Orthogonality of Chebyshev Polynomials of the Second Kind/Equality

Theorem

Let $\map {U_n} x$ denote the Chebyshev polynomials of the second kind of order $n$.

$\ds \int_{-1}^1 \sqrt {1 - x^2} \, \paren {\map {U_n} x}^2 \rd x = \dfrac \pi 2$

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 30$: Chebyshev Polynomials: Orthogonality: $30.39$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 31$: Chebyshev Polynomials: Orthogonality: $31.39.$