91Ó°ÊÓ

In complex analysis, continuity is a fundamental property of functions. A function is continuous at a point if the limit of the function as it approaches that point is equal to the function's value at that point. This concept helps understand how the function behaves locally.
For the function \(f(z) = \frac{1}{1+z^2}\), it is a rational function, meaning it is expressed as the ratio of two polynomials. Rational functions in complex analysis are continuous everywhere except where the denominator is zero.
In this specific problem, we are asked to check for continuity in the entire complex plane \(\mathbb{C}\). To do this, we identify where the denominator, \(1+z^2\), becomes zero, hence finding the potential points of discontinuity. Solving \(1+z^2=0\), we get \(z = \pm i\).
Therefore, \(f(z)\) is continuous everywhere on \(\mathbb{C}\), except at the points where \(z = i\) and \(z = -i\). These points are known as singularities, and specifically, they are poles of order 1 for this function. Away from these singularities, the function behaves in a "smooth" manner, without any jumps or breaks.

Power series play a significant role in complex analysis as they allow complex functions to be represented in terms of simpler polynomial functions. A power series is an infinite sum of terms in the form \(a_n z^n\), where \(a_n\) are the coefficients, and \(z\) is the complex variable.
To find the power series expansion of \(\frac{1}{1+z^2}\) around \(z_0 = 0\), utilize the geometric series formula, \(\frac{1}{1-z} = \sum_{n=0}^\infty z^n\), which converges when \(|z|<1\). By substituting \(z^2\) with \(-z^2\), the series becomes \(\frac{1}{1-(-z^2)} = \sum_{n=0}^\infty (-1)^n z^{2n}\).
Each term \((-1)^n z^{2n}\) simplifies the function representation and displays how the function behaves in the vicinity of \(z_0 = 0\). This power series representation sheds light on the nature of the complex function by allowing calculations involving derivatives and integrals to be more straightforward.

The radius of convergence is an essential concept involved when discussing power series. It determines the region within which the series representation of a function reliably converges to the function itself.
The convergence of a power series \(\sum_{n=0}^\infty a_n z^n\) is limited to the region \(|z| < R\), where \(R\) is the radius of convergence. This is calculated using the Cauchy-Hadamard theorem, given by \( R = \frac{1}{\limsup_{n\to\infty} \sqrt[n]{|a_n|}} \).
For the function \(\frac{1}{1+z^2}\), the series \(\sum_{n=0}^\infty (-1)^n z^{2n}\) indicates that the radius of convergence \(R = 1\). This is because the closest singularity from the origin, the point of expansion, is at \(z = \pm i\), which are at a distance 1 from the origin on the complex plane.
Therefore, the power series is valid only inside the disk \(|z| < 1\). As for the modified function \(\frac{1}{1+\lambda^2 z^2}\), the radius of convergence is determined by the parameter \(\lambda\), specifically \(R = \frac{1}{|\lambda|}\).
This underscores the idea that the singular points and their distances from the center of expansion play a critical role in defining the limits of convergence, which cannot be concluded only from a real line perspective but must consider the full complex plane.