InverseHaversine

InverseHaversine[z]

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 .
  • 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.

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  (22)

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  (3)

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

InverseHaversine is defined for all complex values:

The real range:

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, https://reference.wolfram.com/language/ref/InverseHaversine.html.

Text

Wolfram Research (2008), InverseHaversine, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseHaversine.html.

BibTeX

@misc{reference.wolfram_2020_inversehaversine, author="Wolfram Research", title="{InverseHaversine}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/InverseHaversine.html}", note=[Accessed: 03-December-2020 ]}

BibLaTeX

@online{reference.wolfram_2020_inversehaversine, organization={Wolfram Research}, title={InverseHaversine}, year={2008}, url={https://reference.wolfram.com/language/ref/InverseHaversine.html}, note=[Accessed: 03-December-2020 ]}

CMS

Wolfram Language. 2008. "InverseHaversine." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseHaversine.html.

APA

Wolfram Language. (2008). InverseHaversine. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseHaversine.html