Article 24
Series for Gudermanian and its Inverse.

Substitute for sechu,secv in (49), (48) their expansions, Art. 16, and integrate, then

gdu = u 1 6u3 + 1 24u5 61 5040u7 +  (50) gd1v = v + 1 6v3 + 1 24v5 61 5040v7 +  (51)

No constants of integration appear, since gdu vanishes with u, and gd1v with v. These series are seldom used in computation, as gdu is best found and tabulated by means of tables of natural tangents and hyperbolic sines, from the equation

gdu = tan1(sinhu),

and a table of the direct function can be used to furnish the numerical values of the inverse function; or the latter can be obtained from the equation,

gd1v = sinh1(tanv) = cosh1(secv).

To obtain a logarithmic expression for gd1v, let

gd1v = u,v = gdu,  therefore secv = coshu,tanv = sinhu, secv + tanv = coshu + sinhu = eu, eu = 1 + sinv cosv = 1 cos(1 2π + v) sin(1 2π + v) = tan 1 4π + 1 2v, u = gd1v = log e tan 1 4π + 1 2v.  (52)
Prob. 63.
Evaluate gd u u u3 u 0, gd 1v v v3 v 0.
Prob. 64.
Prove that gd u sin u is an infinitesimal of the fifth order, when u0.
Prob. 65.
Prove the relations 1 4π + 1 2vtan 1eu, 1 4π 1 2v = tan 1eu.