THEOREM 1 (The Residue Theorem) Suppose that $f$ is analytic on a simply-connected domain $D$ except for a finite number of isolated singularities at points $z_1,z_2,\ldots, z_N$ of $D$. Let $\gamma$
be a piecewise smooth positively oriented simple closed curve in $D$ that does not pass through any of the points $z_1,z_2,\ldots, z_N$. Then
$$
\int_\gamma f(z) dz = 2\pi i \sum_{z_k \text{ inside }\gamma} \Res(f; z_k),
$$
where the sum is taken over all those singularities $z_k$ of $f$ that lie inside $\gamma$.

THEOREM 1A (The Residue Theorem) Suppose that $f$ is analytic on a simply-connected external domain $D$ except for a finite number of isolated singularities at points $z_1,z_2,\ldots, z_{N-1}, z_N=\infty$ of $D$. Let $\gamma$
be a piecewise smooth negatively oriented simple closed curve in $D$ that does not pass through any of the points $z_1,z_2,\ldots, z_N$. Then
$$
\int_\gamma f(z) dz = 2\pi i \sum_{z_k \text{ outside }\gamma} \Res(f; z_k),
$$
where the sum is taken over all those singularities $z_k$ of $f$ that lie outside $\gamma$.

THEOREM
If $f$ has only a finite number of singularities $z_1,z_2,\ldots, z_{N-1}, z_N=\infty$ on $\mathbb{C}$ , then
$$
\sum_{k=1,\ldots, N} \Res(f; z_k)=0.
$$

PROPOSITION Suppose $P$ and $Q$ are polynomials that are real-valued on the real axis of degrees $m$ and $n$ respectively $n\ge m+2$. Suppose $Q(x)\ne 0$ for all $x\in \mathbb{R}$. Then
\begin{align*}
\int_{-\infty}^\infty \frac{P(x)}{Q(x)}dx =&
2\pi i \sum_{z_j\in U } \Res (\frac{P}{Q}; z_j)=\\
-&2\pi i \sum_{z_k\in V } \Res (\frac{P}{Q}; z_k),
\end{align*}
with the sum taken over all poles of $P/Q$ that lie in the upper half-plane $U=\{z\colon \Im z>0\}$ or in the lower half-plane $V=\{z\colon \Im z<0\}$.

Content

Integrals of Rational Functions

Integrals over the Real Axis Involving Trigonometric Functions

Integrals of Trigonometric Functions over $[0, 2\pi]$

Integrals Involving $\log x$ or Fractional Powers of $x$