Prove the nth Hermite polynomial has n real zeros

Thread starter Happiness
Start date Dec 15, 2015
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Polynomial

In summary, the nth Hermite polynomial is a mathematical function often used in physics and engineering. It is defined as Hn(x) = (-1)^n e^(x^2) (d^n/dx^n)(e^(-x^2)) and has n real zeros. This can be proven through mathematical induction or using the properties of Hermite polynomials and the fundamental theorem of algebra. The fact that it has n real zeros has various applications in quantum mechanics, signal processing, image processing, and data analysis.

Dec 15, 2015

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Dec 15, 2015

Avodyne

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http://math.stackexchange.com/questions/104845/the-roots-of-hermite-polynomials-are-all-real

1. What is the nth Hermite polynomial?

The nth Hermite polynomial is a type of mathematical function that is commonly used in the field of physics and engineering. It is named after the French mathematician Charles Hermite and is denoted by Hn(x).

2. How is the nth Hermite polynomial defined?

The nth Hermite polynomial can be defined as:
Hn(x) = (-1)^n e^(x^2) (d^n/dx^n)(e^(-x^2))

3. How many real zeros does the nth Hermite polynomial have?

The nth Hermite polynomial has n real zeros. This means that for every value of n, there are exactly n points on the x-axis where the polynomial crosses or touches the x-axis.

4. How can the fact that the nth Hermite polynomial has n real zeros be proven?

There are several ways to prove that the nth Hermite polynomial has n real zeros. One way is to use mathematical induction, where the statement is proved for a base case (usually n=1 or n=2) and then shown to be true for all subsequent cases. Another way is to use the properties of Hermite polynomials and the fundamental theorem of algebra.

5. What are some applications of the nth Hermite polynomial having n real zeros?

The property of the nth Hermite polynomial having n real zeros has various applications in physics and engineering. It is used to represent the wave functions of quantum harmonic oscillators and to solve differential equations in quantum mechanics. It is also used in signal processing and image processing, as well as in modeling financial data and analyzing data in statistics.