Laplace Transform of Sine
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Theorem
Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
- $\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$.
Proof 1
\(\ds \map {\laptrans {\sin {a t} } } s\) | \(=\) | \(\ds \int_0^{\to +\infty} e^{-s t} \sin {a t} \rd t\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to \infty} \int_0^L e^{-s t} \sin {a t} \rd t\) | Definition of Improper Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to \infty} \intlimits {\frac {e^{-s t} \paren {-s \sin a t - a \cos a t} } {\paren {-s}^2 + a^2} } 0 L\) | Primitive of $e^{a x} \sin b x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to \infty} \paren {\dfrac {e^{-s L} \paren {-s \sin a L - a \cos a L} } {s^2 + a^2} - \dfrac {e^{-s \times 0} \paren {-s \, \map \sin {0 \times a} - a \, \map \cos {0 \times a} } } {s^2 + a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to \infty} \paren {\dfrac {s \sin 0 + a \cos 0} {s^2 + a^2} - \dfrac {e^{-s L} \paren {s \sin a L + a \cos a L} } {s^2 + a^2} }\) | Exponential of Zero and rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {s \sin 0 + a \cos 0} {s^2 + a^2} - 0\) | Exponential Tends to Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {s^2 + a^2}\) | Sine of Zero is Zero, Cosine of Zero is One |
$\blacksquare$
Proof 2
\(\ds \laptrans {e^{i a t} }\) | \(=\) | \(\ds \frac 1 {s - i a}\) | Laplace Transform of Exponential | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {s + i a} {s^2 + a^2}\) | multiplying top and bottom by $s + i a$ |
Also:
\(\ds \laptrans {e^{i a t} }\) | \(=\) | \(\ds \laptrans {\cos a t + i \sin a t}\) | Euler's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \laptrans {\cos a t} + i \laptrans {\sin a t}\) | Linear Combination of Laplace Transforms |
So:
\(\ds \laptrans {\sin a t}\) | \(=\) | \(\ds \map \Im {\laptrans {e^{i a t} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {s + i a} {s^2 + a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {s^2 + a^2}\) |
$\blacksquare$
Proof 3
\(\ds \laptrans {\sin a t}\) | \(=\) | \(\ds \laptrans {\frac {e^{i a t} - e^{-i a t} } {2 i} }\) | Euler's Sine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 i} \paren {\laptrans {e^{i a t} } - \laptrans {e^{-i a t} } }\) | Linear Combination of Laplace Transforms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 i} \paren {\frac 1 {s - i a} - \frac 1 {s + i a} }\) | Laplace Transform of Exponential | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 i} \paren {\frac {s + i a - s + i a} {s^2 + a^2} }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 i} \paren {\frac {2 i a} {s^2 + a^2} }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {s^2 + a^2}\) | simplifying |
$\blacksquare$
Proof 4
By definition of the Laplace transform:
- $\ds \laptrans {\sin a t} = \int_0^{\to +\infty} e^{-s t} \sin a t \rd t$
From Integration by Parts:
- $\ds \int f g' \rd t = f g - \int f' g \rd t$
Here:
\(\ds f\) | \(=\) | \(\ds \sin a t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds f'\) | \(=\) | \(\ds a \cos a t\) | Derivative of $\sin a x$ | ||||||||||
\(\ds g'\) | \(=\) | \(\ds e^{-s t}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds g\) | \(=\) | \(\ds -\frac 1 s e^{-s t}\) | Primitive of $e^{a x}$ |
So:
\(\text {(1)}: \quad\) | \(\ds \int e^{-s t} \sin a t \rd t\) | \(=\) | \(\ds -\frac 1 s e^{-s t} \sin a t + \frac a s \int e^{-s t} \cos a t \rd t\) |
Consider:
- $\ds \int e^{-s t} \cos a t \rd t$
Again, using Integration by Parts:
- $\ds \int h j\,' \rd t = h j - \int h' j \rd t$
Here:
\(\ds h\) | \(=\) | \(\ds \cos a t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds h'\) | \(=\) | \(\ds -a \sin a t\) | Derivative of Cosine Function | ||||||||||
\(\ds j\,'\) | \(=\) | \(\ds e^{-s t}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds j\) | \(=\) | \(\ds -\frac 1 s e^{-s t}\) | Primitive of Exponential Function |
So:
\(\ds \int e^{-s t} \cos a t \rd t\) | \(=\) | \(\ds -\frac 1 s e^{-st} \cos a t - \frac a s \int e^{-s t} \sin a t \rd t\) |
Substituting this into $(1)$:
\(\ds \int e^{-s t} \sin a t \rd t\) | \(=\) | \(\ds -\frac 1 s e^{-s t} \sin a t + \frac a s \paren {-\frac 1 s e^{-s t} \cos a t - \frac a s \int e^{-s t} \sin a t \rd t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 s e^{-s t} \sin a t - \frac a {s^2} e^{-s t} \cos a t - \frac {a^2} {s^2} \int e^{-s t} \sin a t \rd t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {1 + \frac {a^2} {s^2} } \int e^{-s t} \sin a t \rd t\) | \(=\) | \(\ds -e^{-s t} \paren {\frac 1 s \sin a t + \frac a {s^2} \cos a t}\) |
Evaluating at $t = 0$ and $t \to +\infty$:
\(\ds \paren {1 + \frac {a^2} {s^2} } \laptrans {\sin a t}\) | \(=\) | \(\ds \intlimits {-e^{-s t} \paren {\frac 1 s \sin a t + \frac a {s^2} \cos a t} } {t \mathop = 0} {t \mathop \to +\infty}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0 - \paren {-1 \paren {\frac 1 s \times 0 + \frac a {s^2} \times 1} }\) | Boundedness of Real Sine and Cosine, Complex Exponential Tends to Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {s^2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \laptrans {\sin a t}\) | \(=\) | \(\ds \frac a {s^2} \paren {1 + \frac {a^2} {s^2} }^{-1}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {s^2} \paren {\frac {s^2} {a^2 + s^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac a {s^2 + a^2}\) |
$\blacksquare$
Proof 5
From Laplace Transform of Second Derivative:
- $(1): \quad \laptrans {\map {f} t} = s^2 \laptrans {\map f t} - s \, \map f 0 - \map {f'} 0$
under suitable conditions.
Then:
\(\ds \map f t\) | \(=\) | \(\ds \sin a t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {f'} t\) | \(=\) | \(\ds a \cos a t\) | |||||||||||
\(\ds \map {f} t\) | \(=\) | \(\ds -a^2 \sin a t\) | ||||||||||||
\(\ds \map f 0\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \map {f'} 0\) | \(=\) | \(\ds a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \laptrans {-a^2 \sin a t}\) | \(=\) | \(\ds s^2 \laptrans {\sin a t} - s \times 0 - a\) | from $(1)$, substituting for $\map f t$, $\map {f'} 0$ and $\map f 0$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds -a^2 \laptrans {\sin a t}\) | \(=\) | \(\ds s^2 \laptrans {\sin a t} - a\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \laptrans {\sin a t}\) | \(=\) | \(\ds \dfrac a {s^2 + a^2}\) | rearranging |
$\blacksquare$
Also see
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Laplace Transforms of some Elementary Functions: $5$.
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.32$