Laplace Transform of Cosine
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Theorem
Let $\cos$ be the real cosine function.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
- $\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$.
Proof 1
\(\ds \map {\laptrans {\cos {a t} } } s\) | \(=\) | \(\ds \int_0^{\to +\infty} e^{-s t} \cos {a t} \rd t\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to \infty} \int_0^L e^{-s t} \cos {a t} \rd t\) | Definition of Improper Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to \infty} \intlimits {\frac {e^{-s t} \paren {-s \cos a t + a \sin a t} } {\paren {-s}^2 + a^2} } 0 L\) | Primitive of $e^{a x} \cos b x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to \infty} \paren {\frac {e^{-s L} \paren {-s \cos a L + a \sin a L} } {s^2 + a^2} - \frac {e^{-s \times 0} \paren {-s \, \map \cos {0 \times a} + a \, \map \sin {0 \times a} } } {s^2 + a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{L \mathop \to \infty} \paren {\frac {s \, \map \cos {0 \times a} - a \, \map \sin {0 \times a} } {s^2 + a^2} - \frac {e^{-s L} \paren {-s \cos a L + a \sin a L} } {s^2 + a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {s \, \map \cos {0 \times a} - a \, \map \sin {0 \times a} } {s^2 + a^2} - 0\) | Exponential Tends to Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {s \cos 0 - a \sin 0} {s^2 + a^2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac s {s^2 + a^2}\) | 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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {s + i a} {s^2 + a^2}\) | multiply 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:
\(\ds \laptrans {\cos a t}\) | \(=\) | \(\ds \map \Re {\laptrans {e^{i a t} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\frac {s + i a} {s^2 + a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac s {s^2 + a^2}\) |
$\blacksquare$
Proof 3
\(\ds \laptrans {\cos a t}\) | \(=\) | \(\ds \laptrans {\frac {e^{i a t} + e^{-i a t} } 2}\) | Euler's Cosine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\laptrans {e^{i a t} } + \laptrans {e^{-i a t} } }\) | Linear Combination of Laplace Transforms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac 1 {s - i a} + \frac 1 {s + i a} }\) | Laplace Transform of Exponential | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac {s + i a + s - i a} {s^2 + a^2} }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac s {s^2 + a^2}\) | simplifying |
$\blacksquare$
Proof 4
By definition of the Laplace Transform:
- $\ds \laptrans {\cos at} = \int_0^{\to +\infty} e^{-s t} \cos at \rd t$
From Integration by Parts:
- $\ds \int f g' \rd t = f g - \int f'g \rd t$
Here:
\(\ds f\) | \(=\) | \(\ds \cos at\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds f'\) | \(=\) | \(\ds -a \sin a t\) | Derivative of Cosine Function | ||||||||||
\(\ds g'\) | \(=\) | \(\ds e^{-s t}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds g\) | \(=\) | \(\ds -\frac 1 s e^{-s t}\) | Primitive of Exponential Function |
So:
\(\text {(1)}: \quad\) | \(\ds \int e^{-s t} \cos a t \rd t\) | \(=\) | \(\ds -\frac 1 s e^{-s t} \cos a t - \frac a s \int e^{-s t} \sin a t \rd t\) |
Consider:
- $\ds \int e^{-s t} \sin a t \rd t$
Again, using Integration by Parts:
- $\ds \int h j \,' \rd t = h j - \int h'j \rd t$
Here:
\(\ds h\) | \(=\) | \(\ds \sin at\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds h'\) | \(=\) | \(\ds a \cos at\) | Derivative of Sine Function | ||||||||||
\(\ds j\,'\) | \(=\) | \(\ds e^{-s t}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds j\) | \(=\) | \(\ds -\frac 1 s e^{-s t}\) | Primitive of Exponential Function |
So:
\(\ds \int e^{-s t} \sin a t \rd t\) | \(=\) | \(\ds -\frac 1 s e^{-s t} \sin a t + \frac a s \int e^{-s t} \cos a t \rd t\) |
Substituting this into $(1)$:
\(\ds \int e^{-s t} \cos a t \rd t\) | \(=\) | \(\ds -\frac 1 s e^{-s t} \cos a t - \frac a s \paren {-\frac 1 s e^{-s t} \sin a t + \frac a s \int e^{-s t} \cos a t \rd t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 s e^{-s t} \cos a t + \frac a {s^2} e^{-s t} \sin a - \frac {a^2} {s^2} \int e^{-s t} \cos a t \rd t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {1 + \frac {a^2} {s^2} } \int e^{-s t} \cos a t \rd t\) | \(=\) | \(\ds -\frac 1 s e^{-s t} \cos a t + \frac a {s^2} e^{-s t} \sin a t\) |
Evaluating at $t = 0$ and $t \to +\infty$:
\(\ds \paren {1 + \frac {a^2} {s^2} } \laptrans {\cos at}\) | \(=\) | \(\ds \intlimits {-e^{-s t} \paren {\frac 1 s \cos a t - \frac a {s^2} \sin a t} } {t \mathop = 0} {t \mathop \to +\infty}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0 - \paren {-1 \paren {\frac 1 s \times 1 + \frac a {s^2} \times 0} }\) | Boundedness of Real Sine and Cosine, Complex Exponential Tends to Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 s\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \laptrans {\cos at}\) | \(=\) | \(\ds \frac 1 s \paren {1 + \frac {a^2} {s^2} }^{-1}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 s \paren {\frac {s^2} {a^2 + s^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac s {s^2 + a^2}\) |
$\blacksquare$
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: $6$.
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.33$