attempts to determine Α1,2 (should s1 ≥ 2), one may to advantage differentiate both sides of Eq. (2.220) with regard to x so as to obtain an independent relationship, viz.
– by resorting to the rule of differentiation of a product; the said rule, coupled with the rule of differentiation of a sum, allows subsequent transformation of Eq. (2.225) to
The rule of differentiation of a power may now be invoked to transform Eq. (2.226) to
after setting x = r1 again, Eq. (2.227) becomes
(2.228)
which dramatically simplifies to
due to the nil value of r1 − r1 and any significant powers thereof. After splitting
(2.230)
– where
(2.231)
for being independent of r as counting variable; A1,1 may now be eliminated via Eq. (2.223), viz.
(2.232)
where cancelation of identical extended products between numerator and denominator gives rise to
Isolation of Α1,2 from Eq. (2.233) is finally in order, according to
(2.234)
generalization then ensues as
(2.235)
Although seldom of relevance, higher order constants, say Al,3, Al,4, …, may be calculated in a similar fashion – by resorting to higher order derivatives of Eq. (2.220), but accordingly requiring more cumbersome algebraic manipulations.
2.2.5 Power
According to Newton’s theorem, it is possible to expand the nth power (with n integer) of a sum of (real) terms, x and y, via a sum of a finite number of products of integer powers of x and y, with exponents adding up to n in every case; more specifically,
Equation (2.236) is also known as binomial formula, or binomial identity – and x and y may represent numbers, or instead functions. The first description of the above formula in its entirety, for nonnegative integer n, was due to Blaise Pascal – a French physicist, mathematician, and philosopher who lived in the seventeenth century; while Greek mathematician Euclid referred to its second order form in the fourth century BCE, and Indian mathematician Pingala mentioned higher orders one century later. However, it was Isaac Newton – who extended it to every real exponent in 1665, that has received credit for such a relationship thereafter.
The simplest case pertains indeed to n = 2, and accordingly reads
– being also known as another notable case of multiplication; for positive values of x and y, this can be graphically illustrated as in Fig. 2.9. The area (x + y)2 of the larger square of side x + y may indeed be obtained via addition of the area of two smaller squares of sides x and y, i.e. x2 and y2, respectively, to the area xy of each of two rectangles of sides x and y.
Figure 2.9 Geometric demonstration of Newton’s binomial formula at two dimensions – resorting to squares of side x and area x2, side y and area y2, and side x + y and area (x + y)2, complemented with two rectangles of sides x and y and area xy.
If y is replaced by −y, then Eq. (2.237) gives rise to
– also known as another notable case of multiplication; however, a simple geometrical interpretation is no longer possible (due to the negative
