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, so the Gudermannian function is defined by
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(1)
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 |  | ![2tan^(-1)[tanh(1/2x)].](/images/equations/Gudermannian/Inline11.svg) |
(2)
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For real 
, this definition is also equal to
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(3)
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(4)
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The Gudermannian is implemented in the Wolfram Language as Gudermannian[z].
The derivative of the Gudermannian is
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(5)
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For real 
, an indefinite integral
is
![intgd(x)dx=-1/2pix+i[Li_2(-ie^x)-Li_2(ie^x)]+C,](/images/equations/Gudermannian/NumberedEquation2.svg) |
(6)
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where 
is the dilogarithm on its principal
branch and 
is a constant of integration.
The Gudermannian has Maclaurin series
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(7)
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The Gudermannian connects the trigonometric and hyperbolic functions via
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(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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The Gudermannian is related to the exponential function by
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(14)
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(15)
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(16)
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(Beyer 1987, p. 164; Zwillinger 1995, p. 485).
Other fundamental identities are
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(17)
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(18)
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(Zwillinger 1995, p. 485).
If 
, then
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(19)
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(20)
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(21)
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(22)
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(Beyer 1987, p. 164; Zwillinger 1995, p. 530), where the last identity has been corrected.
An additional identity is given by
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(23)
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(M. Somos, pers. comm., Apr. 15, 2006).
The Gudermannian function can also be extended to the complex plane, as illustrated above.
