Inverse Gudermannian

The inverse function of the Gudermannian  gives the vertical position  in the Mercator projection in terms of the latitude  and may be defined for  by

			(1)
			(2)
			(3)
			(4)
			(5)

The inverse Gudermannian is implemented in the Wolfram Language as InverseGudermannian[z].

Its derivative is given by

(6)

It has Maclaurin series

(7)

(OEIS A091912 and A136606).

REFERENCES:

Beyer, W. H. "Gudermannian Function." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 164, 1987.

Sloane, N. J. A. Sequences A091912 and A136606 in "The On-Line Encyclopedia of Integer Sequences."

Zwillinger, D. (Ed.). "Gudermannian Function." §6.9 in CRC Standard Mathematical Tables and Formulae, 31st ed. Boca Raton, FL: CRC Press, pp. 530-532, 1995.