Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.1 DIFFERENTIATION

The pattern emerging is clear and strongly suggests that the results generalise to

f(n)=

∑n

r=0

n!

r!(n−r)!

u(r)v(n−r)=

∑n

r=0

nC

ru

(r)v(n−r), (2.14)

where the fractionn!/[r!(n−r)!] is identified with the binomial coefficientnCr

(see chapter 1). Toprovethat this is so, we use the method of induction as follows.

Assume that (2.14) is valid fornequal to some integerN.Then

f(N+1)=

∑N

r=0

NCrd

dx

(

u(r)v(N−r)

)

=

∑N

r=0

NC

r[u

(r)v(N−r+1)+u(r+1)v(N−r)]

=

∑N

s=0

NCsu(s)v(N+1−s)+

N∑+1

s=1

NCs− 1 u(s)v(N+1−s),

where we have substituted summation indexsforrin the first summation, and

forr+ 1 in the second. Now, from our earlier discussion of binomial coefficients,

equation (1.51), we have

NC

s+

NC

s− 1 =

N+1C

s

and so, after separating out the first term of the first summation and the last

term of the second, obtain

f(N+1)=NC 0 u(0)v(N+1)+

∑N

s=1

N+1C

su

(s)v(N+1−s)+NC

Nu

(N+1)v(0).

ButNC 0 =1=N+1C 0 andNCN=1=N+1CN+1, and so we may write

f(N+1)=N+1C 0 u(0)v(N+1)+

∑N

s=1

N+1C

su

(s)v(N+1−s)+N+1C

N+1u

(N+1)v(0)

=

N∑+1

s=0

N+1C

su

(s)v(N+1−s).

This is just (2.14) withnset equal toN+ 1. Thus, assuming the validity of (2.14)

forn=Nimplies its validity forn=N+ 1. However, whenn=1equation

(2.14) is simply the product rule, and this we have already proved directly. These

results taken together establish the validity of (2.14) for allnand prove Leibnitz’

theorem.