# Elliptic functions and integrals

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It is important to understand how mathematicians thought differently at different periods. Early algebraists had to prove their formulas by geometry. Similarly early workers with integration considered their problems solved if they could relate an integral to a geometric object.

Many integrals arose from attempts to solve mechanical problems. For example the period of a simple pendulum was found to be related to an integral which expressed arc length but no form could be found in terms of 'simple' functions. The same was true for the deflection of a thin elastic bar.

The study of elliptical integrals can be said to start in 1655 when Wallis began to study the arc length of an ellipse. In fact he considered the arc lengths of various cycloids and related these arc lengths to that of the ellipse. Both Wallis and Newton published an infinite series expansion for the arc length of the ellipse.

At this point we should give a definition of an elliptic integral. It is one of the form

*r*(

*x*, √

*p*(

*x*) )

*dx*

*r*(

*x*,

*y*) is a rational function in two variables and

*p*(

*x*) is a polynomial of degree 3 or 4 with no repeated roots.

In 1679 Jacob Bernoulli attempted to find the arc length of a spiral and encountered an example of an elliptic integral.

Jacob Bernoulli, in 1694, made an important step in the theory of elliptic integrals. He examined the shape the an elastic rod will take if compressed at the ends. He showed that the curve satisfied

*ds*/

*dt*= 1/√(1 -

*t*

^{4})

*x*

^{2}+

*y*

^{2})

^{2}= (

*x*

^{2}-

*y*

^{2})

*x*of

*dt*/√(1 -

*t*

^{4})

This is a particularly simple case of an elliptic integral. Notice for example that it is similar in form to the function sin^{-1}(*x*) which is given by the integral from 0 to *x* of

*dt*/√(1 -

*t*

^{2})

In the year 1694 Jacob Bernoulli considered another elliptic integral

*t*

^{2}

*dt*/√(1 -

*t*

^{4})

^{-1}.

**Article by:** *J J O'Connor* and *E F Robertson*

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