Question

Verify that the eigenfunctions $$ 1, \cos \frac{\pi x}{L}, \cos \frac{2 \pi x}{L}, \ldots, \cos \frac{n \pi x}{L}, \ldots $$ of Problem 2 are orthogonal on \([0, L]\).

Short answer

Question: Verify that the given eigenfunctions are orthogonal on the interval \([0, L]\). Answer: The inner product of any two different eigenfunctions is zero, which verifies that the given eigenfunctions are orthogonal on the interval \([0, L]\).

Step-by-step solution

Define the inner product for the given eigenfunctions

For the given eigenfunctions, we can define the inner product for any two eigenfunctions, say, \(\phi_m(x)\) and \(\phi_n(x)\) as the integral of their product on the interval \([0, L]\): $$ \langle\phi_m, \phi_n\rangle = \int_{0}^{L} \phi_m(x) \phi_n(x) dx $$ To show that the eigenfunctions are orthogonal, we need to prove that this inner product is zero for \(m \neq n\).

Write down the general expression for the eigenfunctions

The general expression for the given eigenfunctions is: $$ \phi_n(x) = \cos{\frac{n\pi x}{L}} $$ where \(n\) is a positive integer. Now let's find the inner product of any two different eigenfunctions \(\phi_m(x)\) and \(\phi_n(x)\) where \(m \neq n\).

Calculate the inner product of two different eigenfunctions

To find the inner product \(\langle \phi_m, \phi_n \rangle\), we need to integrate the product of \(\phi_m(x)\) and \(\phi_n(x)\) over the interval \([0, L]\): $$ \langle \phi_m, \phi_n \rangle = \int_{0}^{L} \cos{\frac{m\pi x}{L}} \cos{\frac{n\pi x}{L}} dx $$

Use the product-to-sum trigonometric identity

Using the product-to-sum trigonometric identity, we can rewrite the product of the two cosine functions inside the integral: $$ \cos A \cos B = \frac{1}{2}\left(\cos (A - B) + \cos (A + B)\right) $$ Applying this identity to our integral, we get: $$ \langle \phi_m, \phi_n \rangle = \frac{1}{2} \int_{0}^{L} \left(\cos{\left(\frac{(m-n)\pi x}{L}\right)} + \cos{\left(\frac{(m+n)\pi x}{L}\right)}\right) dx $$

Integrate the two cosine terms

Now, let's integrate the two cosine terms separately: $$ \int_{0}^{L} \cos{\left(\frac{(m-n)\pi x}{L}\right)} dx = \left[\frac{L}{(m-n)\pi} \sin{\left(\frac{(m-n)\pi x}{L}\right)}\right]_{0}^{L} $$ and $$ \int_{0}^{L} \cos{\left(\frac{(m+n)\pi x}{L}\right)} dx = \left[\frac{L}{(m+n)\pi} \sin{\left(\frac{(m+n)\pi x}{L}\right)}\right]_{0}^{L} $$

Evaluate the integrals at the limits

Evaluating the two integrals at the limits \(x=0\) and \(x=L\), we get: $$ \left[\frac{L}{(m-n)\pi} \sin{\left(\frac{(m-n)\pi x}{L}\right)}\right]_{0}^{L} = \frac{L}{(m-n)\pi} \sin{(m-n)\pi} - \frac{L}{(m-n)\pi} \sin{0} = 0 $$ and $$ \left[\frac{L}{(m+n)\pi} \sin{\left(\frac{(m+n)\pi x}{L}\right)}\right]_{0}^{L} = \frac{L}{(m+n)\pi} \sin{(m+n)\pi} - \frac{L}{(m+n)\pi} \sin{0} = 0 $$ Since both integrals result in zero, the inner product is also zero: $$ \langle \phi_m, \phi_n \rangle = 0 $$

Conclusion

We have shown that the inner product of any two different eigenfunctions is zero, which verifies that the given eigenfunctions are orthogonal on the interval \([0, L]\).

Key concepts

Inner Product

The inner product in the context of functions is a vital concept that allows us to compare two functions and measure their 'similarity' in terms of integration over a specific interval. For functions \( \( \phi_m(x) \) \)) and \( \( \phi_n(x) \) \) that belong to a space of square-integrable functions, their inner product over an interval \( [0, L] \) is given by the integral of their product across this interval:
\[\langle\phi_m, \phi_n\rangle = \int_{0}^{L} \phi_m(x) \phi_n(x) dx\]
This operation acts similarly to the dot product for vectors but catered for continuous functions. It's a crucial step in evaluating the orthogonality of eigenfunctions, which has profound implications in various fields such as quantum mechanics and signal processing.

Trigonometric Identities

Understanding trigonometric identities aids in simplifying complex trigonometric expressions. In our context, the product-to-sum identities allow us to transform the product of two cosine terms into a sum of two distinct cosine terms:
\[\cos A \cos B = \frac{1}{2}\left(\cos (A - B) + \cos (A + B)\right)\]
By using these identities, we can break down the integral of the product of two cosines into simpler components that can be more easily integrated. This step is crucial when proving the orthogonality of eigenfunctions, as it leads to simpler expressions whose integrals can determine whether the functions are orthogonal.

Eigenfunction Orthogonality

Eigenfunction orthogonality is a central concept in mathematical physics and engineering. It conveys that certain functions, when multiplied together and integrated over a certain interval, give a result of zero unless they are the same function. This property is analogous to perpendicular vectors having a dot product of zero. For eigenfunctions \( \( \phi_m(x) \) \)and \( \( \phi_n(x) \) \), the condition for orthogonality over an interval \( [0, L] \) is given by:
\[\langle \phi_m, \phi_n \rangle = 0\]\
The implication of this property is profound, as it allows the formation of basis sets in function spaces, making it easier to solve differential equations and expand functions into series, much like Fourier series.

Integral Calculus

In the realm of integral calculus, we focus on finding the antiderivatives of functions and computing the area beneath their curves within certain limits. This process is exemplified when integrating trigonometric functions. For instance, evaluating the integrals arising from orthogonality checks on eigenfunctions involves calculating the definite integral of cosine functions, such as:
\[\int_{0}^{L} \cos{\left(\frac{(m-n)\pi x}{L}\right)} dx\]
In practice, the evaluation of these integrals at specific limits, often using trigonometric identities and fundamental theorems of calculus, can determine if the integrals vanish or not, which in turn reveals whether the functions being tested are orthogonal. This application is not only a cornerstone in the study of mathematical function properties but is also underlying many practical techniques in science and engineering simulations.