The Elliptic Integrals

In our earlier studies of the nonlinear Pendulum problem (Example 2.6), we arrived at an integral expression, which did not appear to be tabulated. The physics of this problem is illustrated in Fig. 4.1, where R denotes the length of (weightless) string attached to a mass m, 6 denotes the subtended angle, and g is the acceleration due to gravity. Application of Newton s law along the path 5 

If we inquire as to the time required to move from angle zero to a, then this is exactly 1 /4 the pendulum period, so that 

The integral term is called the complete Elliptical integral of the first kind and is widely tabulated (Byrd and Friedman 1954). In general, the incomplete elliptic integrals are defined by 

The series expansions of the elliptic integrals are useful for computation. For the range 0 1, the complete integral expansions are 

It is useful to compare the nonlinear solution of the pendulum problem to an approximate solution for small angles, sin 6 d, hence Eq. 4.27 becomes 

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Related to the elliptic integral of the third kind are the Lame functions, which arise in the generalisation of spherical harmonics to confocal ellipsoidal coordinates. Applications of these in molecular electrostatics can be found... 

Many biomolecules are not spherical but can be approximated by an ellipsoid of revolution with uj uj = ug. In these cases the elliptic integral (22) can be expressed in a closed form. The most interesting shape would be a very elongated (i.e. rod-like) ellipsoid with a " a = Og. Then one has simply... 

Numerical solution forthe period) Redo Exercise 6.7.4 using either numerical integration of the differential equation, or numerical evaluation of the elliptic integral. Specifically, compute the period T(a), where a runs from 0 to 180 in steps of 10°. 

In this case, the elliptic integral of the second kind is converted to a complete integral ... 

The elliptic functions are thus related to the elliptic integrals the same as the trigonometrical functions are related to the inverse trigonometrical functions, for, as we have seen, if... 

Appendix A. The elliptic integral of second kind Appendix B. The roots of equation (25)... 

Here F k) and E k) are complete elliptic integrals of the first and second kind. When fc 1 the elliptic integral F k) diverges logarithmically, i.e.. 

F(cp, 0) is the elliptic integral of the first kind 9 is related with 0 as follows ... 

Here F(k) and E (k) are complete elliptic integrals of the first and second kind. When k— I the elliptic integral F(k) diverges logarithmically, i.e. the helix pitch P(H)—>oo. Simultaneously, when fe l, the integral E k)- and Eq. (57) give the critical field for the untwisting of the helix ... 

Ek = 4/o cosHka) + Jssh sin-(A rt) with the gap 2Jssh = 8assH - With the elliptic integral... 

The right-hand side of Equation (6.18) is an elliptic integral of the first kind, which is tabulated for values of/c = sin0 j < 1. For relatively small reorientation angles, the elliptic integral can be expanded as a series. Equation (6.18) can be approximated to the second-order. 

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