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Inverse Haversine

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The inverse haversine function hav^(-1)(z) is defined by

hav^(-1)(z)=2sin^(-1)(sqrt(z)).
(1)

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

It has derivative

d/(dz)hav^(-1)(z)=1/(sqrt((1-z)z))
(2)

and indefinite integral

inthav^(-1)(z)dz=(2z-1)sin^(-1)(sqrt(z))+sqrt(z(1-z))+C,
(3)

where C is a constant of integration.

The inverse haversine has the series expansion

hav^(-1)(x)=sum_(n=0)^(infty)1/(2^(2n-1)(2n+1))(2n; n)x^(n+1/2)
(4)
=2x^(1/2)+1/3x^(3/2)+3/(20)x^(5/2)+5/(56)x^(7/2)+...
(5)

(OEIS A143581 and A143582).

See also

Haversine

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References

Sloane, N. J. A. Sequences A143581 and A143582 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Inverse Haversine

Cite this as:

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

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