Reviewing Laplace Transforms, Cauchy-Reimann conditions

Ok, I saw this discussed in both Truxal and in Ogata, so it must be worth working to understand it. Be aware that much of this article will contain language very close to that used by Truxal and by Ogata. So, don't quote me.

In order for functions in the complex plane to be useful to us engineers, the function must be"analytic." This means that the derivative of the function must be unique.

Points at which complex functions are not analytic are called "singular." These include zeroes and poles. Think about a zero on a surface in the s-plane.
Delta s can approach the zero from any direction. The derivative could parallel to the real axis, by letting omega be constant and letting delta sigma approach zero, or parallel to the imaginary axis by letting sigma be constant and letting delta omega approach zero.