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I have a very simple question.
In Oppenheim book, it says that:
If CT Fourier transform of $x(t)$ is $X(j\omega)$ then, CT Fourier transform of $x^*(t)$ is $X^*(-j\omega)$.
What I can't understand is what is $X^*(-j\omega)$? Or what is $X^*(j\omega)$? Can you tell me what I should understand when I see $X^*(-j\omega)$ or $X^*(j\omega)$?
For example $x(t) = (a + bj)t$ and $X(j\omega) = (c + dj)\omega$ what should be $X^*(-j\omega)$?
Thanks
$X^*(j\omega)$ is the complex conjugate of $X(j\omega)$. So if
$$X(j\omega)=X_R(\omega)+jX_I(\omega)$$
then
$$X^*(j\omega)=X_R(\omega)-jX_I(\omega)$$
and
$$X^*(-j\omega)=X_R(-\omega)-jX_I(-\omega)$$
To gain some more understanding, consider the definition of the Fourier transform:
$$X(j\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt$$
So we have
$$X(-j\omega)=\int_{-\infty}^{\infty}x(t)e^{j\omega t}dt$$
and
$$X^*(-j\omega)=\int_{-\infty}^{\infty}x^*(t)e^{-j\omega t}dt=\mathcal{F}\{x^*(t)\}$$
as claimed in the quote of your question.
