Laplace Transform of Sine Integral Function/Proof 2

Theorem

$\laptrans {\map \Si t} = \dfrac 1 s \arctan \dfrac 1 s$

where:

$\laptrans f$ denotes the Laplace transform of the function $f$

$\Si$ denotes the sine integral function

Proof

\(\ds \laptrans {\dfrac {\sin t} t}\)

\(=\)

\(\ds \arctan \dfrac 1 s\)

Laplace Transform of $\dfrac {\sin t} t$

\(\ds \leadsto \ \ \)

\(\ds \laptrans {\int_{u \mathop \to 0}^{u \mathop = t} \frac {\sin u} u \rd u}\)

\(=\)

\(\ds \dfrac 1 s \arctan \dfrac 1 s\)

Laplace Transform of Integral

\(\ds \leadsto \ \ \)

\(\ds \laptrans {\map \Si t}\)

\(=\)

\(\ds \dfrac 1 s \arctan \dfrac 1 s\)

Definition of Sine Integral

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transform of Integrals: $18$