The ubiquitous "helper function" $\frac{f(z) - f(a)}{z - a}$ I've been looking at basic complex analysis recently, and have noticed (am imagining?) something which I've never really paid attention to before: The "helper function" 
$$g(z) = \frac{f(z) - f(a)}{z - a}$$
At first, I didn't give it a second thought really - just used in the definition of the derivative.
Then I saw that it was used again in the proof of the Cauchy Integral Formula, then again in the derivation of Taylor and Laurent series.
Then finally in the proof of the Schwarz lemma in the form $\displaystyle g(z) = \frac{f(z)}{z}$.
So now I'm a little suspicious.
I will ask with hesitation...is there anything more going on here? Is it just a coincidence that it seems so useful? Where else can I expect this guy to pop up?
 A: It's a shift function on the power series centered at $a$; it takes the 0th term and deletes it, and makes the linear term a constant, and keeps going all the way down the line.  
A: Well, it's essentially a "gradient" function, in the sense that $g(z)$ measures the gradient of the line (or technically the secant) drawn from $(a, f(a))$ to $(z, f(z))$, and for $z$ in the neighbourhood of $a$ it approximates the derivative of $f$ at $a$ and hence $f(a)+(z-a)g(z)$, being the equation of the secant, will approximate the curve in that neighbourhood - which for sufficiently well-behaved curves will give you an idea of what the curve is doing in that neighbourhood.
So you can expect it to show up in cases where it's appropriate to approximate the curve as a secant line, or where you want to remove first-order behaviour (e.g. removing singularities) to look at higher order stuff.
A: It will pop up in any discussion that has to do with the change between two values of a function. That's a lot of discussions. 
