In signal processing, we define an analytic signal as a complex-valued signal which has no frequency components for $\omega<0$. It can be shown that the real part and the imaginary part of an analytic signal $x(t)=x_R(t)+jx_I(t)$ are related by the Hilbert transform:

\begin{align} x_I(t) & = \mathscr{H}\{x_R(t)\}\tag{1} \\ x_R(t) & = -\mathscr{H}\{x_I(t)\}\tag{2} \end{align}

In complex analysis, an analytic function (or holomorphic function) is a complex function of a complex variable that is infinitely differentiable, and its Taylor series about a given point converges to the function in some neighbourhood of that point.

The question is if there is some relationship between these two apparently unrelated meanings of the word "analytic", and if so, what is this relationship?

EDIT: I've just found a very similar question on the mathematics site, but the answers there are very concise without any details.

In signal processing, we define an analytic signal as a complex-valued signal which has no frequency components for $\omega<0$. It can be shown that the real part and the imaginary part of an analytic signal $x(t)=x_R(t)+jx_I(t)$ are related by the Hilbert transform:

\begin{align} x_I(t) & = \mathscr{H}\{x_R(t)\}\tag{1} \\ x_R(t) & = -\mathscr{H}\{x_I(t)\}\tag{2} \end{align}

In complex analysis, an analytic function (or holomorphic function) is a complex function of a complex variable that is infinitely differentiable, and its Taylor series about a given point converges to the function in some neighbourhood of that point.

The question is if there is some relationship between these two apparently unrelated meanings of the word "analytic", and if so, what is this relationship?

EDIT: I've just found a very similar question on the mathematics site, but the answers there are very concise without many details.

In signal processing, we define an analytic signal as a complex-valued signal which has no frequency components for $\omega<0$. It can be shown that the real part and the imaginary part of an analytic signal $x(t)=x_R(t)+jx_I(t)$ are related by the Hilbert transform:

\begin{align} x_I(t) & = \mathscr{H}\{x_R(t)\}\tag{1} \\ x_R(t) & = -\mathscr{H}\{x_I(t)\}\tag{2} \end{align}

In complex analysis, an analytic function (or holomorphic function) is a complex function of a complex variable that is infinitely differentiable, and its Taylor series about a given point converges to the function in some neighbourhood of that point.

The question is if there is some relationship between these two apparently unrelated meanings of the word "analytic", and if so, what is this relationship?

In signal processing, we define an analytic signal as a complex-valued signal which has no frequency components for $\omega<0$. It can be shown that the real part and the imaginary part of an analytic signal $x(t)=x_R(t)+jx_I(t)$ are related by the Hilbert transform:

\begin{align} x_I(t) & = \mathscr{H}\{x_R(t)\}\tag{1} \\ x_R(t) & = -\mathscr{H}\{x_I(t)\}\tag{2} \end{align}

In complex analysis, an analytic function (or holomorphic function) is a complex function of a complex variable that is infinitely differentiable, and its Taylor series about a given point converges to the function in some neighbourhood of that point.

The question is if there is some relationship between these two apparently unrelated meanings of the word "analytic", and if so, what is this relationship?

EDIT: I've just found a very similar question on the mathematics site, but the answers there are very concise without any details.