Function Repository Resource:

# HeumanLambda

Evaluate the Heuman lambda function

Contributed by: Jan Mangaldan
 ResourceFunction["HeumanLambda"][ϕ,m] gives the Heuman lambda function .

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The Heuman lambda function is defined as .
Argument conventions for elliptic integrals are discussed in "Elliptic Integrals and Elliptic Functions".
ResourceFunction["HeumanLambda"][ϕ,m] has branch cut discontinuities at and at .
For certain special arguments, ResourceFunction["HeumanLambda"] automatically evaluates to exact values.
ResourceFunction["HeumanLambda"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (3)

Evaluate numerically:

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Plot over a subset of the reals:

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### Scope (5)

Evaluate for complex arguments and parameters:

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Evaluate to high precision:

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

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Simple exact values are generated automatically:

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Parity transformation and quasiperiodicity relations are automatically applied:

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### Applications (3)

Apsidal angle of a gyroscopic pendulum, plotted as a function of the initial angle θ and the coefficient of stability μ:

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Visualize the solid angle subtended by a circular disk:

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Evaluate the solid angle:

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Compare with the result of NIntegrate:

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Visualize the gravitational attraction of a point to a semi-infinite right circular cylinder:

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Evaluate the vertical component of gravitational attraction, assuming the cylinder has the density of iron:

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Compare with the result of NIntegrate:

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### Properties and Relations (2)

HeumanLambda can be expressed in terms of Legendre-Jacobi elliptic integrals:

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Certain cases of EllipticPi can be expressed in terms of EllipticK and HeumanLambda:

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## Version History

• 1.0.0 – 01 June 2021