# InverseHaversine

gives the inverse haversine function .

# Details • 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 .
• 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.
• InverseHaversine can be used with Interval and CenteredInterval objects. »

# Examples

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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(30)

### Numerical Evaluation(6)

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:

InverseHaversine can be used with Interval and CenteredInterval objects:

### 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:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot the higher derivatives:

The formula for the  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: