gives the inverse haversine function .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The inverse haversine function is defined by .
  • All results are given in radians.
  • For real between and , the results are always in the range to .
  • InverseHaversine[z] has branch cut discontinuities in the complex plane running from to and to .
  • InverseHaversine can be evaluated to arbitrary numerical precision.
  • InverseHaversine automatically threads over lists.


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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at a singular point:

Scope  (29)

Numerical Evaluation  (5)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

InverseHaversine threads elementwise over lists and matrices:

Specific Values  (3)

Values of InverseHaversine at fixed points:

The value of InverseHaversine at zero:

Find a value of for which using Solve:

Visualization  (2)

Plot the InverseHaversine function:

Plot the real part of InverseHaversine[x+y]:

Plot the imaginary part of InverseHaversine[x+y]:

Function Properties  (10)

InverseHaversine is defined for all real values from the interval [0,1]:

InverseHaversine is defined for all complex values:

The real range:

InverseHaversine is not an analytic function:

Nor is it meromorphic:

InverseHaversine is non-decreasing over its real domain:

InverseHaversine is injective:

InverseHaversine is not surjective:

InverseHaversine is non-negative over its real domain:

InverseHaversine does have both singularity and discontinuity in (-,0] and [1,):

InverseHaversine is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot the higher derivatives:

The formula for the ^(th) derivative:

Integration  (3)

Compute the indefinite integral using Integrate:

The definite integral:

More integrals:

Series Expansions  (3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The Taylor expansion at a generic point:

InverseHaversine can be applied to a power series:

Applications  (2)

Distance between two points on a sphere:

Distance between two cities in kilometers:

Modeling Lévy's second arc sine law:

Properties & Relations  (2)

Derivative of inverse haversine function:

Integral of inverse haversine function:

Use FunctionExpand to expand InverseHaversine:

Wolfram Research (2008), InverseHaversine, Wolfram Language function,


Wolfram Research (2008), InverseHaversine, Wolfram Language function,


Wolfram Language. 2008. "InverseHaversine." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2008). InverseHaversine. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2021_inversehaversine, author="Wolfram Research", title="{InverseHaversine}", year="2008", howpublished="\url{}", note=[Accessed: 19-May-2022 ]}


@online{reference.wolfram_2021_inversehaversine, organization={Wolfram Research}, title={InverseHaversine}, year={2008}, url={}, note=[Accessed: 19-May-2022 ]}