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Chapter 3: Problem 8

Let \(T_{n}(x)\) denote the degree \(n\) Chebyshev polynomial. Find a formula for \(T_{n}(0)\).

Short Answer

Expert verified
Answer: The formula for the Chebyshev polynomial of degree \(n\) evaluated at \(0\) is given by $T_n(0) = \begin{cases} (-1)^{n/2}, & \text{if n is even}\\ 0, & \text{if n is odd} \end{cases}$.

Step by step solution

01

Review Chebyshev Polynomial Recursion Formula

The Chebyshev polynomials satisfy the following recursion formula: \(T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)\), with \(T_0(x) = 1\) and \(T_1(x) = x\).
02

Find \(T_n(0)\) for \(n=0,1,2,3\)

Using the initial conditions and the recursion formula, we can find the first few Chebyshev polynomials evaluated at \(0\). \(T_0(0) = 1\) \(T_1(0) = 0\) For \(n = 2\), using the recursion formula, \(T_{2}(0) = 2\cdot 0\cdot T_{1}(0) - T_0(0) = -1\) For \(n = 3\), using the recursion formula, \(T_{3}(0) = 2\cdot 0\cdot T_{2}(0) - T_1(0) = 0\)
03

Observe a Pattern

From the first few values of \(T_n(0)\), we can observe a pattern: - For even \(n\), \(T_n(0) = (-1)^{n/2}\). - For odd \(n\), \(T_n(0) = 0\).
04

Write the General Formula for \(T_n(0)\)

Based on the pattern observed in Step 3, we can provide the formula for \(T_n(0)\): $T_n(0) = \begin{cases} (-1)^{n/2}, & \text{if n is even}\\ 0, & \text{if n is odd} \end{cases}$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Recursion Formula
The recursion formula is a fundamental concept in mathematics, particularly in polynomial evaluation and sequence determination. It enables us to define complex objects in terms of simpler ones. For the Chebyshev polynomials, the recursion formula is particularly elegant:
For each integer value of n greater than 1, Chebyshev's polynomials follow the relationship
\[\begin{equation} T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x) \end{equation}\]
Polynomial Evaluation
Polynomial evaluation is the process of computing the value of a polynomial for a given value of its variable. It's critical in various areas of mathematics, including algebra, calculus, and numerical analysis. When evaluating the Chebyshev polynomial at zero, you apply the recursion formula repeatedly to find the value at each step, reducing higher-degree polynomials to more manageable terms. For example,
\[\begin{equation} T_2(0) = 2 \times 0 \times T_1(0) - T_0(0) = -1 \end{equation}\]
This allows for the calculation of the polynomial's value at any integer step without having to solve the entire polynomial from scratch.
Pattern Recognition in Mathematics
Pattern recognition plays a pivotal role in mathematical analysis and problem-solving. It is the identification of particular patterns within a sequence or a collection of objects. Once a pattern is recognized, it can greatly simplify problem-solving and predictions for subsequent terms. For Chebyshev polynomials evaluated at zero, a simple pattern emerges which states that for even n, the value is
\[\begin{equation} (-1)^{n/2} \end{equation}\]
and for odd n it is zero. Recognizing this allows mathematicians to bypass computationally heavy recursion entirely for large values of n.
Theoretical Numerical Analysis
Theoretical numerical analysis provides mathematicians and scientists with frameworks and tools to approximate solutions to complex problems that are difficult to solve analytically. It involves various algorithms and error analysis to assure the quality and reliability of the results. In the context of Chebyshev polynomials, numerical analysis methods would apply the recursion formula to compute polynomial values efficiently, offering powerful means to handle vast computations systematically and understanding the underlying behavior of polynomials as functions.

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Most popular questions from this chapter

(a) Given the data points \((1,0),(2, \ln 2),(4, \ln 4)\), find the degree 2 interpolating polynomial. (b) Use the result of (a) to approximate \(\ln 3\). (c) Use Theorem \(3.3\) to give an error bound for the approximation in part (b). (d) Compare the actual error to your error bound.

Describe the character drawn by the following three-piece Bezier curve: \((0,1)(0,1)(0,0)(0,0)\) \((0,0)(0,1)(1,1)(1,0)\) \((1,0)(1,1)(2,1)(2,0)\)

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