Proving A_n(r) with Integer Coeff Polynomials: Q&A

  • Thread starter physicsRookie
  • Start date
In summary, the significance of proving A_n(r) with integer coefficient polynomials lies in its application in various mathematical fields and its ability to study the behavior of polynomials. A_n(r) is defined as a set of polynomials with degree n and integer coefficients, where the value of r is a root of the polynomial. It can be proved using various techniques such as mathematical induction and proof by contradiction. The use of integer coefficients is important as it allows for the generalization of results to all real numbers. While A_n(r) can also be proved using other types of polynomials, using integer coefficients is the most commonly used method due to its simplicity and wider applicability.
  • #1
physicsRookie
5
0
Define a sequence
[tex]A_n(r) = \int_{-1}^1(1-x^2)^n \cos(rx)\, dx, \qquad n \in \mathbb{N}, r \in \mathbb{R}.
[/tex]

Prove that
[tex]A_n(r) = \frac{n!}{r^{2n+1}}[P_n(r)\sin r - Q_n(r)\cos r] [/tex]

where [tex]P_n[/tex] and [tex]Q_n[/tex] are two polynomials with integer coefficients. What is the degree of [tex]P_n[/tex] and of [tex]Q_n[/tex]?


Can anyone help me? Thanks.
 
Physics news on Phys.org
  • #2
Induction. And an educated guess for the degrees, followed by induction.
 
  • #3


Sure, I can help you with this question. Let's break it down step by step.

First, let's recall the definition of the sequence A_n(r) given in the question. It is defined as the integral of the function (1-x^2)^n multiplied by the cosine of rx, over the interval [-1,1]. This sequence depends on two variables, n and r, and we can see that n must be a positive integer and r can be any real number.

Next, we want to prove that A_n(r) can be expressed as a polynomial with integer coefficients, multiplied by a sine and cosine term. In other words, we want to show that A_n(r) has the form:

A_n(r) = P(r)sin(r) - Q(r)cos(r)

where P(r) and Q(r) are polynomials with integer coefficients. To do this, we will use mathematical induction.

First, let's consider the base case where n = 1. In this case, we have:

A_1(r) = \int_{-1}^1(1-x^2)\cos(rx)\, dx

To evaluate this integral, we can use integration by parts with u = (1-x^2) and dv = cos(rx)dx. This gives us:

A_1(r) = (1-x^2)\frac{\sin(rx)}{r}\bigg|_{-1}^1 - \int_{-1}^1 \frac{-2x}{r}\sin(rx)\, dx

Simplifying this expression, we get:

A_1(r) = \frac{2\sin(r)}{r}

We can see that this is in the form we wanted, with P(r) = 0 and Q(r) = 2/r. Therefore, the base case is true.

Next, let's assume that the statement is true for some positive integer k. In other words, we assume that:

A_k(r) = P_k(r)\sin(r) - Q_k(r)\cos(r)

Now, we want to prove that this statement is also true for k+1. We have:

A_{k+1}(r) = \int_{-1}^1 (1-x^2)^{k+1}\cos(rx)\, dx

Using integration by parts again, we get:

A_{k+1}(r) = (1
 

1. What is the significance of proving A_n(r) with integer coefficient polynomials?

The significance of proving A_n(r) with integer coefficient polynomials lies in its application in various mathematical fields such as number theory, algebra, and geometry. It allows for the study and understanding of the behavior of polynomials with integer coefficients, which have important implications in problem-solving and proof techniques.

2. What is the definition of A_n(r) with integer coefficient polynomials?

A_n(r) with integer coefficient polynomials is a set of polynomials with degree n and integer coefficients, where the value of r is a root of the polynomial. In other words, when the polynomial is evaluated at r, the result is equal to 0.

3. How is A_n(r) with integer coefficient polynomials proved?

A_n(r) with integer coefficient polynomials can be proved using various techniques such as mathematical induction, proof by contradiction, and proof by construction. The specific method used depends on the problem at hand and the preference of the mathematician.

4. What is the significance of using integer coefficients in proving A_n(r)?

The use of integer coefficients in proving A_n(r) is significant because it allows for the generalization of the results to all real numbers. This is because any real number can be expressed as a combination of integers and rational numbers, and by using integer coefficients, the proof becomes applicable to all real numbers.

5. Can A_n(r) with integer coefficient polynomials be proved using other types of polynomials?

Yes, A_n(r) can also be proved using other types of polynomials, such as rational coefficients or complex coefficients. However, using integer coefficients is the most commonly used method due to its simplicity and applicability to a wider range of problems.

Similar threads

Replies
1
Views
1K
  • Calculus
Replies
4
Views
1K
Replies
4
Views
188
Replies
9
Views
847
Replies
1
Views
734
  • Differential Equations
Replies
1
Views
1K
Replies
5
Views
270
  • Calculus and Beyond Homework Help
Replies
3
Views
503
Replies
3
Views
1K
Back
Top