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

\(\displaystyle \map {\laptrans {\cos {a t} } } s\) \(=\) \(\displaystyle \int_0^{\to +\infty} e^{-s t} \cos {a t} \rd t\) $\quad$ Definition of Laplace Transform $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \intlimits {\frac {e^{-s t} \paren {-s \cos a t + a \sin a t} } {\paren {-s}^2 + a^2} } {t \mathop = 0} {t \mathop \to +\infty}\) $\quad$ Primitive of $e^{a x} \cos b x$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 0 - \frac {-s \, \map \cos {0 \times a} + a \, \map \sin {0 \times a} } {s^2 + a^2}\) $\quad$ Exponential Tends to Zero, Exponential of Zero $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {s \cos 0 - a \sin 0} {s^2 + a^2}\) $\quad$ simplifying $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac s {s^2 + a^2}\) $\quad$ Sine of Zero is Zero, Cosine of Zero is One $\quad$

$\blacksquare$


Proof 2

\(\displaystyle \laptrans {e^{i a t} }\) \(=\) \(\displaystyle \frac 1 {s - i a}\) $\quad$ Laplace Transform of Exponential $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {s + i a} {s^2 + a^2}\) $\quad$ multiply top and bottom by $s + i a$ $\quad$

Also:

\(\displaystyle \laptrans {e^{i a t} }\) \(=\) \(\displaystyle \laptrans {\cos a t + i \sin a t}\) $\quad$ Euler's Formula $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \laptrans {\cos a t} + i \laptrans {\sin a t}\) $\quad$ Linear Combination of Laplace Transforms $\quad$

So:

\(\displaystyle \laptrans {\cos a t}\) \(=\) \(\displaystyle \map \Re {\laptrans {e^{i a t} } }\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \map \Re {\frac {s + i a} {s^2 + a^2} }\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac s {s^2 + a^2}\) $\quad$ $\quad$

$\blacksquare$


Proof 3

\(\displaystyle \laptrans {\cos a t}\) \(=\) \(\displaystyle \laptrans {\frac {e^{i a t} + e^{-i a t} } 2}\) $\quad$ Cosine Exponential Formulation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\laptrans {e^{i a t} } + \laptrans {e^{-i a t} } }\) $\quad$ Linear Combination of Laplace Transforms $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\frac 1 {s - i a} + \frac 1 {s + i a} }\) $\quad$ Laplace Transform of Exponential $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\frac {s + i a + s - i a} {s^2 + a^2} }\) $\quad$ simplifying $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac s {s^2 + a^2}\) $\quad$ simplifying $\quad$

$\blacksquare$


Proof 4

By definition of the Laplace Transform:

$\displaystyle \laptrans {\cos at} = \int_0^{\to +\infty} e^{-s t} \cos at \rd t$


From Integration by Parts:

$\displaystyle \int f g' \rd t = f g - \int f'g \rd t$

Here:

\(\displaystyle f\) \(=\) \(\displaystyle \cos at\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle f'\) \(=\) \(\displaystyle -a \sin a t\) $\quad$ Derivative of Cosine Function $\quad$
\(\displaystyle g'\) \(=\) \(\displaystyle e^{-s t}\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle g\) \(=\) \(\displaystyle -\frac 1 s e^{-s t}\) $\quad$ Primitive of Exponential Function $\quad$

So:

\((1):\quad\) \(\displaystyle \int e^{-s t} \cos a t \rd t\) \(=\) \(\displaystyle -\frac 1 s e^{-s t} \cos a t - \frac a s \int e^{-s t} \sin a t \rd t\) $\quad$ $\quad$


Consider:

$\displaystyle \int e^{-s t} \sin a t \rd t$

Again, using Integration by Parts:

$\displaystyle \int h j \,' \rd t = h j - \int h'j \rd t$

Here:

\(\displaystyle h\) \(=\) \(\displaystyle \sin at\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle h'\) \(=\) \(\displaystyle a \cos at\) $\quad$ Derivative of Sine Function $\quad$
\(\displaystyle j\,'\) \(=\) \(\displaystyle e^{-s t}\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle j\) \(=\) \(\displaystyle -\frac 1 s e^{-s t}\) $\quad$ Primitive of Exponential Function $\quad$

So:

\(\displaystyle \int e^{-s t} \sin a t \rd t\) \(=\) \(\displaystyle -\frac 1 s e^{-s t} \sin a t + \frac a s \int e^{-s t} \cos a t \rd t\) $\quad$ $\quad$

Substituting this into $(1)$:

\(\displaystyle \int e^{-s t} \cos a t \rd t\) \(=\) \(\displaystyle -\frac 1 s e^{-s t} \cos a t - \frac a s \paren {-\frac 1 s e^{-s t} \sin a t + \frac a s \int e^{-s t} \cos a t \rd t}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle -\frac 1 s e^{-s t} \cos a t + \frac a {s^2} e^{-s t} \sin a - \frac {a^2} {s^2} \int e^{-s t} \cos a t \rd t\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {1 + \frac {a^2} {s^2} } \int e^{-s t} \cos a t \rd t\) \(=\) \(\displaystyle -\frac 1 s e^{-s t} \cos a t + \frac a {s^2} e^{-s t} \sin a t\) $\quad$ $\quad$


Evaluating at $t = 0$ and $t \to +\infty$:

\(\displaystyle \paren {1 + \frac {a^2} {s^2} } \laptrans {\cos at}\) \(=\) \(\displaystyle \intlimits {-e^{-s t} \paren {\frac 1 s \cos a t - \frac a {s^2} \sin a t} } {t \mathop = 0} {t \mathop \to +\infty}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 0 - \paren {-1 \paren {\frac 1 s \times 1 + \frac a {s^2} \times 0} }\) $\quad$ Boundedness of Real Sine and Cosine, Complex Exponential Tends to Zero $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 s\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \laptrans {\cos at}\) \(=\) \(\displaystyle \frac 1 s \paren {1 + \frac {a^2} {s^2} }^{-1}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 s \paren {\frac {s^2} {a^2 + s^2} }\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac s {s^2 + a^2}\) $\quad$ $\quad$

$\blacksquare$


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