**APPENDIX A**

**AUXILIARY MATERIAL**

# A.1 Some useful Taylor series

We collect here some useful Taylor series used throughout the text, *z* denotes a complex variable, and the expression on the right denotes the convergence region.

**1.**Finite geometric series:

(A.1)

**2.**Geometric series:

**3.**Any branch of the logarithm has the expansion:

(A.3)

# A.2 “” notation

**Definition A.1** *Let F*(*x*) *be a real function. We say that “F*(*x*) *is of order f*(*x*) *as x tends to a” and write F*(*x*) = *f*(*x*)) *as x* → *a, if and only if there exist two constants C* > 0 *and* δ > 0 *such that* |*F*(*x*)| ≤ *C* |*f*(*x*)| *when* |*x − a*| < δ.

For example, if *F*(*x*) is *n* + 1 times differentiable around *x* = *a*, then by Taylor’s theorem,

where the remainder term satisfies *R*_{n}(*x*) = ((*x − a*)^{n+1}). Here, and many times when it is clear by context, we omit “as ...

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