Inverse Gudermannian
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) = [画像:int_0^xsectdt]
(1)
= 2tanh^(-1)[tan(1/2x)]
(2)
= [画像: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
| [画像: d/(dx)gd^(-1)(x)=secx. ] |
(6)
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It has Maclaurin series
| gd^(-1)(x)=x+1/6x^3+1/(24)x^5+(61)/(5040)x^7+(277)/(72576)x^9+... |
(7)
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See also
GudermannianExplore with Wolfram|Alpha
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More things to try:
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 GudermannianCite this as:
Weisstein, Eric W. "Inverse Gudermannian." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InverseGudermannian.html