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1. ### How to determine Tchebysheff polynomial general expression

Hello! Tchebysheff polynomials are often defined with trigonometric functions: T_m (x) = \begin{cases} \cos(m \arccos (x)) & -1 \le x \le 1\\ \mathrm{cosh} (m \mathrm{arccosh} (x)) & x > 1\\(-1)^m \mathrm{cosh} (m \mathrm{arccosh} |x|) & x < 1 \end{cases} But they are also polynomials, and...
2. ### Approximation in orthogonal functions with pulse shaping

Hello! I have tried for a whole afternoon to solve this problem but I didn't succeed. Let \cos(2 \pi (f_0 + i/T_N) t + \phi_i) and \cos(2 \pi (f_0 + j/T_N) t + \phi_j) be two quasi-orthogonal functions: \int_{0}^{T_N} \cos(2 \pi (f_0 + i/T_N) t + \phi_i) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt...
3. ### Variational expression: a demonstration

Oh, so it had been a misunderstanding. My professor presented the first expression, K = \displaystyle \frac{\left[\int_a^b E(x)e_n(x)dx\right]^2}{\left[\int_a^b E(x)e_1(x)dx\right]^2} as a variational, stationary expression. This is why I wrote the post. So, the K expression is not...
4. ### Variational expression: a demonstration

Even if in this textbook the calculation are quite different, the results that I should demonstrate are the same: the statement is: K «is stationary for small arbitrary variations in the electric field distribution about its correct value. It, therefore, follows that a first-order approximation...
5. ### Variational expression: a demonstration

With a more accurate example, I tried these realistic values for each integral: a = 1.9;\\ b=2.1;\\ w = 4 and I had to consider E_0(x) = \sqrt{(x - a)(b - x)} instead of E(x) to make this example (I need an expression for E_0(x)!). n = 3, 5 gave a difference bc - ad which is nonzero...
6. ### Variational expression: a demonstration

Sorry if I'm late (I didn't receive notifications by e-mail). I tried to do some numerical examples too, with functions very similar to your ones. In the particular problem, it should be e_n(x) = \sin(n \pi x/w)\\ e_1(x) = \sin(\pi x/w)\\ E(x) = \displaystyle \sqrt{(x - a)(b - x)} with...
7. ### Variational expression: a demonstration

First of all, thank you for your reply! Maybe you wanted to write: \begin{array}{l}a = \int_a^b E_0(x) e_n(x) \, dx \\ b = \int_a^b h(x) e_n(x) \, dx \\ c = \int_a^b E_0(x) e_1(x) \, dx \\ d = \int_a^b h(x) e_1(x) \, dx \end{array} Substituting: bc - ad = 0 \Rightarrow \int_a^b h(x) e_n(x)...
8. ### Variational expression: a demonstration

Homework Statement Hello! My problem is with a variational expression. I have a quantity, say L, which could be determined by the ratio: K = \displaystyle \frac{\left[\int_a^b E(x)e_n(x)dx\right]^2}{\left[\int_a^b E(x)e_1(x)dx\right]^2} Where e_1(x), e_n(x) are known functions and n...