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:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix Haversine function using MatrixFunction:

Haversine can be used with Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

The values of Haversine at fixed points:

Value at zero:

Find the first positive extremum of Haversine using Solve:

Substitute in the result:

Visualize the result:

Plot the Haversine function:

Plot the real part of Haversine(z):

Plot the imaginary part of Haversine(z):

Polar plot with :

Haversine is defined for all real and complex values:

Haversine achieves all values between zero and one, inclusive, on the reals:

The range for complex values is the whole plane:

Haversine is periodic with period :

Expand using ComplexExpand assuming real variables x and y:

Haversine has the mirror property :

Haversine is an analytic function:

Haversine is neither non-decreasing nor non-increasing:

Haversine is not injective:

Haversine is not surjective:

Haversine is non-negative:

Haversine has no singularities or discontinuities:

Haversine is neither convex nor concave:

TraditionalForm formatting:

The first derivative:

Higher derivatives:

Plot the higher derivatives:

The formula for the derivative:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

The definite integral:

More integrals:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The general term in the series expansion using SeriesCoefficient:

The first-order Fourier series:

This happens to be the complete series:

The Taylor expansion at a generic point:

Haversine can be applied to a power series:

Haversine can be represent in terms of Sin:

Series representation:

Haversine can be represented in terms of MeijerG:

Haversine can be represented as a DifferentialRoot: