# Haversine

Haversine[z]

gives the haversine function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The haversine function is defined by .
• The argument of haversine is assumed to be in radians. (Multiply by Degree to convert from degrees.)
• Haversine[z] is the entire function of z with no branch cut discontinuities.
• Haversine can be evaluated to arbitrary numerical precision.
• Haversine automatically threads over lists.

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at 0:

## Scope(31)

### 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:

Haversine threads elementwise over lists and matrices:

### Specific Values(3)

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:

### Visualization(3)

Plot the Haversine function:

Plot the real part of Haversine(x+y):

Plot the imaginary part of Haversine(x+y):

Polar plot with :

### Function Properties(6)

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 :

### Differentiation(3)

The first derivative:

Higher derivatives:

Plot the higher derivatives:

The formula for the  derivative:

### Integration(3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

The definite integral:

More integrals:

### Series Expansions(4)

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:

### Function Representations(4)

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:

## Applications(1)

Distance between two points on a sphere:

Distance between two cities in kilometers (assuming spherical Earth):

## Properties & Relations(2)

Derivative of haversine function:

Integral of haversine function:

Use FunctionExpand to expand Haversine in terms of standard trigonometric functions: