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# EllipticE

 EllipticE[m]gives the complete elliptic integral . EllipticE[, m]gives the elliptic integral of the second kind .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• For , .
• .
• EllipticE[m] has a branch cut discontinuity in the complex m plane running from to .
• EllipticE[, m] has a branch cut discontinuity running along the ray from to infinity.
• For certain special arguments, EllipticE automatically evaluates to exact values.
• EllipticE can be evaluated to arbitrary numerical precision.
 Scope   (14)
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:
Find series expansions at branch points:
Find limits at branch cuts:
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 expansion:
Expand in series with respect to the modulus:
EllipticE can be applied to a power series:
 Applications   (8)
Compute elliptic integrals:
Plot an incomplete elliptic integral over the complex plane:
Perimeter length of an ellipse:
Series expansion for almost equal axes lengths:
Compare with an approximation by Ramanujan:
Arc length of a hyperbola as a function of the angle of a point on the hyperbola:
Plot the arc length as a function of the angle:
Vector potential of a ring current in cylindrical coordinates:
The vertical and radial components of the magnetic field:
Plot magnitude of the magnetic field:
Inductance of a solenoid of radius r and length a with fixed numbers of turns per unit length:
Inductance per unit length of the infinite solenoid:
Calculate the surface area of a triaxial ellipsoid:
The area of an ellipsoid with half axes 3, 2, 1:
Calculate the area by integrating the differential surface elements:
Parametrization of a mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):
Plot the resulting balloon:
Calculate the ratio of the main curvatures:
Express the radius of the original sheets in terms of the radius of the inflated balloon:
Expand special cases:
Expand special cases under argument restrictions:
Numerically find a root of a transcendental equation:
Integrals:
Laplace transforms:
Limits on branch cuts:
EllipticE is automatically returned as a special case for some special functions:
The defining integral converges only under additional conditions:
Different conventions exist for the second argument:
In traditional form the vertical separator must be used:
Nested derivatives and integrals:
Plot EllipticE at integer points:
Calculate EllipticE through an analytically continued Taylor series:
Riemann surface of EllipticE: