Question

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 $$ \lim _{\mid x \rightarrow \infty} z^{-n} f(z)=c $$ 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 $$ \gamma_{r}(t)=f\left(r e^{t}\right) $$ for \(0 \leq r<\infty, 0 \leq t \leq 2 m\), and prove the following statements about the curves \(\gamma_{r}:\) (a) Ind \(\left(\gamma_{0}\right)=0 .\) (b) Ind \(\left(\gamma_{,}\right)=n\) for all sufficiently large \(r\). (c) Ind \(\left(\gamma_{,}\right)\) is a continuous function of \(r\), on \([0, \infty)\). \([\operatorname{In}(b)\) and \((c)\), use the last part of Exercise 26.] Show that \((a),(b)\), and \((c)\) are contradictory, since \(n>0\).

Short answer

Question: Prove that there exists at least one complex number \(z\) such that \(f(z) = 0\) for a given continuous complex function \(f(z)\) that satisfies the conditions in the exercise. Hint: Assume \(f(z) \neq 0\) for all \(z\), and try proving statements (a), (b), and (c) for the given \(\gamma_r(t) = f(r e^t)\) function. Show that these statements lead to a contradiction.

Step-by-step solution

Define the function \(\gamma_{r}(t)\)

Define the function \(\gamma_r(t)\) using the given exercise: $$\gamma_{r}(t) = f\left(r e^{t}\right)$$ where \(0 \leq r < \infty, 0 \leq t \leq 2m\).

Prove statement (a)

Prove that Ind\((\gamma_0) = 0\). For \(r=0\), the function \(\gamma_{r}(t)\) becomes \(\gamma_0(t) = f(0)\). Since the index of any constant in the complex plane is zero, Ind\((\gamma_0) = 0\).

Prove statement (b)

Prove that Ind\((\gamma_{r}) = n\) for all sufficiently large \(r\). For a sufficiently large \(r\), we have \begin{align*} \text{Ind}\left(\gamma_{r}\right) &= \text{Ind}(z^{-n} f(z)) \\ &= \text{Ind}\left(\frac{1}{z^{n}}\right) \\ &= n \end{align*} Therefore, Ind\((\gamma_{r})=n\) for all sufficiently large \(r\).

Prove statement (c)

Prove that Ind\((\gamma_{r})\) is a continuous function of \(r\) on \([0,\infty)\). According to the last part of Exercise 26, if \(f(z)\) is continuous and nonzero, then Ind\((f(zr))\) is a continuous function of \(r\). We assume that \(f(z) \neq 0\) for all \(z\), so Ind\((\gamma_r)\) is a continuous function of \(r\) on \([0, \infty)\).

Show that statements (a), (b), and (c) are contradictory

From statement (a), we know that Ind\((\gamma_0) = 0\). From statement (b), Ind\((\gamma_{r}) = n\) for all sufficiently large \(r\). According to statement (c), Ind\((\gamma_{r})\) is a continuous function of \(r\) on \([0, \infty)\). However, the only way for a continuous function to go from a value of 0 to a value of \(n\) is if the function takes all values between 0 and \(n\). This contradicts our assumption that \(f(z) \neq 0\) for all \(z\). Therefore, our assumption is false, and there exists at least one complex number \(z\) such that \(f(z) = 0\).

Key concepts

Complex Functions

Complex functions are a type of function that input and output complex numbers. These functions play a significant role in complex analysis—a branch of mathematics dealing with functions of complex numbers.
The complex plane, used to visualize these numbers, consists of a real axis and an imaginary axis. Complex functions can be seen as transformations on this plane, mapping one complex number to another.
Key concepts in complex functions include holomorphic functions (functions that are differentiable in a complex sense) and analytic functions (functions that can be represented by a power series). Both properties are essential to understanding how complex functions behave.
When studying complex functions, it's essential to recognize terms like polar form (expressing complex numbers as a magnitude and an angle) and Euler’s formula, which provides a deep connection between complex exponentials and trigonometric functions.
Understanding these basic ideas equips students to explore more complex topics such as residues, contour integrations, and the fascinating domain of Riemann surfaces.

Limit at Infinity

In complex analysis, the concept of a "limit at infinity" can be a bit more intricate compared to real analysis. It helps us understand how functions behave as the input grows very large or very small, essentially moving towards infinity in the complex plane.
In our problem, the expression \( \lim_{|z| \to \infty} z^{-n} f(z) = c \) involves considering the function \( f(z) \) at infinity. This means as the modulus of \( z \) becomes indefinitely large, we analyze the behavior of the function.
Key to solving these problems is understanding that complex infinity captures more complexity than real infinity. One often visualizes this using the "extended complex plane," where the point at infinity is a single, unique spot.
An important aspect is the idea that if a function approaches a certain value as it approaches infinity, this hints at properties like boundedness or specific asymptotic behavior, all of which are crucial in proving theorems in complex analysis.

Index of a Curve

The index of a curve, often written as Ind(\(\gamma\)), is an essential concept in complex analysis. It measures how many times a curve winds around a particular point in the complex plane. This winding number can be positive or negative, depending on the direction of winding.
In our problem, assertions about Ind(\(\gamma_r\)) show how the curve related to \(f(z)\) behaves in the complex plane. Step (a) in the solution shows that a constant function results in a zero index because the curve doesn't wind around any point.
As \( r \) becomes large, the solution demonstrates that the index becomes \( n \). This shift from 0 to \( n \) reveals the continuous nature of the index and indicates a contradiction to the function being non-zero everywhere.
Understanding curve indices helps mathematical problems discern function properties like roots and zeros, providing insights into underlying complex structures.

Continuous Functions

Continous functions in complex analysis are functions where small changes in the input result in small changes in the output. They are the complex counterpart to what students learn in real analysis.
In the given exercise, continuity of \( f(z) \) plays a pivotal role. It ensures no abrupt changes in function values across the complex plane, an aspect used to argue Ind(\(\gamma_r\)) is continuous.
The continuous nature of \(f(z)\) provides a bridge between the points in the problem—showing that if the index starts at 0 and continuously transforms to \( n \), there must be intermediate values present, leading to contradiction if assuming \( f(z) eq 0 \) always.
Such properties make continuous functions foundational in proving many characteristics of complex functions and their general behavior across domains.