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Haversine


Haversine

The haversine, also called the haversed sine, is a little-used entire trigonometric function defined by

hav(z)=1/2vers(z)
(1)
=1/2(1-cosz)
(2)
=sin^2(1/2z),
(3)

where versin(z) is the versine, cosz is the cosine, and sinz is the sine.

The haversine is implemented in the Wolfram Language as Haversine[z].

HaversineReIm
HaversineContours

The haversine can be extended to the complex plane as illustrated above.

Its derivative is given by

 d/(dz)hav(z)=1/2sinz,
(4)

and its indefinite integral by

 inthav(z)dz=1/2(z-sinz)+C.
(5)

It has Maclaurin series

hav(z)=sum_(k=1)^(infty)((-1)^(k-1))/(2(2k)!)z^(2k)
(6)
=1/4z^2-1/(48)z^4+1/(1440)z^6-1/(80640)z^8+....
(7)

See also

Covercosine, Coversine, Excosecant, Exsecant, Hacoversine, Havercosine, Vercosine, Versine

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 78, 1972.Smart, W. M. Text-Book on Spherical Astronomy, 6th ed. Cambridge, England: Cambridge University Press, p. 18, 1960.

Referenced on Wolfram|Alpha

Haversine

Cite this as:

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

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