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Math Help - Laplace transform - Hypberbolic cosine

  1. #1
    Newbie mukmar's Avatar
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    Laplace transform - Hypberbolic cosine

    Problem:
    Find the laplace transform of \cosh bt where b \in \mathbb{R}

    Attempt:
    \mathcal{L}\{\cosh bt\} = F(s) = \frac{1}{2}\left[\displaystyle\int_0^{\infty} e^{(b - s)t}dt + \displaystyle\int_0^{\infty} e^{(-b - s)t}dt \right]

     = \frac{1}{2}\left[\displaystyle\lim_{A \to \infty} \frac{1}{b - s}e^{(b - s)t}\big|^A_0  + \displaystyle\lim_{A \to \infty} \frac{1}{- b - s}e^{(-b - s)t}\big|^A_0 \right]

    So the first limit converges when t = A if b - s < 0, therefore s > b.

    The second limit converges when t = A if - b - s < 0, therefore s > -b.

    So the solution is F(s) = \frac{s}{s^2 - b^2 }, with the conditions s > -b and s > b.

    But the textbook solution requires s > |b|, which is the same as s > b and  s < -b.

    So I'm wondering where my error is. Any hints?
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  2. #2
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    s > |b| means s > b > -s
    b > -s is the same as -b < s right?

    You have the right answer, and the book does too.
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  3. #3
    Newbie mukmar's Avatar
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    Thanks, that was one of those dumb oversights on my part.
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