What is $v_{infty}$?

So, you can calculate the trajectory of a celestial projectile by using this equation:

$$e=frac{rv_{infty}}{mu},$$

where $mu$ is the central body’s Gravitational Parameter, so it is definable by $mu=GM$. $r$ is the orbiting body’s current distance from the sun. That leaves us with only one variable left to uncover; $v_{infty}$ (the Hyperbolic Excess Velocity). If you do a bit of research, you get an equation defining it as:

$v_{infty}$ definition

Where you have to know variables considering Earth, and an outside planet. But, what if we need to find out the Hyperbolic Excess Velocity of an outside planet, having nothing to do with the Earth or another outside planet?

In an answer to this question, I want a reliable explanation of how to calculate $v_{infty}$, not referring to another form of the equation for eccentricity, isolating $v_{infty}$. I also want an explanation of what the Hyperbolic Excess Velocity is, in its essence, and an example of how to calculate it.