Symbolics > Working with Symbolics > Calculus > Example: Symbolic Elliptic Integral Functions
Example: Symbolic Elliptic Integral Functions
The following elliptic integral functions appear in many symbolic calculations.
* 
The elliptic integral functions are not part of the PTC Mathcad Prime set of built-in functions.
EllipticK: The Complete Elliptic Integral of the First Kind
1. Show the definition of the complete elliptic integral of the first kind, EllipticK(m).
Click to copy this expression
2. Evaluate EllipticK numerically.
Click to copy this expression
3. Plot the numerical values of EllipticK in the range of 0<m<1.
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
Click to copy this expression
The integral equals π/2 when m=0, and approaches 12 as m approaches 1. The horizontal marker shows the value of Elliptick(l/10), or:
Click to copy this expression
EllipticF: The Incomplete Elliptic Integral of the First Kind
1. Show the definition of the incomplete elliptic integral of the first kind, EllipticF(x, m).
Click to copy this expression
2. Evaluate EllipticF numerically.
Click to copy this expression
3. Show the relationship between EllipticF and EllipticK.
Click to copy this expression
Click to copy this expression
The two integrals are equal.
EllipticE: The Elliptic Integral of the Second Kind
1. Show the definition of the complete elliptic integral of the second kind, EllipticE(m):
Click to copy this expression
Alternatively, the function is given by:
Click to copy this expression
2. Evaluate EllipticE numerically.
Click to copy this expression
3. Show the definition of the incomplete elliptic integral of the second kind, EllipticE(x, m):
Click to copy this expression
4. Evaluate EllipticEi numerically.
Click to copy this expression
5. Show the relationship between EllipticE and EllipticEi.
Click to copy this expression
Click to copy this expression
The two integrals are equal.
EllipticP: The Elliptic Integral of the Third Kind
1. Show the definition of the complete elliptic integral of the third kind, EllipticPi(n, m):
Click to copy this expression
2. Evaluate EllipticP(n, m) numerically.
Click to copy this expression
3. Show the definition of the incomplete elliptic integral of the third kind, EllipticPi(x, n, m):
Click to copy this expression
4. Evaluate EllipticPi numerically.
Click to copy this expression
5. Show the relationship between EllipticP and EllipticPi.
Click to copy this expression
Click to copy this expression
The two integrals are equal at x=π/2.
Was this helpful?