Definite Integral to Infinity of Power of x over 1 + 2 x Cosine Beta + x Squared

Theorem

$\ds \int_0^\infty \dfrac {x^m \rd x} {1 + 2 x \cos \beta + x^2} = \frac \pi {\sin m x} \frac {\sin m \beta} {\sin \beta}$

Proof

This theorem requires a proof. In particular: write $x^2 + 2 x \cos \beta + 1 \equiv \paren {x + e^{i \beta} } \paren {x + e^{-i \beta} }$ then use partial fractions. Ugly so will do later. 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): $\S 15$: Definite Integrals involving Rational or Irrational expressions: $15.21$