exists a point at which f is a conformal map or else stipulate that In particular, the alternate condition in (5) that is sufficient since it implicitly stipulates that be conformal. Proof that each characterization makes sense Some of these definitions require justification to demonstrate that they are well-defined. For example, when the value of the function is defined as the result of a limiting process (i.e. an infinite sequence or series), it must be demonstrated that such a limit always exists. Characterization 2 Since it follows from the ratio test that converges for all x. Characterization 3 Since the integrand is an integrable function of , the integral expression is well-defined. It must be shown that the function from to defined by is a bijection. Since is positive for positive , this function is strictly increasing, hence injective. If the two integrals hold, then it is surjective as well. Indeed, these integrals do hold; they follow from the integral test and the divergence of the harmonic series. Equivalence of the characterizations The following proof demonstrates the equivalence of the first three characterizations given for e above. The proof consists of two parts. First, the equivalence of characterizations 1 and 2 is established, and then the equivalence of characterizations 1 and 3 is established. Arguments linking the other characterizations are also given. Characterization 1 ⇔ characterization 2 The following argument is adapted from a proof in Rudin, theorem 3.31, p. 63–65. Let be a fixed non-negative real number. Define By the binomial theorem, (using x ≥ 0 to obtain the final inequality) so that where ex is in the sense of definition 2. Here, limsups must be used, because it is not known if tn converges. For the other direction, by the above expression of tn, if 2 ≤ m ≤ n, Fix m, and let n approach infinity. Then (again, liminf's must be used because it is not known if tn converges). Now, taking the above inequality, letting m approach infinity, and putting it together with the other inequality, this becomes so that This equivalence can be extended to the negative real numbers by noting and taking the limit as n goes to infinity. The error term of this limit-expression is described by where the polynomial's degree (in x) in the term with denominator nk is 2k. Characterization 1 ⇔ characterization 3 Here, the natural logarithm function is defined