For every n ≥ 1, consider the entire function \(p_n(z)=\sum_{k=0}^n \frac{z^k}{k !}\). Which of the following statements are true?
Analyzing the Properties of \(p_n(z)\)
The function \(p_n(z)\) is defined as the sum \(\sum_{k=0}^n \frac{z^k}{k !}\) for \(n \ge 1\). This expression is the \(n\)-th partial sum of the Taylor series expansion of the exponential function \(e^z\) around \(z=0\). Let's analyze each given statement.
\(p_n(z) = 1 + z + \frac{z^2}{2!} + \dots + \frac{z^n}{n!}\)
Convergence of \(p_n(z)\)
Statement 1: The sequence of functions \((p_n)_{n \ge 1}\) converges to an entire function uniformly on compact subsets of \(\mathbb{C}\).
The Taylor series for the exponential function \(e^z\), which is \(\sum_{k=0}^\infty \frac{z^k}{k !}\), converges for all \(z \in \mathbb{C}\). The function \(e^z\) is an entire function.
A fundamental result in complex analysis states that a power series converges uniformly on any compact subset contained within its circle of convergence. The circle of convergence for the Taylor series of \(e^z\) is the entire complex plane (\(|z| < \infty\)). Therefore, the sequence of partial sums, \(p_n(z)\), converges uniformly to \(e^z\) on any compact subset of \(\mathbb{C}\). Since \(e^z\) is an entire function, this statement is true.
Zeros of \(p_n(z)\)
Statement 2: For all \(n \ge 1\), \(p_n\) has a zero in the set \(\{z \in \mathbb{C} : |z| \le 2023\}\).
\(p_n(z)\) is a polynomial of degree \(n\). By the Fundamental Theorem of Algebra, every polynomial of degree \(n \ge 1\) has exactly \(n\) complex roots (zeros), counted with multiplicity.
For small values of \(n\), the zeros are indeed within the disk \(|z| \le 2023\):
- For \(n=1\), \(p_1(z) = 1+z\). The zero is \(z=-1\), and \(|-1|=1 \le 2023\).
- For \(n=2\), \(p_2(z) = 1+z+\frac{z^2}{2}\). The zeros are \(z = -1 \pm i\). The modulus is \(|-1 \pm i| = \sqrt{(-1)^2 + (\pm 1)^2} = \sqrt{2} \approx 1.414\), which is less than or equal to 2023.
However, as \(n \to \infty\), the sequence \(p_n(z)\) converges to \(e^z\). The function \(e^z\) has no zeros in the entire complex plane. By Hurwitz's theorem, if a sequence of analytic functions converges uniformly on compact sets to an analytic function that has no zeros in a domain, then for sufficiently large \(n\), the functions in the sequence will also have no zeros in that domain. Since \(e^z\) has no zeros in \(\mathbb{C}\), and \(p_n \to e^z\) uniformly on compact sets, for any compact set \(K\), there exists \(N\) such that for all \(n > N\), \(p_n(z)\) has no zeros in \(K\). The disk \(\{z \in \mathbb{C} : |z| \le 2023\}\) is a compact set. Therefore, for sufficiently large \(n\), \(p_n(z)\) will have no zeros in this disk. Thus, the statement that \(p_n\) has a zero in this disk for all \(n \ge 1\) is false.
Sequence of Zeros Tending to Infinity
Statement 3: There exists a sequence \((z_n)\) of complex numbers such that \(\displaystyle \lim _{n \rightarrow \infty}\left|z_n\right|=\infty\) and \(p_n(z_n) = 0\) for all \(n \ge 1\).
This statement is about the behavior of the zeros of \(p_n(z)\) as \(n\) increases. As discussed for statement 2, \(p_n(z)\) converges to the zero-free function \(e^z\). If there was a subsequence of zeros \(z_{n_k}\) such that \(|z_{n_k}|\) was bounded, then this subsequence would have a convergent subsequence \(z_{n_{k_j}} \to z_0\) by Bolzano-Weierstrass. Since \(p_{n_{k_j}}(z_{n_{k_j}}) = 0\) and \(p_n \to e^z\) uniformly on compact sets, we would have \(p_{n_{k_j}}(z_{n_{k_j}}) \to e^{z_0}\). Thus, \(e^{z_0} = 0\), which is a contradiction because \(e^z\) has no zeros. Therefore, the magnitude of all zeros of \(p_n(z)\) must tend to infinity as \(n \to \infty\). This implies that for each \(n\), we can pick a zero \(z_n\) of \(p_n(z)\) such that the sequence \((|z_n|)\) tends to infinity. This statement is true.
Minimum Modulus of Zeros
Statement 4: Let \(S_n\) denote the set of all the zeros of \(p_n\). If \(a_n=\min _{z \in S_n}|z|\), then \(a_n \rightarrow \infty\) as \(n \rightarrow \infty\).
This statement focuses on the minimum magnitude among all zeros of \(p_n(z)\). It claims that even the zero closest to the origin moves away from the origin and its distance tends to infinity as \(n\) increases.
This is a stronger statement than option 3. It implies that all zeros of \(p_n(z)\) eventually lie outside any fixed disk centered at the origin. This is indeed a known property of the zeros of the partial sums of the exponential series. The zeros of \(p_n(z)\) are known to satisfy \(|z| > n\) for large \(n\), and more precise bounds show that the minimum modulus tends to infinity. Thus, \(a_n = \min_{z \in S_n}|z|\) tends to infinity as \(n \to \infty\). This statement is true.
Based on the analysis, statements 1, 3, and 4 are true, while statement 2 is false.
Important Questions from Complex Analysis
If f (z) \( = \begin{cases} \frac {|z|}{Re(z)} &, \quad \text{if } {} \text{Re} (z)\neq 0 \\ 0 &, \quad \text{if } \text{ Re (z) = 0} \end{cases} \), then-
For z ∈ ℂ let f(z) = \(\left\{ \begin{matrix} \rm \frac{\bar{z}^2}{z}\ if \ z \ne 0,\\\ \rm 0 \ \ otherwise. \end{matrix} \right.\)
Then which of the following statements is false?
Let f, g be entire functions such that \(\lim _{z \rightarrow \infty} \frac{f(z)}{z^n}=\lim _{z \rightarrow \infty} \frac{g(z)}{z^n}=1\) for some fixed positive integer n. Which of the following statements is true?
A function f : \(\mathbb{C}\)\(\longmapsto \mathbb{C}\) is said to be analytic at ∞, if the function g defined by \(g(w)=f\left(\frac{1}{w}\right)\) is analytic at 0 with an appropriate value given for g(0). Which of the following statements is true?
For z ∈ \(\mathbb{C}\), let ℜz denotes its real part. Let f be an entire function satisfying |f(z)| ≤ |z| |ℜz| on \(\mathbb{C}\). Which of the following statements are true?
