Phase information of laplace transform

Question

When we do a fourier transform of a function which has even and odd parts we get a frequency domain representation which has both real and imaginary parts. We can then represent this as plot of magnitude and phase. In the case of the laplace transform we generate a magnitude surface but we seem to disgard the phase surface ? Doesnt this phase surface hold valuable information. Is there some branch of maths that looks at the laplace phase surface ? Ive never heard of anyone mentioning this ?

Answer 1

When we look at a transfer function:

$$\mathscr{H}\left(\text{s}\right):=\frac{\alpha_1\cdot\text{s}^\text{n}+\alpha_2\cdot\text{s}^{\text{n}-1}+\dots+\alpha_\text{q}\cdot\text{s}^\text{m}}{\beta_1\cdot\text{s}^\text{a}+\beta_2\cdot\text{s}^{\text{a}-1}+\dots+\beta_\text{z}\cdot\text{s}^\text{b}}\tag1$$

We know that for the phase we can write:

$$\arg\left(\mathscr{H}\left(\omega\text{j}\right)\right)=\arg\left(\frac{\alpha_1\cdot\left(\omega\text{j}\right)^\text{n}+\alpha_2\cdot\left(\omega\text{j}\right)^{\text{n}-1}+\dots+\alpha_\text{q}\cdot\left(\omega\text{j}\right)^\text{m}}{\beta_1\cdot\left(\omega\text{j}\right)^\text{a}+\beta_2\cdot\left(\omega\text{j}\right)^{\text{a}-1}+\dots+\beta_\text{z}\cdot\left(\omega\text{j}\right)^\text{b}}\right)$$ $$\arg\left(\alpha_1\cdot\left(\omega\text{j}\right)^\text{n}+\alpha_2\cdot\left(\omega\text{j}\right)^{\text{n}-1}+\dots+\alpha_\text{q}\cdot\left(\omega\text{j}\right)^\text{m}\right)-\arg\left(\beta_1\cdot\left(\omega\text{j}\right)^\text{a}+\beta_2\cdot\left(\omega\text{j}\right)^{\text{a}-1}+\dots+\beta_\text{z}\cdot\left(\omega\text{j}\right)^\text{b}\right)\tag2$$