11 ARGUMENT PRINCIPLE 5 f; h are analytic functions on and inside �, except for some finite poles. There are no poles of f and h on �. h < f everywhere on �. Then Ind.f ◦�; 0/ = Ind..f + h/◦�; 0/: That is, Z P Z P f ,� * f ,� = f +ℎ,� * f +ℎ,� (4) Proof. In class we gave a heuristic proof involving a person walking a dog around f ◦� on a leash of length h◦�. Here is the analytic proof. The argument principle requires the function to have no zeros or poles on �. So we first show that this is true of f , f + h, .f + h/_f . The argument is goes as follows. Zeros: The fact that 0 f h < f on � implies f has no zeros on �. It also implies f + h has no zeros on �, since the value of h is never big enough to cancel that of f . Since f and f + h have no zeros, neither does .f + h/_f . Poles: By assumption f and h have no poles on �, so f + h has no poles there. Since f has no zeros on �, .f + h/_f has no poles there. Now we can apply the argument principle to f and f + h f ¨ 1 dz f◦�; Z P : �i Ê f = Ind. 0/ = f ,� * f ,� (5) 2 � f h ¨ 1 . + / dz f h ◦�; Z P : �i Ê f h = Ind.. + / 0/ = f +ℎ,� * f +ℎ,� (6) 2 � + 0 1 h h h f + h Next, by assumption < 1, so ◦� is inside the unit circle.
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