All Questions
Tagged with chebyshev-polynomials calculus
8
questions
2
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3
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127
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Finding the Chebyshev polynomials $T_n$ by elementary means
Suppose that one person called the Student—virtually, an advanced schoolchild—obtained a tip that the Chebyshev polynomials of the first kind exist and unique for each $n$. By the Chebyshev polynomial ...
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How to prove by induction all Chebyshev polynomials of the first kind $T_n$ when $n\geq 1$ have a positive leading coefficient??
Use the first kind Chebyshev polynomial $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$ to show how the leading coefficient is always positive $(1, 2, 4, 8, 16, 32...)$ when $n\geq 1$ using proof by induction
$$...
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Finding Chebychev coefficients
To find Chebyshev coefficients I need to compute the following polynomial with $T_k(x)$ being the Chebyshev polynomial. We use that $T_k(\cos(z))=\cos(kz)$
$$
2\int_0^1 T_k(x)\frac{1}{\sqrt{1-x^2}} dx=...
- 31
1
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1
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181
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Is this integral evaluation legitimate?
I would like to evaluate the following integral:
$$\int_{-1}^{1}\frac{1}{\sqrt{1-x^{2}}}\cos (2\arccos (x))\cos (3\arccos (x)){\mathrm{d} x}$$
Some experience from taking a Numerical Methods course ...
- 436
3
votes
1
answer
996
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On the extrema of Chebyshev polynomials of the second kind
I wish to prove that the magnitude of extreme values of $U_n(x)$, the Chebyshev polynomial of the second kind, is monotonically increasing on $[-1,1]$. By symmetry it suffices to prove it over $[0,1]$....
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2
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1
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2k
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Trigonometric Function Simplification: $T_2 (x) = \cos (2 \arccos x)$
Let $T_n (x) = \cos (n \arccos x)$ where $x$ is a real number, $x \in [–1, 1]$ and $n$ is a positive integer.
Show that $$T_2 (x) = 2x^2 – 1.$$
My attempt:
$T_2 (x) = \cos (2 \arccos x)$
Because ...
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1
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1
answer
44
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Integral calculation/transformation $\varphi_n= 2x\varphi_n$
Please help with below integral transformation.
\begin{align*}
T_n(x) & = \cos(n\cos^{-1} x)\\
w(t) & = (1 - x^2)^{-\frac{1}{2}}\\
\varphi_n(x) & = \int_{-1}^{1} \frac{T_n(t) - T_n(x)}{t -...
- 429
3
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0
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Definite integral including the Chebyshev polynomial
I would like to know the proof of
$$ \int_a^b \frac{T_n(x/a)T_n(x/b)\, dx}{x(b^2-x^2)^{1/2}(x^2-a^2)^{1/2}}=\frac{\pi}{2 ab}, 0<a<b, n \in \Bbb N $$
where $T_n(x)$ is the Chebyshev polynomial of ...
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