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I am given a question of Fourier Transform:

$$ e^{2(t-1)}u(t-1) $$ My teacher solved it by using the formula which I couldn't understand so I tried to apply the properties on it. Now I have solved it by the following method:

$$ e^{2(t)}u(t) \rightarrow \frac{1}{2+j\omega} $$

Now we know that: $$\delta(t-t_0) \rightarrow e^{-j\omega t_0} $$ So I used the above property on $u(t-1)$ and got the following answer which is same as my teacher got, which is:

$$ \frac{e^{-j\omega}}{2+j\omega} $$

Is my method correct?

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Assuming that your definition of the Fourier transform is $$ \hat f(\omega) = \int_{\mathbb R} f(t)e^{j\omega t}\ \mathsf dt, $$ then yes, your answer is correct. We can use a change of variables $s=t+1$ to compute \begin{align} \hat f(\omega) &= \int_1^\infty e^{-2(t+1)}e^{j\omega t}\ \mathsf dt\\ &= e^{-j\omega}\int_0^\infty e^{-2s}e^{j\omega s}\ \mathsf ds\\ &= e^{-j\omega}\cdot\frac{1}{2+j\omega}. \end{align}

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