Question

Prove that the functions defined by the two series $$ 1+a z+a^{2} z^{2}+\cdots \quad \text { and } \quad \frac{1}{1-z}-\frac{(1-a) z}{(1-z)^{2}}+\frac{(1-a)^{2} z^{2}}{(1-z)^{3}}-\cdots $$ are analytic continuations of one another.

Short answer

The series \(1+az+a^{2}z^{2}+\cdots\) and \( \frac{1}{1-z}-\frac{(1-a)z}{(1-z)^{2}}+\frac{(1-a)^{2}z^{2}}{(1-z)^{3}}-\cdots \) are analytic continuations of one another, since substituting the binomial expansion into the second series and simplifying yields the first series. Thus, the domain has been maximized while preserving analyticity.

Step-by-step solution

Recognize the Properties of Power Series

In both series we have powers of \(z\) multiplied by coefficients. When comparing terms in both series, for all powers of \(z\), we notice that the coefficients are related in a specific way.

Identify an Expansion Process

Starting from \( \frac{1}{1-z}-\frac{(1-a)z}{(1-z)^{2}}+\frac{(1-a)^{2}z^{2}}{(1-z)^{3}}-\cdots \), expand the terms using the binomial formula, \( \frac{1}{(1-z)^n} = \sum_{k=0}^{\infty} {n+k-1 \choose k} z^k \). Substitute the infinite sum for each term and simplify.

Simplify the Resulting Series

When we simplify series in step 2, all terms with power \(z^n\), where \(n = 0, 1, 2, ...\), should lead to \(a^n\), and hence matches with our original series \(1+az+a^{2}z^{2}+\cdots\) .

Understanding Analytic Continuation

An analytic continuation of a function consists of maximizing its domain while preserving analyticity. This is satisfied by the given series in the exercise.

Key concepts

Power Series

In mathematics, especially in the realm of complex analysis, a power series is a series of the form \( \[a_0 + a_1z + a_2z^2 + \. \cdot \cdot \cdot\] \), where \( a_n \) are coefficients and \( z \) is a complex variable. A power series is centered around a point (often zero) and can be used to represent functions in a region where the series converges.

Power series are highly valuable as they allow us to analyze functions that can be difficult to study otherwise. When a power series converges, it represents an analytic function in its region of convergence, which means the function is smooth and differentiable within that area.

One of the fundamental operations in working with power series is finding their radius of convergence. This radius determines the interval or disk within which the series will converge to a function. Moreover, power series can be manipulated algebraically, such as adding, subtracting, multiplying, and even differentiating or integrating term by term within the radius of convergence. These properties make power series crucial for solving differential equations, calculating integrals, and much more in various fields, particularly in mathematical physics.

Binomial Formula

The binomial formula, also known as the binomial theorem, is a fundamental result in algebra that expresses the expansion of a power of a binomial \( (1+z)^n \). According to this formula, the expansion is given by \( (1+z)^n = 1 + nz + \. \cdot \cdot \cdot + {n\choose k}z^k + \. \cdot \cdot \cdot + z^n \), where \( {n\choose k} \) denotes a binomial coefficient.

Interestingly, the binomial formula can be extended to negative and fractional exponents, provided \( |z| < 1 \). This is how we acquire a power series expansion for \( \frac{1}{(1-z)^n} \) for any complex number \( z \) with \( |z| < 1 \) and any real or complex number \( n \). The expanded binomial formula plays a pivotal role in the analytic continuation, allowing functions defined as power series within their radius of convergence to be expressed in other regions beyond their circle of convergence.

Mathematical Physics Foundations

The discussion of power series and analytic continuation also plays a significant role in the foundations of mathematical physics. These concepts provide powerful tools to deal with complex functions, which are ubiquitous in physics.

Mathematical physics uses power series to solve physical problems, such as quantum mechanics' wave functions or general relativity's spacetime equations. In these fields, physical phenomena are described by differential equations that can often be solved, or at least approximated, using power series expansions of the solutions. Analytic continuation is then used to extend these solutions to wider domains, thus revealing new physical insights and predictive capabilities. These mathematical techniques allow physicists to approximate behaviors near singularities, such as black holes, and model the spread of waves or quantum states in various media. Without the foundational understanding of power series and analytic continuation, many complex systems within mathematical physics would be much harder to explore.

Properties of Power Series

The effectiveness of power series in mathematical analysis comes from their properties. Properties of power series include convergence within a radius, term-by-term differentiation and integration, and the ability to approximate functions and their behavior.

For a power series centered at \( a \), such as \( \sum_{n=0}^{\infty} b_n(z-a)^n \), the radius of convergence is the distance from \( a \) to the nearest point where the function represented by the series ceases to be analytic.

• Uniform convergence: Within its radius of convergence, a power series converges uniformly, allowing one to integrate or differentiate the power series term by term.
• Algebra of power series: Power series can be added, subtracted, multiplied, and divided (when the divisor series isn't zero at \( a \) and within the common radius of convergence).
• Term-by-term operations: Within its radius, term-by-term differentiation and integration of the power series yield another power series with the same radius of convergence.
• Analyticity: The function represented by a convergent power series is analytic, meaning it is infinitely differentiable and thus smooth within its radius of convergence.

Understanding these properties allows students to work effectively with power series and apply them to a wide range of problems in both pure and applied mathematics.