Integral to Infinity of Sine p x Cosine q x over x/Mistake

Source Work

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables

Chapter $15$: Definite Integrals

Definite Integrals involving Trigonometric Functions: $15.34$

2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.)

Chapter $18$: Definite Integrals

Definite Integrals involving Trigonometric Functions: $18.34$

Mistake

$\ds \int_0^\infty \frac {\sin p x \cos q x} x \rd x = \begin {cases} 0 & : p > q > 0 \\ \\ \dfrac \pi 2 & : 0 < p < q \\ \\ \dfrac \pi 4 & : p = q > 0 \end {cases}$

Correction

As demonstrated in Integral to Infinity of $\dfrac {\sin p x \cos q x} x$, this is incorrect.

It should be:

$\ds \int_0^\infty \frac {\sin p x \cos q x} x \rd x = \begin {cases} \dfrac \pi 2 & : p > q > 0 \\ \\ 0 & : 0 < p < q \\ \\ \dfrac \pi 4 & : p = q > 0 \\ \end {cases}$

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: $18.34$