The Gudermannian function, named after Christoph Gudermann (1798 – 1852), relates the circular and hyperbolic trigonometric functions without using complex numbers.
It is defined by
:
The following identities also hold:
:
The inverse Gudermannian function is given by
:
The derivatives of the Gudermannian and its inverse are
:
ee also
*Hyperbolic secant distribution
*Mercator projection
*Tangent half-angle formula
*Tractrix
*Trigonometric identity
References
* CRC "Handbook of Mathematical Sciences" 5th ed. pp 323–5.
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