# Fourier Transforms, symmetry, real/imaginary

I was hoping to clarify if the following was correct:

-a real function (neither even nor odd) in time exhibits conjugate symmetry in frequency, so the real part of the frequency response is even, and the imaginary portion is odd

-a real, even function has a frequency response that is strictly real and even

-an imaginary, odd function has a frequency response that is strictly imaginary and odd

-also: can anything be said if the function were imaginary in time (neither odd nor even)? would it be anti-conjugate symmetric, so the frequency response would be such that the real part is odd and the imaginary portion is even?

$$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt\tag{1}$$
$$X^*(\omega)=\int_{-\infty}^{\infty}x^*(t)e^{j\omega t}dt\tag{2}$$
Now if $$x(t)$$ is purely imaginary we have $$x^*(t)=-x(t)$$, and, consequently,
$$X^*(\omega)=-\int_{-\infty}^{\infty}x(t)e^{j\omega t}dt=-X(-\omega)\tag{3}$$