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).
2. Evaluate EllipticK numerically.
3. Plot the numerical values of EllipticK in the range of 0<m<1.
 s 0 d 0.0025 l 9.9975 u s d l EK0 400 u EllipticK 0 EK1 400 u EllipticK
u 10
EK2 400 u EllipticK
u 15
EK3 400 u EllipticK
u 30
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:
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).
2. Evaluate EllipticF numerically.
3. Show the relationship between EllipticF and EllipticK.
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):
Alternatively, the function is given by:
2. Evaluate EllipticE numerically.
3. Show the definition of the incomplete elliptic integral of the second kind, EllipticE(x, m):
4. Evaluate EllipticEi numerically.
5. Show the relationship between EllipticE and EllipticEi.
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):
2. Evaluate EllipticP(n, m) numerically.
3. Show the definition of the incomplete elliptic integral of the third kind, EllipticPi(x, n, m):
4. Evaluate EllipticPi numerically.
5. Show the relationship between EllipticP and EllipticPi.
The two integrals are equal at x=π/2.