Question

Show that the function \(f_{1}(z)=1 /\left(z^{2}+1\right)\), where \(z \neq \pm i\), is the analytic continuation into \(\mathbb{C}-\\{i,-i\\}\) of the function \(f_{2}(z)=\sum_{n=0}^{\infty}(-1)^{n} z^{2 n}\), where \(|z|<1\).

Short answer

Yes, the function \(f_1(z)=1 /\left(z^{2}+1\right)\) is the analytic continuation of the function \(f_2(z)=\sum_{n=0}^{\infty}(-1)^{n} z^{2 n}\) into \(\mathbb{C}-\{i,-i\}\).

Step-by-step solution

Expansion of the function \(f_1(z)\)

Begin with the function \(f_1(z) = 1/(z^2 + 1)\). Using binomial series to expand the function, you get: \(f_1(z) = \sum_{n=0}^{\infty}(-1)^n z^{2n}\). This is valid when \(|z| < 1\).

Comparison of function \(f_1(z)\) and \(f_2(z)\)

Now, compare this expansion with the function \(f_2(z)=\sum_{n=0}^{\infty}(-1)^{n} z^{2 n}\). It's clear they are the same, hence the function \(f_1(z)\) is the analytic continuation of the function \(f_2(z)\). However, we are not done yet. The analytic continuation should be valid in \(\mathbb{C} - \{i, -i\}\)

Check for validity in the complex plane without \(i\) and \(-i\)

Now, checking the analytic continuation of \(f_1(z)\) and \(f_2(z)\) in the plane \(\mathbb{C} - \{i, -i\}\), we see that the function \(f_1(z)\) is analytic except at the points \(z=i\) and \(z=-i\), where it has simple poles. The function \(f_2(z)\) has radius of convergence 1 (since it's a geometric series with common ratio \(z^2\), and converges when \(|z^2|<1\), i.e., \(|z|<1\)). Hence, merging them gives \(f_1(z)\) as the analytic continuation of \(f_2(z)\) on \(\mathbb{C} - \{i, -i\}\).

Key concepts

Complex Analysis

Complex analysis is the study of functions that are defined and analytic in the complex plane. To understand this, think of working not just with real numbers but with complex numbers, which include an imaginary component denoted as \(i\). Analytic functions have derivatives at every point in some region of the complex plane. This means they are smooth and well-behaved in that area.
In the exercise, we explore whether a certain function, \(f_1(z)\), remains "analytic" across a larger portion of the complex plane by extending the domain of another function, \(f_2(z)\). This process is known as analytic continuation. Here, \(f_1(z)\) is analytic except at points where \(z = i\) and \(z = -i\) because it has simple poles—these are points where the function can't be defined in a traditional sense. However, apart from those, it maintains its "analytic" nature everywhere else in the complex plane, effectively demonstrating how we can "continue" \(f_2(z)\) into a larger set.

Functional Expansion

Functional expansion refers to the process of representing a function as an infinite series to make it easier to analyze or compute. In this exercise, we used binomial series to expand the function \(f_1(z) = \frac{1}{z^2 + 1}\) into an infinite series. This allows us to express \(f_1(z)\) as a series similar to \(f_2(z)\).
This technique is crucial because it helps us see relationships between different functions, especially when attempting analytic continuation. By looking at the series, we can compare \(f_1(z)\) with other known series like that of a geometric series and leverage properties of convergence to analyze them further. Moreover, such expansions reveal the function's behavior within specific domains, confirming they are indeed identical on overlapping regions of their domains.

Geometric Series

A geometric series is a series with a constant ratio between successive terms. Its general form appears as \(a + ar + ar^2 + ar^3 + \dots\). In a geometric series, if the absolute value of the common ratio \(r\) is less than 1, the series converges to \(\frac{a}{1-r}\).
In the problem, the function \(f_2(z)\) is expressed as a geometric series: \(\sum_{n=0}^{\infty}(-1)^{n} z^{2n}\). Given that \(|z|<1\), this ensures the series converges smoothly, defining \(f_2\) within its domain. This series provided an accessible way to explore how \(f_2(z)\) behaves, proving it actually maps perfectly onto \(f_1(z)\) except near \(z=i\) and \(z=-i\) without complications from those singular points where poles exist. By understanding the geometric series, you get a glimpse into how infinite sums can represent broader categories of functions with specific domain limitations due to convergence constraints.