Laplace Transform of t cosine a t

Theorem

$\map {\laptrans {t \cos a t} } s = \dfrac {s^2 - a^2} {\paren {s^2 + a^2}^2}$

where:

$s$ is a complex number with $\map \Re s > a$

$\laptrans f$ denotes the Laplace transform of $f$ evaluated at $s$.

$\ds \map {\laptrans {t \cos a t} } s$

$=$

$\ds \map {\dfrac \d {\d s} } {\map {\laptrans {-\cos a t} } s}$

$\ds $

$=$

$\ds -\map {\dfrac \d {\d s} } {\dfrac s {s^2 + a^2} }$

$\ds $

$=$

$\ds -\dfrac {\paren {s^2 + a^2} \dfrac \d {\d s} s - s \map {\dfrac \d {\d s} } {s^2 + a^2} } {\paren {s^2 + a^2}^2}$

$\ds $

$=$

$\ds -\dfrac {s^2 + a^2 - 2 s^2} {\paren {s^2 + a^2}^2}$

$\ds $

$=$

$\ds \dfrac {s^2 - a^2} {\paren {s^2 + a^2}^2}$

simplifying

$\blacksquare$

1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text B$: Table of Special Laplace Transforms: $22.$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.46$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 33$: Laplace Transforms: Table of Special Laplace Transforms: $33.46.$