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...
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...
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...
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...
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...
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...
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)...
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...