- #1

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- Homework Statement
- find Laplace transform of e^(at)/t

- Relevant Equations
- L(e^at/t)

hi guys

i am facing a little problem calculating this Laplace transform ## \mathscr{L}(\frac{e^{\alpha t}}{t})## , when calculate it using the method of the inverse Laplace transform its equal to

$$ ln{\frac{1}{s-\alpha}}$$

but then when i try to use the theorem

$$ \mathscr{L}(\frac{f(t)}{t})=\int_{s}^{\infty}F(s)ds=\int_{s}^{\infty}\mathscr{L}(f(t))ds = \int_{s}^{\infty}\frac{1}{s-\alpha}ds$$

$$=lim_{s→∞}(s-\alpha)-ln(|s-\alpha|)$$

it seems that there is a term that will blow up to infinity!

what i am missing here?!

i am facing a little problem calculating this Laplace transform ## \mathscr{L}(\frac{e^{\alpha t}}{t})## , when calculate it using the method of the inverse Laplace transform its equal to

$$ ln{\frac{1}{s-\alpha}}$$

but then when i try to use the theorem

$$ \mathscr{L}(\frac{f(t)}{t})=\int_{s}^{\infty}F(s)ds=\int_{s}^{\infty}\mathscr{L}(f(t))ds = \int_{s}^{\infty}\frac{1}{s-\alpha}ds$$

$$=lim_{s→∞}(s-\alpha)-ln(|s-\alpha|)$$

it seems that there is a term that will blow up to infinity!

what i am missing here?!

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