Integral to Infinity of Sine p x Cosine q x over x
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Theorem
- $\ds \int_0^\infty \frac {\sin p x \cos q x} x \rd x = \begin {cases} \dfrac \pi 2 & : p > q > 0 \\ \\ 0 & : 0 < p < q \\ \\ \dfrac \pi 4 & : p = q > 0 \end {cases}$
Proof
\(\ds \int_0^\infty \frac {\sin p x \cos q x} x \rd x\) | \(=\) | \(\ds \int_0^\infty \frac 1 2 \cdot \frac {\sin \paren {\paren {p + q} x} + \sin \paren {\paren {p - q} x} } x \rd x\) | Werner Formula for Sine by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int_0^\infty \frac {\sin \paren {\paren {p + q} x} } x \rd x + \frac 1 2 \int_0^\infty \frac {\sin \paren {\paren {p - q} x} } x \rd x\) | Linear Combination of Definite Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{t \mathop \to \infty} \frac 1 2 \int_0^t \frac {\sin \paren {\paren {p + q} x} } x \rd x + \lim_{t \mathop \to \infty} \frac 1 2 \int_0^t \frac {\sin \paren {\paren {p - q} x} } x \rd x\) | Definition of Improper Integral on Closed Interval Unbounded Above | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{t \mathop \to \infty} \frac 1 2 \int_0^{t \paren {p + q} } \frac {\sin u} u \rd u + \lim_{t \mathop \to \infty} \frac 1 2 \int_0^t \frac {\sin \paren {\paren {p - q} x} } x \rd x\) | Integration by Substitution: $u = \paren {p + q} x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{t \mathop \to \infty} \frac 1 2 \int_0^{t \paren {p + q} } \frac {\sin u} u \rd u + \lim_{t \mathop \to \infty} \frac 1 2 \int_0^{t \paren {p - q} } \frac {\sin v} v \rd v\) | Integration by Substitution: $v = \paren {p - q} x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \int_0^\infty \frac {\sin u} u \rd u + \frac 1 2 \lim_{t \mathop \to \infty} \int_0^{t \paren {p - q} } \frac {\sin v} v \rd v\) | Definition of Improper Integral on Closed Interval Unbounded Above: by hypothesis $p + q$ is (strictly) positive | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4 + \frac 1 2 \lim_{t \mathop \to \infty} \int_0^{t \paren {p - q} } \frac {\sin v} v \rd v\) | Dirichlet Integral |
Case $p > q > 0$:
\(\ds \) | \(\) | \(\ds \frac \pi 4 + \frac 1 2 \lim_{t \mathop \to \infty} \int_0^{t \paren {p - q} } \frac {\sin v} v \rd v\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4 + \frac 1 2 \int_0^\infty \frac {\sin v} v \rd v\) | Definition of Improper Integral on Closed Interval Unbounded Above: by hypothesis $p - q$ is (strictly) positive | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4 + \frac 1 2 \cdot \frac \pi 2\) | Dirichlet Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 2\) | simplifying |
$\Box$
Case $0 < p < q$:
\(\ds \) | \(\) | \(\ds \frac \pi 4 + \frac 1 2 \lim_{t \mathop \to \infty} \int_0^{t \paren {p - q} } \frac {\sin v} v \rd v\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4 + \frac 1 2 \int_0^{-\infty} \frac {\sin v} v \rd v\) | Definition of Improper Integral on Closed Interval Unbounded Below: by hypothesis $p - q$ is (strictly) negative | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4 - \frac 1 2 \int_0^\infty \frac {\sin w} w \rd w\) | Integration by Substitution: $w = -v$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4 - \frac 1 2 \cdot \frac \pi 2\) | Dirichlet Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | simplifying |
$\Box$
Case $p = q > 0$:
\(\ds \) | \(\) | \(\ds \frac \pi 4 + \frac 1 2 \lim_{t \mathop \to \infty} \int_0^{t \paren {p - q} } \frac {\sin v} v \rd v\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4 + \frac 1 2 \lim_{t \mathop \to \infty} \int_0^0 \frac {\sin v} v \rd v\) | $p - q = 0$ by hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4 + \frac 1 2 \cdot 0\) | Definite Integral on Zero Interval, independently of $t$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Trigonometric Functions: $15.34$
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 18$: Definite Integrals: Definite Integrals involving Trigonometric Functions: $18.34$