Gudermannian function

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Gudermannian function
Mplwp gudermann piaxis.svg
General information
General definition
Domain, Codomain and Image
Domain(−, +) 
Codomain[–π/2, π/2] 
Basic features
Specific values
At zero0
Value at +∞+π/2
Value at −∞π/2
Specific features
Fixed point0

  • For real numbers.

The Gudermannian function, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic functions without explicitly using complex numbers.

It is defined for all by[1][2][3]

The inverse of the Gudermannian function, is sometimes known as the Lambertian function and written [4]


Alternative definitions[edit]

Some identities[edit]


Graph of the inverse Gudermannian function

Some identities[edit]



The function was introduced by Johann Heinrich Lambert in the 1760s at the same time as the hyperbolic functions. He called it the "transcendent angle," and it went by various names until 1862 when Arthur Cayley suggested it be given its current name as a tribute to Gudermann's work in the 1830s on the theory of special functions.[5] Gudermann had published articles in Crelle's Journal that were later collected in a book[6] which expounded and to a wide audience (although represented by the symbols and ).

The notation was introduced by Cayley[7] where he starts by calling the inverse of the integral of the secant function:

and then derives "the definition" of the transcendent:

observing immediately that it is a real function of


  • On a Mercator projection a line of constant latitude is parallel to the equator (on the projection) and is displaced by an amount proportional to the inverse Gudermannian of the latitude.
  • The Gudermannian also appears in a moving mirror solution of the dynamical Casimir effect.[10]

See also[edit]


  1. ^ Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W., eds. (2010). NIST Handbook of Mathematical Functions. Cambridge University Press.
  2. ^ Handbook of Mathematical Sciences (5th ed.). Boca Raton, FL: CRC Press. pp. 323–325.
  3. ^ Weisstein, Eric W. "Gudermannian". MathWorld.
  4. ^ Lee, Laurence Patrick (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monograph. Vol. 16. University of Toronto Press.
  5. ^ Becker, George F.; van Orstrand, C.E. (1931). Hyperbolic Functions. Read Books. p. xlix – via Internet Archive (
  6. ^ Gudermann, C. (1833). Theorie der potenzial-oder cyklisch-hyperbolischen Functionen [Theory of the Potential or Cyclic Hyperbolic Functions].
  7. ^ Cayley, A. (1862). "On the transcendent gd. u". Philosophical Magazine. 4th Series. 24 (158): 19–21. doi:10.1080/14786446208643307.
  8. ^ Osborne, P. (2013). "The Mercator projections". p. 74 – via
  9. ^ Robertson, John S. (1997). "Gudermann and the simple pendulum". The College Mathematics Journal. 28 (4): 271–276. doi:10.2307/2687148. JSTOR 2687148.
  10. ^ Good, Michael R.R.; Anderson, Paul R.; Evans, Charles R. (2013). "Time dependence of particle creation from accelerating mirrors". Physical Review D. 88 (2): 025023. arXiv:1303.6756. Bibcode:2013PhRvD..88b5023G. doi:10.1103/PhysRevD.88.025023. S2CID 56113100.