# EllipticK EllipticK[m]

gives the complete elliptic integral of the first kind .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• EllipticK is given in terms of the incomplete elliptic integral of the first kind by .
• EllipticK[m] has a branch cut discontinuity in the complex m plane running from to .
• For certain special arguments, EllipticK automatically evaluates to exact values.
• EllipticK can be evaluated to arbitrary numerical precision.
• EllipticK automatically threads over lists.
• EllipticK 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:

Series expansion at Infinity:

## Scope(38)

### Numerical Evaluation(5)

Evaluate to high precision:

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

Evaluate numerically for complex arguments:

Evaluate EllipticK efficiently at high precision:

EllipticK can be used with Interval and CenteredInterval objects:

### Specific Values(5)

Simple exact values are generated automatically:

Some exact values in terms of Gamma after applying FunctionExpand:

Find directional limiting values at branch cuts:

Value at infinity:

Find the root of the equation :

### Visualization(2)

Plot EllipticK:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

EllipticK is defined for all real values less than 1:

EllipticK takes all real positive values:

EllipticK is not an analytic function:

Has both singularities and discontinuities:

EllipticK is not a meromorphic function:

EllipticK is nondecreasing on its domain:

EllipticK is injective:

EllipticK is not surjective:

EllipticK is non-negative on its domain:

EllipticK is convex on its domain:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of EllipticK:

Definite integral over an interval lying on the branch cut:

More integrals:

### Series Expansions(3)

Taylor expansion for EllipticK:

Plot the first three approximations for EllipticK around :

Series expansions at branch points:

EllipticK can be applied to power series:

### Integral Transforms(3)

Compute the Laplace transform using LaplaceTransform:

### Function Representations(5)

Relation to other elliptic integrals:

Relation to the LegendreP:

Represent in terms of MeijerG using:

EllipticK can be represented as a DifferentialRoot:

## Applications(7)

Small-angle approximation to the period of a pendulum:

Plot the period versus the initial angle:

Vector potential due to a circular current flow, in cylindrical coordinates:

The components of the magnetic field:

Plot the magnitude of the magnetic field:

Resistance between the origin and the point in an infinite 3D lattice of unit resistors:

Energy for the Onsager solution of the Ising model:

Plot of the specific heat:

Find the critical temperature:

Calculate a singular value:

Current flow in a rectangular conducting sheet with voltage applied at a pair of opposite corners:

Plot the flow lines with bounds defined via EllipticK:

Construct lowpass elliptic filter for analog signal:

Compute filter ripple parameters and associate elliptic function parameter:

Use elliptic degree equation to find the ratio of the pass and the stop frequencies:

Compute corresponding stop frequency and elliptic parameters:

Compute ripple locations and poles and zeros of the transfer function:

Compute poles of the transfer function:

Assemble the transfer function:

Compare with the result of EllipticFilterModel:

## Properties & Relations(4)

This shows the branch cuts of the EllipticK function:

Numerically find a root of a transcendental equation:

Solve a differential equation:

EllipticK is a particular case of various mathematical functions:

## Possible Issues(3)

Machine-precision evaluation can result in numerically inaccurate answers near branch cuts:

The defining integral converges only under additional conditions:

Different argument conventions exist that result in modified results:

## Neat Examples(1)

Probability that a random walker in a 3D cubic lattice returns to the origin:

Carry out a modeling run of 1000 walks and count how many it returns to the origin:

Compare with the expected count at :