Winding number

Let $f(z)$ be analytic in the annulus $\Omega = \left\{ 1 < \left|z\right| < 2\right\}$. Assume that $f$ has no zeros in $\Omega$. Show that there exists an integer $n$ and an analytic function $g$ in $\Omega$ such that, for all $z \in \Omega$, $f(z) = z^{n} e^{g(z)}$.

Let $S$ be the unilateral shift and let $f$ be an analytic function in a neighborhood of $\mathrm{cl}\ \mathbb {D}$ such that $f(z) \ne 0$ if $\left|z\right| = 1$. Let $\gamma (t) = f(\exp (2\pi i t))$, $0 \le t \le 1$. Show that $\sigma_e(f(S)) = f(\partial \mathbb {D}) = \left\{ \gamma (t): 0 \le t \le 1\right\}$ and that if $\lambda \notin f(\partial \mathbb {D})$, $\operatorname {ind}(f(S) - \lambda ) = -n(\gamma ; \lambda )$, where $n(\gamma ; \lambda ) =$ the winding number of $\gamma$ about $\lambda$. Moreover, show that if $\operatorname {ind}(f(S) - \lambda ) = 0$, then $\lambda \notin \sigma (f(S))$.

What are the components of the set of invertible elements in $C(\partial \mathbb {D})$?

Compute $I=\displaystyle \oint_{C} \dfrac {x\, dy-y\, dx}{x^{2}+y^{2}}$, where $C$ is a simple closed contour traversed in the positive direction and not passing through the origin.

Compute $I=\dfrac {1}{2\pi }\displaystyle \oint_{C} \dfrac {X\, dY-Y\, dX}{X^{2}+Y^{2}}$, where $X=ax+by$, $Y=cx+dy$, and $C$ is a simple closed contour enclosing the origin ($ad-bc\neq 0$).

If the simple contour $C$ encloses the origin, $X=\varphi (x,y)$, $Y=\psi (x,y)$, and the curves $\varphi (x,y)=0$ and $\psi (x,y)=0$ have several simple intersection points inside the contour $C$, compute the integral $I$ (see problem 4321).

Find the circulation $\Gamma$ of the vector $\overrightarrow {a}=\operatorname {grad}\left(\arctan \dfrac {y}{x}\right)$ along a closed contour $C$: (1) $C$ does not encircle the $Oz$ axis; (2) $C$ encircles the $Oz$ axis.

Let

Find all analytic functions $f : \Omega \to \Omega$ with the following property: if $\gamma$ is any cycle in $\Omega$ which is not homologous to zero (mod $\Omega$), then $f * \gamma$ is not homologous to zero (mod $\Omega$).

Find the value of the line integral $\displaystyle \int_{C} \dfrac {x \, dy-y \, dx}{x^{2}+y^{2}}$, taken around a closed contour $C$ in the positive direction.

The curves $X(x,y)=0$ and $Y(x,y)=0$ have several simple points of intersection. Compute the value of the integral $\dfrac {1}{2\pi }\displaystyle \int_{C} \dfrac {X\, dY-Y\, dX}{X^{2}+Y^{2}}$, taken in the positive direction around a closed contour $C$.

We consider a closed, continuous, oriented curve in the $z$-plane that avoids the origin. If, starting from an arbitrary point, $z$ describes the entire curve in the given direction (returning to its starting point) the argument of $z$ changes continuously and its total variation is a multiple, $2\pi n$, of $2\pi$. The integer $n$ is called the winding number of the curve.

Every ray from the origin intersects the curve in question at least $\left|n\right|$ times.

Here $L$ denotes a closed continuous curve without double points and $\mathfrak {D}$ the closed interior of $L$. The winding number of a closed, continuous, oriented curve avoiding the origin is the integer $n$ such that the argument of $w$ changes by $2\pi n$ as $w$ describes the entire curve once in the given direction.

The functions $\varphi (z)$ and $\psi (z)$ are regular in $\mathfrak {D}$, except possibly at finitely many poles, finite and different from $0$ on the boundary $L$ of $\mathfrak {D}$. We define $f(z)=\varphi (z)\psi (z)$. The winding number of the image under $w=f(z)$ of $L$ is equal to the sum of the winding numbers of the images of $L$ under $w=\varphi (z)$ and $w=\psi (z)$ respectively.

The winding number of a closed, continuous, oriented curve avoiding the origin is the integer $n$ such that the argument of $w$ changes by $2\pi n$ as $w$ describes the entire curve once in the given direction.

The function $f(z)$ is single-valued, regular and does not assume a certain value $a$ in the annulus $r<\left|z\right|<R$. All the closed continuous curves without double points that enclose the circle $\left|z\right|=r$ and that lie inside the annulus are mapped by the function $w=f(z)-a$ onto curves in the $w$-plane with the same winding number.

Let $\gamma$ be a continuously differentiable closed curve in the complex plane, with parameter interval $[a, b]$, and assume that $\gamma (t) \neq 0$ for every $t \in [a, b]$. Define the index of $\gamma$ to be

Prove that Ind $(\gamma )$ is always an integer. Hint: There exists $\varphi$ on $[a, b]$ with $\varphi^{\prime } = \gamma^{\prime } / \gamma , \varphi (a) = 0$. Hence $\gamma \exp (-\varphi )$ is constant. Since $\gamma (a) = \gamma (b)$ it follows that $\exp \varphi (b) = \exp \varphi (a) = 1$. Note that $\varphi (b) = 2 \pi i$ Ind $(\gamma )$.

