Are these functions orthonormal?

Question

Are the following set of functions orthonormal over the interval $0$ to $1$? $$Y_r(x) = \sin{\beta_r x}-\sinh{\beta_r x}-\frac{\sin\beta_r-\sinh\beta_r}{\cos\beta_r-\cosh\beta}\left(\cos\beta_r x-\cosh\beta_r x\right)$$ where $\beta_r$ are the positive solutions to:
$$1-\cos\beta_r\cosh\beta_r=0$$ I know that the functions $Y_r(x)$ are orthogonal. I went to check if they were orthonormal by evaluating $\int_0^1 Y_r Y_r \, dx$ numerical and checking if the integral equaled to $1$. For $\beta_r = 4.7300$ I get that the integral is equal to $1.03593$. For Larger values of $\beta_r$ the integral approaches $1$ so that makes me think that the set of functions is orthonormal and I am just encountering numerical error. Any thoughts?

Answer 1

I agree with your computations; these functions are not normalized. In fact, Mathematica is able to compute the exact value of of the the squared norm of the first function to be

$$\int_0^1 Y_1^2(1,x) = \frac{(\sin (\beta_1 )-\sinh (\beta_1 ))^2}{(\cos (\beta_1 )-\cosh (\beta_1))^2}.$$

Since $\beta_1 \approx 4.73$, we get about $1.0359$ for the integral.

I assume this computation is coming out of a Sturm-Liouville problem and I don't find this type of behavior to be at all unusual in that context.