Problem 25 - Advanced Engineering Mathematics

Question

Use binomial series to formally show that $$ \left(1-2 x t+t^{2}\right)^{-1 / 2}=\sum_{n=0}^{\infty} P_{n}(x) $$

Step-by-Step Solution

Answer

Expand \((1 - 2xt + t^2)^{-1/2}\) using the binomial series formula, yielding \(\sum_{n=0}^{\infty} P_n(x)\).

Step 1: Identify the Binomial Series Formula

The binomial series formula for inverse square root is given by the series expansion: \((1 - u)^{-\frac{1}{2}} = \sum_{n=0}^{\infty} \binom{2n}{n} \left(\frac{u}{4}\right)^{n}\). This formula is crucial for obtaining the binomial series expansion of any expression in the form \((1 - u)^{-\frac{1}{2}}\).

Step 2: Substitute the Expression for u

We are aiming to express \((1 - 2xt + t^2)^{-1/2}\) in this binomial form. Recognize that our expression represents \((1 - u)\) with \(u = 2xt - t^2\).

Step 3: Expand the Formula to Match Expression

Substitute \(u = 2xt - t^2\) into the binomial series formula: \((1 - (2xt - t^2))^{-\frac{1}{2}} = \sum_{n=0}^{\infty} \binom{2n}{n} \left(\frac{2xt - t^2}{4}\right)^{n}\). This substitution begins the process of expanding our expression using the binomial theorem.

Step 4: Separate Terms of Polynomial

Separate the terms \(2xt\) and \(-t^2\) in \(u\) to simplify: \((1 - (2xt - t^2))^{-\frac{1}{2}} = (1 - (a - b))^{-\frac{1}{2}}\) where \(a = 2xt\) and \(b = t^2\).

Step 5: Set Up Power Series

Now convert the \((\frac{2xt - t^2}{4})^{n}\) into a power series using the binomial expansion in multiple terms. This involves repeatedly expanding and collecting like terms for \(t\).

Step 6: Definition of Pn(x)

Finally, let \(P_n(x)\) represent the coefficients such that \(\sum_{n=0}^{\infty} P_n(x) t^{n}\) matches the power series. Therefore, \(P_n(x)\) is determined through the expansion process.

Key Concepts

Inverse Square Root Expansion

The concept of inverse square root expansion involves expanding expressions of the form \((1 - u)^{-1/2}\). This type of expansion is particularly useful in approximations and series solutions across various fields of mathematics and physics. It is essentially about expressing complex functions as an infinite sum of simpler terms.In our exercise, we want to express \((1 - 2xt + t^2)^{-1/2}\), which involves recognizing the expression \((1 - u)\) where \(u = 2xt - t^2\). Once this is set, we use the binomial series formula for expansion. This helps in breaking down what seems like a complex expression into manageable and understandable components.

Power Series

A power series is an infinite sum of terms in the form of \(a_n t^n\), where \(a_n\) represents the coefficients and \(t\) is a variable. Power series are like a Swiss army knife for mathematicians due to their ability to represent functions that are difficult to handle otherwise.In this context, we transform the expression \((\frac{2xt - t^2}{4})^{n}\) into a power series. By doing this, we create a series involving powers of \(t\) where \(t\) varies, and we handle each term individually. This is a step-by-step approach where each layer of \(n\) reveals a distinct piece of the larger puzzle that is our original expression.

Binomial Theorem

The binomial theorem is a powerful tool used to expand expressions raised to any power. It's most known from the well-loved binomial expansion: \[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k,\]which swiftly breaks down any such expression into simpler terms.For our exercise, the binomial theorem is adapted into the form \((1 - u)^{-1/2}\), producing the series:\[(1 - u)^{-rac{1}{2}} = \sum_{n=0}^{\infty} \binom{2n}{n} \left(\frac{u}{4}\right)^{n}.\]This facilitates the expansion process, allowing each term in the expansion to be related back to our original expression \((1-2xt+t^2)^{-1/2}\). This reliance on a clear mathematical framework is what helps to keep things structured and understandable.

Coefficient Determination

Determining the coefficients, \(P_n(x)\), in a power series is a key part of solving such problems. Once the power series is set up, each term has a coefficient which, when multiplied by the respective power of \(t\), gives part of the approximation of our function.In the exercise, \(P_n(x)\) are the coefficients derived from the binomial series expansion. Each \(P_n(x)\) corresponds to a specific power of \(t\) in our expanded series. To find these, you collect terms from the expanded form and match them with the original equation. This helps us to associate each coefficient with its respective term, thereby completing the expansion and ensuring that our solution accurately represents the original function.