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# EllipticK

 EllipticK[m]gives the complete elliptic integral of the first kind .
• 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.
Evaluate numerically:
 Scope   (7)
Evaluate numerically for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Simple exact values are generated automatically:
Series expansions at branch points:
Find directional limits at branch cuts:
EllipticK can be applied to power series:
 Applications   (5)
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:
This shows the branch cuts of the EllipticK function:
Numerically find a root of a transcendental equation:
Integrals:
Laplace transforms:
Solve a differential equation:
EllipticK is a particular case of various mathematical functions:
Machine-precision evaluation can result in numerically inaccurate answer near branch cuts:
The defining integral converges only under additional conditions:
Different argument conventions exist that result in modified results:
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 return to the origin:
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