the Taylor coefficients
When a holomorphic function is written as its power series sum a_n (z - z_0)^n, the numbers a_0, a_1, a_2, ... are the Taylor coefficients. They are the fingerprints of the function at the centre z_0: knowing all of them recovers the function completely on its disk. The natural question is how to extract a given a_n, and there are two complementary answers — one from derivatives, one from integrals.
The derivative formula is a_n = f^(n)(z_0) / n!. You get it by differentiating the series n times and evaluating at z_0: every term below the n-th vanishes, the n-th term contributes n! a_n, and everything above vanishes too, leaving a_n = f^(n)(z_0)/n!. So a_0 = f(z_0), a_1 = f'(z_0), a_2 = f''(z_0)/2, and so on. The integral formula, equivalent but often more useful, comes from the Cauchy integral formula: a_n = (1 / 2 pi i) times the integral over a circle around z_0 of f(w) / (w - z_0)^(n+1) dw. This second form is powerful because it does not require you to compute high derivatives directly — a contour integral can be estimated even when derivatives are hard.
The integral form has an immediate and famous consequence: the Cauchy estimates. If |f| <= M on the circle of radius r around z_0, then |a_n| <= M / r^n. Bounding the coefficients bounds the function and vice versa, and from this single inequality flow Liouville's theorem (a bounded entire function is constant) and the maximum modulus principle. So the Taylor coefficients are not just bookkeeping; they are the quantitative handle through which growth, boundedness, and rigidity of holomorphic functions are controlled.
For f(z) = e^z at z_0 = 0, every derivative is again e^z, equal to 1 at 0, so a_n = 1/n! and e^z = sum z^n / n!. For f(z) = 1/(1 - z) at 0, f^(n)(0) = n!, so a_n = n!/n! = 1, recovering sum z^n. The integral form gives the same a_n by a single contour integral, no high derivatives needed.
Two routes to the same coefficient: differentiate n times and divide by n!, or integrate f(w)/(w - z_0)^(n+1) once around a circle.
The Taylor coefficients depend on the centre z_0; the SAME function has different coefficients when re-expanded around a different point. The function is intrinsic, but its coefficients are a local readout tied to where you stand.