Laplace Transform of Sine Integral Function/Proof 1
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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
Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
- $\map f 0 = 0$
and:
\(\ds \map {f'} t\) | \(=\) | \(\ds \dfrac {\sin t} t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds t \map {f'} t\) | \(=\) | \(\ds \sin t\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \laptrans {t \map {f'} t}\) | \(=\) | \(\ds \laptrans {\sin t}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {s^2 + 1}\) | Laplace Transform of Sine | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -\dfrac \d {\d s} \laptrans {\map {f'} t}\) | \(=\) | \(\ds \dfrac 1 {s^2 + 1}\) | Derivative of Laplace Transform | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {\dfrac \d {\d s} } {s \laptrans {\map f t} - \map f 0}\) | \(=\) | \(\ds -\dfrac 1 {s^2 + 1}\) | Laplace Transform of Derivative | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds s \laptrans {\map f t}\) | \(=\) | \(\ds -\int \dfrac 1 {s^2 + 1} \rd s\) | $\map f 0 = 0$, and integrating both sides with respect to $s$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds s \laptrans {\map f t}\) | \(=\) | \(\ds -\arctan s + C\) | Primitive of $\dfrac 1 {x^2 + a^2}$ |
By the Initial Value Theorem of Laplace Transform:
- $\ds \lim_{s \mathop \to \infty} s \laptrans {\map f t} = \lim_{t \mathop \to 0} \map f t = \map f 0 = 0$
which leads to:
- $c = \dfrac \pi 2$
Thus:
\(\ds s \laptrans {\map f t}\) | \(=\) | \(\ds \dfrac \pi 2 - \arctan s\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \arccot s\) | Sum of Arctangent and Arccotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \arctan \dfrac 1 s\) | Arctangent of Reciprocal equals Arccotangent | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \laptrans {\map f t}\) | \(=\) | \(\ds \dfrac 1 s \arctan \dfrac 1 s\) |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Sine, Cosine and Exponential Integrals: $36$: Method $1$