Mathematical Physics Vol 1

5.8 Elliptic integrals

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5.8 Elliptic integrals

It can be observed that the introduction of inverse operations in Mathematics, for example for numbers, leads to numbers of a "new nature". Let’s illustrate this with a few examples: • addition - inverse - subtraction, leads to negative numbers, • multiplication - inverse - division, leads to fractions, • power of a number - inverse - root of a number, leads to irrational numbers, • derivative - inverse - integral, leads to results that cannot be expressed in terms of elemen tary functions, but rather by functional relations between variables. Let us illustrate the last item by several examples. Observe the following integrals, and the corresponding solutions Z d x √ 1 + x 2 = log x + p 1 + x 2 + C , Z d x √ 1 − x 2 = arcsin x + C . These integrals can be observed as special cases of the following integrals

d x p P 2 +( x ) , d x p P 2 − ( x )

J 1 =

(5.191)

J 2 =

where

2 + bx + c , a > 0 , 2 + bx + c , a < 0 ,

P 2 +( x )= ax P 2 +( x )= ax

are second order polynomials. For these polynomials the values of the integrals are

1 √ a log ( 2 ax + b + 2 √ a p ax 2 + bx + c )+ C ,

J 1 =

1

2 ax + b

arcsin −

J 2 =

+ C .

√

√ b 2

− a

− 4 ac

We can see that for relatively simple forms of sub-integral functions 1 / p 1 ± x 2 we obtain relatively complex functions ( logarithm and the inverse trigonometric function - arcsine ). Examples in which roots of polynomials of the third and fourth degree appear are, of course, more complex and lead to new functions. Such functions (integrals) are obtained, for example, when calculating the length of the arc of an ellipse. The solution of this problem leads to integrals of the form Z R ( x , √ P ) d x where R is a rational function, and P is a third or fourth degree polynomial. These integrals are called elliptic integrals .