Elliptic functions and integrals

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The terminology for elliptic integrals and functions has changed during their investigation. What were originally called elliptic functions are now called elliptic integrals and the term elliptic functions reserved for a different idea. We will therefore use modern terminology throughout this article to avoid confusion.

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

where 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 - t4)

then introduced the lemniscate curve

(x2+y2)2 = (x2-y2)

whose arc length is given by the integral from 0 to x of

dt/√(1 - t4)

This integral, which is clearly satisfies the above definition so is an elliptic integral, became known as the lemniscate integral.

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 - t2)

The other good features of the lemniscate integral are the fact that it is general enough for many of its properties to be generalised to more general elliptic functions, yet the geometric intuition from the arc length of the lemniscate curve aids understanding.

In the year 1694 Jacob Bernoulli considered another elliptic integral

t2 dt/√(1 - t4)

and conjectured that it could not be expressed in terms of 'known' functions, sin, exp, sin-1.

References (8 books/articles)

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

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JOC/EFR May 1996

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