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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 this 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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and its indefinite integral is
![intgd(z)dz=-1/2pix+i[Li_2(-ie^x)-Li_2(ie^x)],](/images/equations/Gudermannian/NumberedEquation2.svg) |
(6)
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where 
is the dilogarithm.
It 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).
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The Gudermannian function can also be extended to the complex plane, as illustrated above.
