Gudermannian

The Gudermannian function is the odd function denoted either or which arises in the inverse equations for the Mercator projection. expresses the latitude in terms of the vertical position in the Mercator projection. For real , it is defined by

			(1)
			(2)

For real , this definition is also equal to

			(3)
			(4)

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

Where the complex Gudermannian is analytic, its derivative is

(5)

For real , an indefinite integral is

(6)

where is the dilogarithm on its principal branch and is a constant of integration.

The Gudermannian has Maclaurin series

(7)

(OEIS A091912 and A136606) for .

For real , the Gudermannian connects the trigonometric and hyperbolic functions via

			(8)
			(9)
			(10)
			(11)
			(12)
			(13)

where the cotangent and cosecant identities require .

For real , the Gudermannian is related to the exponential function by

			(14)
			(15)
			(16)

(Beyer 1987, p. 164; Zwillinger 1995, p. 485).

Other fundamental identities are, for real ,

(17)

(18)

for , where is the inverse Gudermannian (Zwillinger 1995, p. 485).

For real and with away from the branch cuts, write . Where the displayed denominators are nonzero,

			(19)
			(20)
			(21)
			(22)

(Beyer 1987, p. 164; Zwillinger 1995, p. 530), where the last identity has been corrected.

Under the same conditions, an additional identity is given by

(23)

(M. Somos, pers. comm., Apr. 15, 2006).

The Gudermannian function can also be extended to the complex plane, as illustrated above.