Editing Gudermannian function (section)

== Taylor series ==
The [[Taylor series]] near zero, valid for complex values <math display=inline>z</math> with <math display=inline>|z| < \tfrac12\pi,</math> are<ref>{{harvp|Legendre|1817}} [https://archive.org/details/exercicescalculi02legerich/page/n165/ §4.2.7(162) pp. 143–144]</ref>

:<math>\begin{align}
\operatorname{gd} z
&= \sum_{k=0}^\infty \frac{E_k}{(k+1)!}z^{k+1}
= z - \frac16z^3 + \frac1{24}z^5 - \frac{61}{5040}z^7 + \frac{277}{72576}z^9 - \dots, \\[10mu]
\operatorname{gd}^{-1} z
&= \sum_{k=0}^\infty \frac{|E_k|}{(k+1)!}z^{k+1}
= z + \frac16z^3 + \frac1{24}z^5 + \frac{61}{5040}z^7 + \frac{277}{72576}z^9 + \dots,
\end{align}</math>

where the numbers <math display=inline>E_{k}</math> are the [[Euler numbers|Euler secant numbers]], 1, 0, -1, 0, 5, 0, -61, 0, 1385 ... (sequences {{OEIS link|A122045}}, {{OEIS link|A000364}}, and {{OEIS link|A028296}} in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]). These series were first computed by [[James Gregory (mathematician)|James Gregory]] in 1671.<ref>{{cite book |editor-last=Turnbull |editor-first=Herbert Westren |year=1939 |title=James Gregory; Tercentenary Memorial Volume |publisher=G. Bell & Sons |page=170 }}</ref>

Because the Gudermannian and inverse Gudermannian functions are the integrals of the hyperbolic secant and secant functions, the numerators <math display=inline>E_{k}</math> and <math display=inline>|E_{k}|</math> are same as the numerators of the [[Hyperbolic functions#Taylor series expressions|Taylor series for {{math|sech}}]] and [[Trigonometric functions#Power series expansion|{{math|sec}}]], respectively, but shifted by one place.

The reduced unsigned numerators are 1, 1, 1, 61, 277, ... and the reduced denominators are 1, 6, 24, 5040, 72576, ... (sequences {{OEIS link|A091912}} and {{OEIS link|A136606}} in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]).