Compute Ind $(\gamma )$ when $\gamma (t) = e^{i n t}, a = 0, b = 2 \pi$. Explain why Ind $(\gamma )$ is often called the winding number of $\gamma$ around 0.

Let $\gamma$ be as in Exercise 23, and assume in addition that the range of $\gamma$ does not intersect the negative real axis. Prove that Ind $(\gamma ) = 0$. Hint: For $0 \leq c < \infty$, Ind $(\gamma +c)$ is a continuous integer-valued function of $c$. Also, Ind $(\gamma +c) \rightarrow 0$ as $c \rightarrow \infty$.

Suppose $\gamma_{1}$ and $\gamma_{2}$ are curves as in Exercise 23, and

Prove that Ind $\left(\gamma_{1}\right) =$ Ind $\left(\gamma_{2}\right)$. Hint: Put $\gamma = \gamma_{2} / \gamma_{1}$. Then $\left|1-\gamma \right| < 1$, hence Ind $(\gamma ) = 0$, by Exercise 24. Also,

Let $\gamma$ be a closed curve in the complex plane (not necessarily diffierentiable) with parameter interval $[0,2 \pi ]$, such that $\gamma (t) \neq 0$ for every $t \in [0,2 \pi ]$.

Choose $\delta > 0$ so that $\left|\gamma (t)\right| > \delta$ for all $t \in [0,2 \pi ]$. If $P_{1}$ and $P_{2}$ are trigonometric polynomials such that $\left|P_{\jmath }(t)-\gamma (t)\right| < \delta / 4$ for all $t \in [0,2 \pi ]$ (their existence is assured by Theorem 8.15), prove that

by applying Exercise 25. Define this common value to be Ind $(\gamma )$. Prove that the statements of Exercises 24 and 25 hold without any differentiability assumption.

Let $f$ be a continuous complex function defined in the complex plane. Suppose there is a positive integer $n$ and a complex number $c \neq 0$ such that

Prove that $f(z) = 0$ for at least one complex number $z$. Note that this is a generalization of Theorem 8.8. Hint: Assume $f(z) \neq 0$ for all $z$, define

for $0 \leq r < \infty , 0 \leq t \leq 2 \pi$, and prove the following statements about the curves $\gamma_{r}$ :

Let $D$ be the closed unit disc in the complex plane. (Thus $z \in D$ if and only if $\left|z\right| \leq 1$). Let $g$ be a continuous mapping of $D$ into the unit circle $T$. (Thus, $\left|g(z)\right| = 1$ for every $z \in D$).

Prove that $g(z) = -z$ for at least one $z \in T$. Hint: For $0 \leq r \leq 1,0 \leq t \leq 2 \pi$, put

and put $\psi (t) = e^{-t t} \gamma_{1}(t)$. If $g(z) \neq -z$ for every $z \in T$, then $\psi (t) \neq -1$ for every $t \in [0,2 \pi ]$. Hence Ind $(\psi ) = 0$, by Exercises 24 and 26. It follows that Ind $\left(\gamma_{1}\right) = 1$. But Ind $\left(\gamma_{0}\right) = 0$. Derive a contradiction, as in Exercise 27.

As in Example 10.36, consider the 1 -form

in $\mathbb {R}^{2}-\left\{ 0\right\}$.

Suppose $\Gamma$ is a closed curve in the plane, with parameter interval $[0,2 \pi ]$. Take $\alpha \notin \Gamma^{*}$. Approximate $\Gamma$ uniformly by trigonometric polynomials $\Gamma_{n}$. Show that $\operatorname {Ind}_{\Gamma_{n}}(\alpha ) = \operatorname {Ind}_{\Gamma_{m}}(\alpha )$ if $m$ and $n$ are sufficiently large. Define this common value to be $\operatorname {Ind}_{\Gamma }(\alpha )$. Prove that the result does not depend on the choice of $\left\{ \Gamma_{n}\right\}$; prove that Lemma 10.39 is now true for closed curves, and use this to give a different proof of Theorem 10.40.

Let $\Omega$ be the plane minus two points, and show that some closed paths $\Gamma$ in $\Omega$ satisfy assumption (1) of Theorem 10.35 without being null-homotopic in $\Omega$.

Suppose $f \in H(\Omega )$, $\Gamma$ is a cycle in $\Omega$ such that $\operatorname {Ind}_{\Gamma }(\alpha ) = 0$ for all $\alpha \notin \Omega ,\left|f(\zeta )\right| \leq 1$ for every $\zeta \in \Gamma^{*}$, and $\operatorname {Ind}_{\Gamma }(z) \neq 0$. Prove that $\left|f(z)\right| \leq 1$.

Suppose $A = C(T)$, the algebra of all continuous complex functions on the unit circle $T$, with the supremum norm. Show that two invertible members of $C(T)$ are in the same coset of $G_{1}$ if and only if they are homotopic mappings of $T$ into the set of all nonzero complex numbers. Deduce from this that $G / G_{1}$ is isomorphic to the additive group of the integers. (The notation is as in Theorem 10.34).

Prove the equality

for any simple piecewise-smooth closed curve $\gamma$ such that the domain bounded by the curve $\gamma$ contains the segment $[0,1]$.