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Inverse Gudermannian

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The inverse function of the Gudermannian y=gd^(-1)phi gives the vertical position y in the Mercator projection in terms of the latitude phi and may be defined for 0<=x<pi/2 by

gd^(-1)(x)=int_0^xsectdt
(1)
=2tanh^(-1)[tan(1/2x)]
(2)
=1/2ln((1+sinx)/(1-sinx))
(3)
=ln[tan(1/4pi+1/2x)]
(4)
=ln(secx+tanx).
(5)

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

Its derivative is given by

d/(dx)gd^(-1)(x)=secx.
(6)

It has Maclaurin series

gd^(-1)(x)=x+1/6x^3+1/(24)x^5+(61)/(5040)x^7+(277)/(72576)x^9+...
(7)

(OEIS A091912 and A136606).

See also

Gudermannian

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References

Beyer, W. H. "Gudermannian Function." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 164, 1987.Sloane, N. J. A. Sequences A091912 and A136606 in "The On-Line Encyclopedia of Integer Sequences."Zwillinger, D. (Ed.). "Gudermannian Function." §6.9 in CRC Standard Mathematical Tables and Formulae, 31st ed. Boca Raton, FL: CRC Press, pp. 530-532, 1995.

Referenced on Wolfram|Alpha

Inverse Gudermannian

Cite this as:

Weisstein, Eric W. "Inverse Gudermannian." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseGudermannian.html

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